NEW REALIZATIONS OF DEFORMED DOUBLE CURRENT ALGEBRAS AND DELIGNE CATEGORIES

In this paper, we propose an alternative construction of a certain class of Deformed Double Current Algebras. We construct them as spherical subalgebras of symplectic reection algebras in the Deligne category. They can also be thought of as ultraproducts of the corresponding spherical subalgebras in finite rank. We also provide new presentations of DDCA of types A and B by generators and relations.


Introduction
Deformed double current algebras (DDCA) of gl m are interpolations with respect to the rank n of Schur algebras associated to symplectic reflection algebras for wreath products S n ˙Γn , where Γ is a finite subgroup of SLp2, Cq ( [Gua10]).They can also be viewed as deformations of enveloping algebras of (generalized) matrix W 1`8 -algebras, and (in some cases) as rational limits of affine Yangians and toroidal quantum groups.DDCA appeared first (in a special case) in the physical paper [BHW95] in 1994.However, the systematic theory of DDCA, including their full definition, was developed only in the last 15 years, in a series of papers by N. Guay and his collaborators [Gua05,Gua07,Gua10,GY17,GHL09].In these papers, presentations of DDCA by generators and relations are given, the Schur-Weyl functor is defined and shown to be an equivalence of categories, and the degeneration of toroidal quantum groups and affine Yangians to DDCA is considered.
The goal of this paper is to give two alternative definitions of DDCA for gl 1 (i.e., of the interpolations of spherical symplectic reflection algebras).The first definition applies to any finite subgroup Γ Ă SL 2 pCq and is based on Deligne categories.Namely, we consider the Deligne category ReppS ν q, ν P C, which is obtained by interpolating the representation categories of the symmetric group S n with respect to n, [Del07].Using this category, we can define the interpolation C ν of the representation category of the symplectic reflection algebra H t,k pS n ˙Γn q attached to Γ in which the integer n is replaced by a complex parameter ν ([Eti14], Subsection 5.3).In the category C ν , we have an object M obtained by interpolating the H t,k pS n ˙Γn q-modules H t,k pS n ˙Γn qe, where e P CrS n ˙Γn s is the projector to the trivial representation, and the DDCA for gl 1 attached to Γ may be defined as D t,k,c,ν pΓq " EndpMq.This definition opens the door for studying the representation theory of D t,k,c,ν pΓq; indeed, if N is another object of C ν then the space HompM, Nq is naturally a (right) module over D t,k,c,ν pΓq.At the same time, it is easy to construct objects of C ν because it is given "by generators and relations"; for instance, if Γ is cyclic then C ν contains the category O which can be studied by methods of the theory of highest weight categories.In fact, in the case Γ " 1 this has already been started in [EA14].
In future publications we plan to apply this approach to the DDCA of gl m for m ą 1.Note that one of its advantages is that it easily applies to the case of m " 1 (discussed in this paper), while this is a difficult case for the approach of [Gua05, Gua07, Gua10, GY17, GHL09] which uses Steinberg-type presentations of gl m .
The second definition of the DDCA (which we show to be equivalent to the first one) is by explicit generators and relations (but different from [Gua05, Gua07, Gua10, GY17, GHL09]), and we give it only for Γ " 1 and Γ " Z{2.This definition is based on deforming the presentation of the Lie algebra po of Hamiltonians on C 2 and of its even part po `by generators and relations.Namely, we show (in part using a computer) that D t,k,c,ν pΓq in the case Γ " 1 is the unique filtered deformation of the enveloping algebra Uppoq for an appropriate filtration.We outline a similar approach for Γ " Z{2, although for larger Γ the relations get too complicated.
The organization of the paper is as follows.
Section 2 contains preliminaries.Section 3 describes generalities on ultraproducts, Deligne categories, and symplectic reflection algebras in complex rank, for simplicity concentrating mostly on the case of the rational Cherednik algebra of type A.
Section 4 explains two definitions of D t,k,ν -the DDCA of type A, both the usual one (as the ultraproduct of spherical rational Cherednik algebras of type A) and the one via Deligne categories, and shows that they are equivalent.In this section we also state and prove the presentation of this algebra by generators and relations, showing that this is the unique filtered deformation of Uppoq.
Finally, in Section 5 we generalize some of our results to DDCA for arbitrary Γ and also state the result about the presentation of the DDCA of type B by generators and relations.
Acknowledgments.This paper owes its existence to Victor Ginzburg, who proposed to study deformed double current algebras in the spring of 2001 and suggested, around the same time, some of the important ideas explored below.We are very grateful to Victor for sharing these ideas and initiating this research.We are also grateful to N. Guay, V. Ostrik, and T. Schedler for useful discussions.The work of P.E. and D.K. was partially supported by the NSF grant DMS-1502244.The computer calculations for this paper were done using MAGMA, [BCP97].

General notation
In what follows we will use a lot of different categories of representations.We will always denote the usual ("finite rank") categories of representations using the boldface font, and use the regular font for the interpolation categories (e.g.ReppS ν q).
For example we will use the following notation for the categories of representations of symmetric groups.For convenience set F 0 " Q. Definition 2.1.1.By ReppS n ; kq denote the category of (possibly infinite dimensional) representations of the symmetric group S n over k.By Rep f pS n ; kq denote the full subcategory of finite dimensional representations.Also for p ě 0 set Rep p pS n q :" ReppS n ; F p q and Rep f p pS n q :" Rep f pS n ; F p q.
We will also fix the notation for the irreducible representations of the symmetric group.
Definition 2.1.2.For a Young diagram λ, by lpλq denote the number of rows of the diagram (the length), by |λ| the number of boxes (the weight) and by ctpλq the content of λ, i.e., ctpλq " ř pi,jqPλ pj ´iq, where pi, jq denotes the box of λ in row i and column j.Definition 2.1.3.For p " 0 or p ą n and a Young diagram λ such that |λ| " n denote by X p pλq the unique simple object of Rep p pS n q corresponding to λ.
For n ą 0 and p ě 0 denote by h p n P Rep p pS n q, or shortly by h n (if there is no ambiguity about the characteristic) the standard permutation representation of S n .
There is an important central element in krS n s: Definition 2.1.4.Denote the central element ř 1ďiăjďn s ij P krS n s by Ω n .Remark 2.1.5.Note that Ω n acts on X p pλq by ctpλq.
As another piece of notation, below we will frequently use the following operation on Young diagrams: Definition 2.1.6.For a Young diagram λ and an integer n ě λ 1 `|λ| denote by λ| n the Young diagram pn ´|λ|, λ 1 , . . ., λ lpλq q, where λ i is the length of the i-th row of λ.
In what follows we will often use the language of tensor categories.Here's what we mean by a tensor category (see Definition 4.1.1 in [EGNO16]): Definition 2.1.7.A tensor category C is a k-linear locally finite abelian rigid symmetric monoidal category, such that End C p½q » k.
We will also fix a notation for the symmetric structure: Definition 2.1.8.For an object X of a tensor category C, we will denote by σ X the map from X b X to itself, given by the symmetric structure, i.e., the map permuting the two copies of X. Oftentimes, when the object we are referring to is obvious from the context, we will denote it simply by σ.
We will also use the notion of the ind-completion of a category.For a general category ind-objects are given by diagrams in the category, with morphisms being morphisms between diagrams.However, in the case of a semisimple category there is a more concrete description.
Definition 2.1.9.For a semisimple category C with the set of simple objects tV α u for α P A the category1 INDpCq is the category D with objects À αPA M α b V α , where M α are (possibly infinite dimensional) vector spaces.The morphism spaces are given by: Thus, in this case, we can think of ind-objects as infinite direct sums of objects of C.
Next we would like to explain a way to define an ind-object of C.
Construction 2.1.10.Suppose 0 " X 0 Ă X 1 Ă X 2 Ă ¨¨¨Ă X i Ă . . . is a nested sequence of objects of C. Then their formal colimit, which we denote by X, is an object of INDpCq.We can write it down explicitly in terms of Definition 2.1.9.Indeed, suppose we have X i " À αPA M i,α b V α .Then it follows that: where Ť iPN X i " lim Ý Ñ X i stands for the colimit along the diagram consisting of points numbered by N and arrows from i to i `1 for all i.
Remark 2.1.11.Suppose that X and Y are two objects constructed via Construction 2.1.10.Then: In case when X is actually an object of C, this simplifies to: Hom INDpCq pX, Y q " ď jPN Hom C pX, Y j q.
In other words, X is a compact object of INDpCq.
Example 2.1.12.We have Rep p pS n q " INDpRep f p pS n qq.Indeed, this holds for the representation category of any finite dimensional algebra.

Wreath products S n ˙Γn
To deal with DDCA with non-trivial Γ we will need to use a certain interpolation of categories of representations of wreath products.Below we will state basic facts about representations of wreath products in finite rank.
Definition 2.2.1.For a finite group Γ, consider the action of S n on Γ n by permutations.
The semiderect product S n ˙Γn is called the wreath product.
Remark 2.2.2.Outside of the present section we will be interested only in Γ Ă SLp2, kq.
However the results stated in the present section hold for any Γ.
We have the following classification of irreducible representations of S n ˙Γn .
Proposition 2.2.3.Suppose k is an algebraically closed field of characteristic charpkq " p ą n, |Γ| or p " 0. Suppose A is the set of indices which goes over all of the irreducible representations of Γ over k, i.e., tW α u αPA is the set of irreducible representations of Γ.Then the set of all irreducible representations of S n ˙Γn over k is in 1-1 correspondence with functions: The representation corresponding to fixed λ is given by: We will use the notations for the representation categories similar to the case of the symmetric group: Also for p ě 0 set Rep p pS n ˙Γn q :" ReppS n ˙Γn ; F p q, Rep f p pS n ˙Γn q :" Rep f pS n ˙Γn ; F p q.

The Cherednik algebra
In this paper we will be mainly interested in rational Cherednik algebras of type A. Thus we will only give definitions of this algebra below.For the definition and theory of general rational Cherednik algebras, see [EM10].
Definition 2.3.1.The rational Cherednik algebra of type A and rank n over a field k, denoted by H t,k pn, kq " H t,k pnq, where t, k P k, is defined as follows.Consider the standard representation of S n acting by permutations on h " k n with the basis given by y i P h, and the dual basis x i P h ˚.Then H t,k pnq is the quotient of krS n s ˙T ph ' h ˚q by the following relations: rx i , x j s " 0, ry i , y j s " 0, ry i , x j s " δ ij pt ´k ÿ m‰i s im q `p1 ´δij qks ij , where s ij denotes the transposition of i and j.
In other words, this is the rational Cherednik algebra corresponding to the root system A n´1 .
This algebra has a filtration determined by degpx i q " degpy i q " 1 and degpgq " 0 for any group element g.The associated graded algebra is: This follows from the fact that the analog of the PBW theorem holds for this algebra: Proposition 2.3.2.The natural map H 0,0 pnq Ñ grpH t,k pnqq is a vector space isomorphism.
Another important object is the spherical subalgebra of the rational Cherednik algebra.
Definition 2.3.3.If charpkq " p ą n or p " 0, denote by B t,k pnq the subalgebra eH t,k pnqe of H t,k pnq, where e P krS n s is the averaging idempotemt.
Note that: grpB t,k pnqq " Sph ' h ˚qSn " krx 1 , . . ., x n , y 1 , . . ., y n s Sn .Remark 2.3.4.One can construct the spherical subalgebra in another way.Indeed, regard k as the trivial representation of S n and apply to it the induction functor Ind

Symplectic reflection algebras
Another entity we are going to use to construct DDCA with non-trivial Γ is symplectic reflection algebras.Below we will give some basic definitions, needed for our purposes.For more on this topic see [EG02].
The symplectic reflection algebra is defined as follows: Definition 2.4.1.Fix a finite subgroup Γ Ă SLp2; kq.Fix numbers t, k P k.Fix numbers c C P k for every conjugacy class C Ă Γ; we will denote the collection of these numbers by c.For every conjugacy class C, set T C :" 1 2 Tr| k 2 γ, where γ P C is an element of the conjugacy class and we take the trace over the tautological representation.Consider V " pk 2 q n , the tautological representation of the wreath product S n ˙Γn .Note that this space has a natural symplectic structure, which we will denote by ω.Let Σ stand for the set of elements of S n ˙Γn conjugate to a transposition.For a conjugacy class C Ă Γ, let Σ C be the set of all elements conjugate to p1, 1, . . ., 1, γq for γ P C.
The symplectic reflection algebra H t,k,c pn, Γq is the quotient of krS n ˙Γn s ˙T pV q by the relations: ωpp1 ´sqy, p1 ´sqxqs, x, y P V.
We can also define the spherical subalgebra of this algebra: Definition 2.4.2.The spherical subalgebra of the symplectic reflection algebra H t,k,c pn, Γq is denoted by B t,k,c pn, Γq and is given by: B t,k,c pn, Γq " eH t,k,c pn, Γqe, where e is the symmetrizer for S n ˙Γn .
We will use the same notation for the categories of representations: Remark 2.4.5.Notice that when Γ " 1 we get back the case of rational Cherednik algebra of type A, i.e., H t,k,H pn, 1q " H t,k pnq.Also, in the case Γ " Z{2Z we get the rational Cherednik algebra of type B.

Ultrafilters
Below we will discuss some basic facts about ultrafilters and ultraproducts.Ultrafilters provide us with a notion of the limit of algebraic structures, which works really well for describing Deligne categories.Thus, we will use this framework extensively in the present paper.
We will define what ultrafilters and ultraproducts are, state their main properties and give some important examples, which will be used later in the paper.The following discussion is an updated version of the corresponding discussion from [Kal19].For more details on this topic in the algebraic context, see [Sch10].

Ultrafilters and ultraproducts: basic definitions
Definition 2.5.1.An ultrafilter F on a set X is a subset of 2 X satisfying the following properties: For any X, there is an obvious family of examples of ultrafilters.Indeed, taking F x " tA P 2 X |x P Au for any x P X gives us an ultrafilter.Such ultrafilters are called principal.Using Zorn's lemma one can show that non-principal ultrafilters F exist iff the cardinality of X is infinite.However the proof is non-constructive.
From now on we will only work with non-principal ultrafilters on X " N.
Definition 2.5.2.For the rest of the paper we will denote by F a fixed non-principal ultrafilter on N.
Note that it doesn't matter which non-principal ultrafilter to take, and all our results do not depend on this choice.Also note that all cofinite sets belong to F .Indeed, if some cofinite set wouldn't belong to F , it would follow that a finite set belongs to F .But from this one can conclude that F is a principal ultrafilter for one of the elements of this set.
Throughout the paper we will use the following shorthand phrase.
Definition 2.5.3.By the statement "A holds for almost all n", where A is a logical statement depending on n, we will mean that A is true for some subset of natural numbers U, such that U P F .
The following is an important lemma describing what happens with the conjuction and disjunction of statements which "hold for almost all n".Lemma 2.5.4. 1) If for two logical statements A and B we know that A holds for almost all n and B holds for almost all n, then A ^B holds for almost all n.
2) If for a finite number of logical statements A i , for i P I, we know that Ž iPI A i holds for almost all n, then there is j P I such that A j holds for almost all n.
Proof. 1) Indeed, we know that there is a set U A P F such that A holds for all n P U A , and the corresponding set for B. Now by definition of the ultrafilter U A X U B P F , and A ^B holds for all n P U A X U B .
2) Suppose that none of the statements A i hold for almost all n.This means that the sets on which A i hold do not belong to F .Thus by definition of the ultrafilter, the sets V i " tn P N| A i does not holdu are in F .Thus V " Ş iPI V i P F .But for any n P V we know that all of the statements A i do not hold.Hence for any n P V we know that Ž iPI A i does not hold.But the set U " tn P N| Ž iPI A i u belongs to F by assumption.So we have V and NzV belonging to F .A contradiction.
We will use these elementary observations quite frequently, sometimes without even mentioning it.Now, define the notion of an ultraproduct.Remark 2.5.6.Thus in a nutshell the ultraproduct consists of "germs" of sequences of elements which are defined for almost all n.Because of this in what follows we will sometimes use "sequence" to mean "sequence defined for almost all n".Remark 2.5.7.Note that for any finite set C, the ultraproduct of its copies ś F C i with C i " C is equal to C. Indeed, for any sequence tc n u nPA , for some A P F , we can define U d " tn P A|d " c n u for any d P C. Then we have Ť dPC U d " A, thus one of the U d 's must belong to F .So it follows that tc n u nPA " tdu nPA for this particular d.

Oftentimes we use the following notation:
Definition 2.5.8.For a sequence tE n u nPN , denote an element te n u nPN P ś F E n by ś F e n .This construction is interesting for us, because it, in a certain sense, preserves a lot of algebraic structures.We will explore this dimension of ultraproducts below.
Example 2.5.9.First, note that the ultraproduct inherits any operation or any relation which is defined on a sequence of sets E n for almost all n.For example, suppose we are given a sequence of k-ary operations ˝n defined for almost all n.Let E :" ś F E n and consider the k-ary operation ˝: E ˆE ˆ¨¨¨ˆE Ñ E defined as ˝pe 1 , e2 , . . ., e k q " ˝pź F e 1 n , . . ., ź F e k n q " ź F ˝n pe 1 n , . . ., e k n q.
Note that this is the same as taking ˝" ś F ˝n P ś F Hom Sets pE ˆk, Eq, so we can call an ultraproduct of ˝n.Now if we have any sequence of relations r n given for almost all n, they can be written as a sequence of k-ary maps with Boolean values.And one can define r to be a relation on E in a similar way rpe 1 , e 2 , . . ., e k q " rp ź For the same reason we can call the relation r the ultraproduct of the relations r n .Note that this means that if the relation r n was true for almost all n (i.e., Impr n q " t1u for almost all n), it follows that r is also true.
One can easily check for oneself that the above examples (2.5.9) can be extended to any collections of sequences of sets, maps between them and relations between maps.That means that if we have a collection of sequences of sets with a certain algebraic structure defined by maps between them, we can form the ultraproducts of these sets and these maps.Moreover if the sequences of maps satisfy a certain collection of relations, the ultraproduct will satisfy them too.
These observations may be formulated in the following way: Theorem 2.5.10.Loś's theorem (Theorem 2.3.2 in [Sch10]) Suppose we have a collection of sequences of sets E pkq i for k " 1, . . ., m, a collection of sequences of elements f prq i for r " 1, . . ., l, and a formula of a first order language φpx 1 , . . ., x l , Y 1 , . . ., Y m q depending on some parameters x i and sets Y j .Denote by n , . . ., E pmq n q is true for almost all n iff φpf p1q , . . ., f plq , E p1q , . . .E pmq q is true.
In the next subsection we will provide a few examples of application of this theorem.One can easily see how the theorem works by working out what happens in these examples on one's own.Many of these examples will be used in the rest of the paper.

Examples of ultraproducts
Example 2.5.11.If E n is a sequence of monoids/groups/rings/fields then ś F E n with operations given by taking the ultraproduct of the operations as elements of the corresponding sets of set-theoretical maps gives us a structure of a monoid/group/ring/field by Loś's theorem.
Example 2.5.12.If V i are finite dimensional vector spaces over a field k, then ś F V n is a vector space over ś F k, which is not necessarily finite dimensional, since the property of being finite dimensional cannot be written in a first-order language.But if the dimensions of V n are bounded, then they are the same for almost all n and hence V has the same dimension (for example, because the ultraproduct of bases is a basis).
Example 2.5.13.Take the ultraproduct of a countably infinite number of copies of Q.Let qpxq P Zrxs be the minimal polynomial for ν.We would like to find an infinite number of pairs ν n , p n such that qpν n q " 0 mod p n .Let us show that there is an infinite number of primes dividing the collection of numbers qplq for l P N, from this it would follow that there is an infinite number of pairs since only a finite number of primes divide each qplq.Suppose it is not so, and there are only k such primes.Fix C such that we have qplq ă C ¨ldegpqq for all positive integer values of l.Denote by Q the number of integers of the form qplq for l P Z ě0 such that qplq ă L. By the above inequality (that is degpqq .On the other hand the number P of numbers less than L divisible only by k fixed primes is less or equal to log 2 pLq k , since each prime number is at least 2. Hence for big enough L we have P ă Q, which contradicts the hypothesis4 . Hence we can take a sequence of distinct primes p n and a sequence of integers ν n tending to infinity such that qpν n q " 0 in F pn and ν n ă p n .It follows that ś F ν n in ś F F pn is a root of qpxq.Hence by an automorphism of C we can send ś F ν n into ν.Example 2.5.15.Suppose C n is a sequence of (locally small) categories.We can define the ultraproduct category p C " ś F C n as the category whose objects are sequences of objects in C n .For clarity we will denote the ultraproduct of objects by5 ś C F .The morphisms in p C are given by and the composition maps are given by the ultraproducts of the composition maps, i.e., p ś F f n q ˝pś F g n q " ś F pf n ˝gn q.By Loś's theorem this data satisfies the axioms of a category.If the categories C n have some structures, for example the structures of an abelian or monoidal category, then p C also has these structures6 .
Usually p C is too big and it is interesting to consider a certain full subcategory C in there, for example by only considering the ultraproducts of sequences of objects of C i bounded in some sense.This will be discussed in more detail in the next subsection.
Remark 2.5.16.Note that taking the ultraproduct of a sequence of algebraic objects as such is different from considering their ultraproduct as a sequence of objects in certain categories.
For example, consider a sequence of countably-dimensional vector spaces V n over k.By Loś's theorem ś F V n is a vector space (although its dimension is more than countable).However, we can also regard V n as objects of the categories C n " Vect k and construct ś C F V n P ś F Vect k .The category ś F Vect k is not equivalent to the category of vector spaces (for example, it is rigid and can have objects of non-integer dimension), so ś C F V n is not a vector space in any sense.Also frequently it is useful to think about an ultraproduct as a certain kind of a limit as n Þ Ñ 8, where n becomes a "free" parameter.
Example 2.5.17.Consider a sequence of finite dimensional algebras A n over Q with a sequence of fixed vector space isomorphisms A n » V .Equivalently, this means that we have a sequence of binary operations µ n : V b V Ñ V which satisfy all the axioms of an algebra.Suppose in some basis (and hence in any basis) the matrices of µ n have entries which depend polynomially on n.
Consider A " ś F A n .By Example 2.5.13 this is an algebra over Cpxq.Since A i are finite dimensional and all isomorphic to V via a fixed isomorphism, we can also conclude that the binary operation on A, which we denote by µ, is given by ś F µ n .Since µ n depended polynomially on n and x " ś F n, it follows that µ is given by the same formulas as the sequence µ n with n substituted by x.In other words, if c γ α,β pnq are the structure constants of µ n in a certain basis then c γ α,β pxq are the structure constants of µ.I.e., n becomes a formal parameter in A.

Restricted ultraproducts
When one works with a sequence of objects which are in some sense infinite dimensional, it's sometimes useful to consider a subobject in the ultraproduct consisting of the sequences of elements which are in a some way bounded.This can be called a restricted ultraproduct.We have already mentioned this in the case of categories in Example 2.5.15.For example, the Deligne category ReppS ν q will be constructed as a full subcategory in a certain ultraproduct category.
In this section we will outline the definitions of the restricted ultraproduct which makes sense in the case of filtered or graded vector spaces and categories.Definition 2.5.18.For a sequence of vector spaces E n with an increasing filtration Definition 2.5.19.For a sequence of vector spaces E n with a grading E n " À 8 k"0 gr k E n , define the restricted ultraproduct ś r F E n to be equal to À k i"0 gr i E n , this construction matches the construction of Definition 2.5.18.
We will use this notion in the case when the dimensions of the space F k E n are finite and stabilize as n Ñ 8 for fixed k.Let us give a few examples.
Example 2.5.20.Consider a countable-dimensional vector space V over k.Consider a sequence of copies of V , i.e., V n " V .Also consider an increasing filtration F j V by finite dimensional subspaces and the same filtration on all V n .We can calculate the restricted ultraproduct of this sequence: Whereas the usual ultraproduct ś F V n is more than countable-dimensional.Example 2.5.21.This is an extension of Example 2.5.17 to an infinite dimensional setting.Consider A n , a sequence algebras over Q with an increasing filtration by finite dimensional subspaces, such that for every k P N there is N k such that for n ą N k all F k A n are isomorphic as vector spaces to a fixed vector space F k A 8 via fixed isomorphisms.I.e., every filtered component stabilizes after a certain point.
This means that we have a collection of sequences of coherent multiplication maps µ k,l n : Let's also suppose that this sequence depends polynomially on n.
Consider A " ś r F A n .Note that as a vector space the restricted ultraproduct equals to: Now as in Example 2.5.17 the ultraproducts µ k,l " ś F µ k,l n define a coherent collection of multiplication maps, the union of which defines a map µ : A ˆA Ñ A. The structure constants of this multiplication can also be obtained by taking the structure constants of A n and plugging in x instead of n.
Note that the same construction works if the structure constants depend on n as rational functions.
This example shows better why it makes sense to think about the ultraproduct as a limit.
We also would like to introduce a related construction, which we will also call a restricted ultraproduct.This will take place in the setting of the ultraproducts of categories.Suppose tD i u is a sequence of artinian abelian categories and D " ś F D i is their ultraproduct (an abelian category which is, in general, not artinian).Suppose C is a full artinian subcategory of D. Using Construction 2.1.10we can obtain ind-objects of C in the following way.
Construction 2.5.22.Suppose we have a sequence of ind-objects X n P INDpD n q such that each X n is equipped with a filtration by objects of D n .I.e., we have It follows that the sequence F i X 8 defines an object X 8 P INDpCq as: We will use a special notation for this construction: Definition 2.5.23.In the setting of Construction 2.5.22,call X 8 the restricted ultraproduct of X n with respect to the fixed filtration.We will write Remark 2.5.24.Let r F ‚ be another filtration on the sequence tX n u such that ś C F F i X n P C, and let r X 8 be the corresponding restricted ultraproduct.Let us say that F, r F are equivalent if for any i there exist rpiq, spiq such that F are equivalent, then we have maps F i X 8 Ñ r F rpiq X 8 and r F i X n Ñ F spiq X 8 , which give rise to maps X 8 Ñ r X 8 and r X 8 Ñ X 8 which are clearly inverse to each other; thus X 8 and r X 8 are naturally isomorphic.This shows that X 8 depends only on the equivalence class of the filtration F .
However, not all filtrations are equivalent.E.g., if X n " k n , F i X n is spanned by the first i `1 standard basis vectors for i ď n ´1, g n P GLpn, kq and r F " g n pF q on X n then in general F, r F are not equivalent.Thus, without specifying a filtration (at least up to equivalence), we cannot define the restricted ultraproduct of X n .

Constructions of the category ReppS ν q
In this section we will discuss a well known construction of the interpolation category for the symmetric group due to Deligne [Del07] and its basic properties.For more on this topic see [CO11,CW12,CO14,Eti14,Eti16].We assume that k has characteristic 0.
We will start by introducing the system of vector spaces which is going to play a role of the homomorphism spaces in the corresponding skeletal category.Although these spaces are best understood using diagrams, we will omit this for the sake of space.We advise anyone seeing Deligne categories for the first time to see [CO11] for a much clearer diagrammatic construction of ReppS ν q.Definition 3.1.1.Denote by kP n,m a vector space over a field k with the basis given by all possible partitions of an n `m-element set.Diagrammatically an element of the basis is represented by two rows of ‚'s, the first of length n and the second of length m, where all ‚'s belonging to the same part of the partition are connected by edges.So, in other words, it is a graph on n `m vertices, the set of connected components of which corresponds to a partition of n `m (The graphs with the same set of connected components represent the same basis element).
Define a map φ n,m,k ν : kP m,k ˆkP n,m Ñ kP n,k for ν P k as follows.Consider two basis elements λ P kP n,m and µ P kP m,k .Take a vertical concatenation of the graphical representations of the corresponding partitions (the last one on top) and identify the rows of length m.After this we are left with a partition of three rows of ‚'s of length n, m and k.Now let's denote by lpµ, λq the number of connected components consisting purely of ‚'s lying in the second row.Also regard a partition of rows n, k consisting of the same connected components as the partition of rows n, m, k but with elements of the second row deleted, and denote it by µ ¨λ.Then φ n,m,k ν pµ, λq " ν lpµ,λq µ ¨λ.Define kP n pνq to be kP n,n with a structure of an algebra given by the map φ n,n,n ν .This algebra is called the partition algebra and it was introduced by Purdon in [Pur91].
The spaces kP n,m can be seen as limits of the homomorphism spaces Hom S N ph bn N , h bm N q, where h N is the permutation representation of S N .
Using this we can define a preliminary skeletal7 category Rep 0 pS ν ; kq: Definition 3.1.2.For ν P k we denote by Rep 0 pS ν ; kq a skeletal rigid symmetric monoidal k-linear category with objects given by elements of Z ě0 , which can be graphically represented by rows of ‚'s, and denoted by rns.
The set of morphisms Hom Rep 0 pSν ;kq prns, rmsq is equal to kP n,m and the composition maps are given by φ n,m,k ν .
Tensor product on objects is defined by the horizontal concatenation of rows and on morphisms by the horizontal concatenation of diagrams.All objects rns are self-dual.
Using this we can define the Deligne category ReppS ν ; kq itself: Definition 3.1.3.For ν P k, the Deligne category ReppS ν ; kq is the Karoubian envelope of the additive envelope of Rep 0 pS ν ; kq.
This means that we add all possible direct sums and direct summands into our category.
Below we will list a few pieces of notation and results concerning Deligne categories.They are well known and can be found for example in [CO11,Eti14].Definition 3.1.4.The object r1s is called the permutation representation and is denoted by h.The object r0s is called the trivial representation and is denoted by k (by a slight abuse of notation).
The important properties of ReppS ν ; kq are listed below: Proposition 3.1.5.a) For ν R Z ě0 ReppS ν ; kq is a semisimple tensor category.b) For ν R Z ě0 simple objects of ReppS ν ; kq are in 1-1 correspondence with Young diagrams of arbitrary size.They are denoted by X pλq.Moreover X pλq is a direct summand in r|λ|s.c) The categorical dimension of h is ν and of k is 1. d) All X pλq are self-dual.

The Deligne category enjoys a certain universal property:
Proposition 3.1.6.(8.3 in [Del07]) For any k-linear Karoubian symmetric monoidal category T , the category of k-linear symmetric monoidal functors from ReppS ν ; kq to T is equivalent to the category T f ν of commutative Frobenius algebras in T of dimension ν.The equivalence sends a functor F to the object F phq.
The important consequence of this result is that for every commutative Frobenius algebra A in a Karoubian symmetric category T of dimension ν, we have a symmetric monoidal functor from ReppS ν ; kq to T which sends h to A.
Remark 3.1.7.Here by a commutative Frobenius algebra in T we mean an object A with the following structure.It is an associative commutative algebra with the corresponding algebraic structure given by µ A , 1 A , and if we define a map: Tr Ý Ñ ½ is required to be non-degenerate, i.e., it corresponds to an isomorphism between A and A ˚under the identification of Hom T pA b A, ½q with Hom T pA, A ˚q.
In the rest of the paper we will use Deligne categories over the following fields: Definition 3.1.8.For ν P C set ReppS ν q :" ReppS ν ; Cq.For ν P Cpνq set Rep ext pS ν q :" ReppS ν ; Cpνqq.

Deligne categories
ReppS ν q and ReppS ν ˙Γν q as ultraproducts 3.2.1The category ReppS ν q as an ultraproduct In this section we will show how to construct ReppS ν q using ultraproducts, and discuss some important consequences of this construction.This method is very useful, because it allows one to transfer all kinds of constructions and their properties from the case of finite rank categories almost automatically.The main ideas of this approach were contained in [Del07], [Har16] 8 .The idea is to construct the category ReppS ν q for non-integer ν as a full subcategory in the ultraproduct category following Example 2.5.15.We have the following result (See the introduction of [Del07] or Theorem 1.1 in [Har16]): Then the full subcategory of the ś F Q-linear category p C generated by h ν under taking tensor products, direct sums and direct summands is equivalent to the C-linear category ReppS ν q, in a way consistent with the fixed isomorphism ś F Q » C. b) Suppose ν P C is algebraic but not a nonnegative integer.Fix a sequence of distinct primes p n , a sequence of integers ν n , and an isomorphism Then the full subcategory of the ś F F pn -linear category p C generated by h ν under taking tensor products, direct sums and direct summands is equivalent to the C-linear category ReppS ν q, in a way consistent with the fixed isomorphism The required isomorphism of fields exists by Example 2.5.13.So we have a Karoubian symmetric monoidal category p C linear over C, with an object ś C F h n of dimension ν.Since every h n is a commutative Frobenius algebra, it follows by Loś's theorem that h ν is also a commutative Frobenius algebra.Hence by Proposition 3.1.6we obtain a symmetric monoidal functor F : ReppS ν q Ñ p C which takes h to h ν .Since ReppS ν q is generated by h under taking tensor products, direct sums and direct summands, it follows that the image of ReppS ν q under F is the full subcategory C in p C generated by h ν under taking tensor products, direct sums and direct summands.So we know that F : ReppS ν q Ñ C is essentially surjective.Now it is enough to prove that it is fully faithful.
Note that it is enough to prove that ź F Hom Sn ph br n , h bs n q " Hom ReppSν q prrs, rssq, and that the composition maps are the same.Indeed, if this is true, both categories can be obtained as the Karoubian envelopes of the additive envelopes of the categories consisting of all rrs or h br ν respectively.But this follows from Theorem 2.6 in [CO11].Indeed, there it is stated that there is an isomorphism between QP r,s and Hom Sn ph br n , h bs n q for n ą r `s.So for almost all n we have Hom Sn ph br n , h bs n q " QP n,m .Also Proposition 2.8 in the same article states that under this isomorphism the composition rule on Hom Sn ph br n , h bs n q transforms into the composition rule on QP r,s in the definition of Rep 0 pS ν q.So it follows that, indeed, ś F Hom Sn ph br n , h bs n q " Hom ReppStq prrs, rssq, and the composition rule is the same.b) Again the required isomorphism exists by Example 2.5.14.The rest of the proof is the same since the representation theory of S n is the same in zero characteristic and in characteristic p ą n, and p n ą ν n for almost all n.
Remark 3.2.2.Note that for the purposes of this theorem we could also have used the categories Rep pn pS νn q.
We can also formulate a similar result for Rep ext pS ν q: Then the full subcategory of the 8 For the similar discussion about ReppGL ν q see [Del07], [Har16], [Kal19].
category p C generated by h ν under taking tensor products, direct sums and direct summands is equivalent to the Cpνq-linear category ReppS ν q, in a way consistent with the fixed isomorphism ś F Q » Cpνq.Proof.This follows from the above Theorem and the fact that C » Cpνq (see Example 2.5.13).
Remark 3.2.4.As mentioned in the beginning of Section 2.1, to treat the algebraic and transcendental cases simultaneously, it's useful to agree on the convention that by F 0 we will mean Q, and so the case ν n " n, p n " 0 in the setting of part pbq of the Theorem 3.2.1 gives us transcendental ν.Also below we will always assume that the sequences p n and ν n are the sequences from Theorem 3.2.1 or Corollary 3.2.3corresponding to the given ν.Finally, we will work only with ν P CzZ ě0 .Now we would like to explain why this construction of the Deligne categories is quite useful.To begin with, we would like to construct the simple objects X pλq as ultraproducts.This is easy to do, using the notation from Definition 2.1.6:Proposition 3.2.5.The irreducible object X pλq of ReppS ν q can be obtained as an ultraproduct of irreducible objects of Rep f pn pS νn q as X pλq " ś C F X νn pλ| νn q.Proof.From Section 3.3 of [CO11] we know that the algebras kP r pνq for ν ‰ 0, 1, . . ., 2r have the same set of idempotents obtained by specialization from idempotents of kpxqP r pxq.Now by construction all simple objects of ReppS ν q are given by the primitive idempotents of End Rep 0 pSν ;kq prrsq " kP r pνq.And by Theorem 3.2.1,kP r pνq » ś F F pn P r pν n q in such a way that basis elements are ultraproducts of basis elements.Thus it follows that idempotents in kP r pνq are given by the ultraproducts of the same idempotents for almost all n.And so the claim follows.
This result allows us to reformulate the definition of ReppS ν q as an ultraproduct.Proposition 3.2.6.In the notation of Theorem 3.2.1 the category ReppS ν q can be described as the full subcategory of p C " Rep f pn pS νn q consisting of sequences of objects Y n " À αPAn X pn pλ n,α q for some indexing sets A n and Young diagrams λ n,α such that both the sequence of |A n | and the sequence of max αPAn p|λ n,α | ´pλ n,α q 1 q, where pλ n,α q 1 is the length of the first row, are bounded for almost all n.
Proof.We know that ReppS ν q is a full subcategory of p C so we just need to match the objects.
On the one hand, suppose Y P ReppS ν q.We know that for some set of Young diagrams µ α with α P A, a finite indexing set, we have Y " À αPA X pµ α q, so from Proposition 3.2.5 it follows that Y " ś C F À αPA X pn pµ α | νn q.Thus we have a required sequence with A n " A and λ α,n " µ α | νn .The sequence |A n | " A is constant, hence so is the sequence max αPAn p|λ n,α | ´pλ n,α q 1 q " max αPA p|µ α |q.
On the other hand, suppose we have a sequence described in the statement of the Theorem.Since we know that |A n | is bounded for almost all n, there is a finite number of options for the cardinality of |A n | for almost all n, thus from part 2 of Lemma 2.5.4 it follows that for almost all n the cardinality is the same.Fix A to be a set of this cardinality.So, for almost all n we have Y n " À αPA X pn pλ n,α q.Suppose max αPAn p|λ n,α | ´pλ n,α q 1 q is bounded by L. Now each λ n,α is a Young diagram of weight ν n with at most L boxes in the rows above the first one.I.e., for n big enough (namely, ν n ą 2L), it follows that each λ n,α " µ n,α | νn where µ n,α is a Young diagram of weight at most L.So for almost all n each Y n is uniquely determined by a collection of |A| Young diagrams of weight at most L. Notice that there is only a finite number of such collections.So by the same Lemma it follows that for almost all n the collection is the same.Denote it by tµ α u αPA .Hence, for almost all n up to a permutation we have Y n " À αPA X pn pµ α | νn q.Hence we have ś C F Y n " À αPA X pµ α q which is indeed an object of ReppS ν q.
So, as promised in Example 2.5.15,ReppS ν q can indeed be described as given by ultraproducts bounded in a certain sense.
We will also need to explain how to interpolate the central element Ω n P krS n s to ReppS ν q.Recall that we can consider the central elements of krS νn s as endomorphisms of the identity functor of Rep pn pS νn q.Definition 3.2.7.Denote by Ω the endomorphism of the identity functor of ReppS ν q given by the restriction of the endomorphism ś F Ω νn .One can easily calculate the action of Ω on simple objects.Proposition 3.2.8.[Eti14] The action of Ω on an object X pλq is given by: Ω| X pλq " ˆctpλq ´|λ| `pν ´|λ|qpν ´|λ| ´1q 2 ˙1Xpλq .
Proof.Since X pλq " ś C F X pn pλ| νn q, one needs to calculate ś F ctpλ| νn q.It's easy to see that each box of λ contributes an extra ´1 to the content of λ| νn , also ν n ´|λ| new boxes in the first row contribute 0 `1 `¨¨¨`pν n ´|λ| ´1q to the content of λ| νn , thus we have: ź which is exactly the value in the statement of the proposition.
Remark 3.2.9.Note that all of the results of this Section work mutatis mutandis for Rep ext pS ν q (see Definition 3.1.8).
Now we would like to give the reader a general idea of how this can be used to transfer constructions and facts from representation theory in finite rank to the context of Deligne categories.
Suppose we have a representation-theoretic structure Y n in each Rep pn pS νn q which can be constructed uniformly in an element-free way for every n.Then we can define the same structure Y in ReppS ν q using the analogs of the same objects and maps.Since the definitions are the same, it would follow that Y " ś F Y n .Now one can try to transfer the properties of Y n to Y.For some it can be as easy as a direct application of Loś's theorem.Others require quite a bit of technical work before one can do that.For some interesting results of this type see [Kal19,HK20].
Oftentimes the structure Y might include some ind-objects of ReppS ν q.This will happen, for example, when we will try to define the rational Cherednik algebra in ReppS ν q.Thus we will deal with ind-objects in the ultraproduct setting in the next subsection.

Ind-objects of ReppS ν q as restricted ultraproducts
In this section we are going to explain how ind-objects of ReppS ν q can be obtained as restricted ultraproducts, thus extending Theorem 3.2.1 in a certain way.
To do that, we will use the result of Construction 2.1.10.
Proposition 3.2.10.Suppose we have a sequence of representations of M n P Rep pn pS νn q, with fixed filtration by subrepresentations of finite length.i.e., we have 9 One can also define, through a more involved construction, the category INDpReppS ν qq as a subcategory of ś F Rep pn pS νn q.Note that this subcategory will not be full.In this way one would also be able to consider , take the ultraproduct directly.It can be shown that this would define the same object M .
Proof.This follows from Construction 2.5.22.
Remark 3.2.11.Note that, using Remark 2.1.11,we conclude that if M P INDpReppS ν qq has finite length, then for any N P INDpReppS ν qq constructed via Proposition 3.2.10,we have: with the filtration arising from the filtration on N.

The category ReppS ν ˙Γν q
In this section we will explain how the category of representations of the wreath product in complex rank can be constructed.
There are several ways to approach this problem.One construction was developed by Knop in [Kno07].Another approach can be found in [Mor12].However, in the present paper we will use a different approach, outlined in [Eti14].For brevity we will only address the case of transcendental ν in this section, although with slight modifications the results can be extended to the algebraic case as well.
Below we will use the notion of a unital vector space.For details see [Eti14].
Definition 3.2.12.A unital vector space V is a vector space together with a unit, i.e., a distinguished non-zero vector denoted by 1 P V .
In [Eti14] it is shown that given a finite dimensional unital vector space V , one can define an ind-object V bν P ReppS ν q.The idea behind this is that, although there is no way to algebraically define x t , there is such a way to define p1 `xq t :" ř mě0 `t m ˘xm .
We can also construct this object via an ultraproduct.Anyone not familiar with [Eti14] might regard this as definition for the purposes of this paper.
Note that the S n -module V bn has a natural filtration induced by the filtration on V given by F 0 V " k1, F 1 V " V .Proposition 3.2.13.For a finite dimensional unital vector space V , the ind-object V bν is given by: Proof.Using the notation of [Eti14], we have: where S λ|n are the corresponding Schur functors, and Thus, we obtain ź C,r Now consider a finite subgroup Γ Ă SLp2, Qq.Proposition 3.2.13allows us to define the following algebra: Definition 3.2.14.An ind-object CrΓs bν is constructed via Proposition 3.2.13starting with QrΓs as a unital vector space.It has the structure of the algebra given by the ultraproduct of the algebra structures on QrΓs bn .Using this, one can define the category ReppS ν ˙Γν q in the following way: Definition 3.2.15.The category ReppS ν ˙Γν q is the category of CrΓs bν -modules in ReppS ν q.I.e., its objects are objects of ReppS ν qq with the structure of a CrΓs bν -module, and its morphisms are morphisms in ReppS ν q which commute with the module structure.
It can be shown that ReppS ν ˙Γν q is equivalent to the wreath product category defined by Knop.We can construct some of the objects of ReppS ν ˙Γν q as ultraproducts.
Proposition 3.2.16.Consider a sequence of modules M n P Rep 0 pS n ˙Γn q whose ultraproduct as S n -modules is a well-defined object of ReppS ν q.Then, this ultraproduct also lies in ReppS ν ˙Γν q.
Proof.Denote M " ś C F M n .Indeed since M n has a structure of a QrΓs bn -module in Rep 0 pS n q, it follows that M has a structure of ś C F QrΓs bn " CrΓs bν -module.Hence it is an object of ReppS ν ˙Γν q.
In this way we can interpolate irreducible objects of Rep 0 pS n ˙Γn q.
Definition 3.2.17.In the notation of Proposition 2.2.3, consider λ to be any function: Denote by X pλq the object of ReppS ν ˙Γν q defined as: where λ n ptrivq " λptrivq| n and λ n pαq " λpαq for all other irreducibles α of Γ.It follows that X pλq is irreducible.
Remark 3.2.18.We leave out the proof of the fact that these ultraproducts indeed define an object of ReppS ν q.This can be done using the results of [Kno07], but we do not need this for this paper.
3.3 Cherednik algebras in complex rank

Cherednik algebra of type A in complex rank
In this subsection we will explain how to construct the interpolation category for the representations of the rational Cherednik algebra of type A. After that we will construct an induction functor interpolating the functors Ind . This will allow us to define the DDC-algebra below.One can find more information about the rational Cherednik algebras in complex rank in [EA14].
The definition of ReppH t,k pνqq mimics the definition of representations in the finite rank in an element-free way: Definition 3.3.1.The category ReppH t,k pνqq is defined as follows.The objects are given by triples pM, x, yq, where M is an ind-object of ReppS ν q, x is a map x : h ˚b M Ñ M and y a map y : h b M Ñ M, both of which are morphisms in INDpReppS ν qq.They also satisfy the following conditions: The morphisms of ReppH t,k pνqq are the morphisms of INDpReppS ν qq which commute with the action-maps x and y.
Also by Rep ext pH t,k pνqq denote the similar category constructed over Rep ext pS ν q.
Note that, since our objects already are "S ν -modules" we don't need to define any additional "S ν -action".
The last formula in the definition may need some explanation.To clarify it, let us apply it to y i b x j b M in the finite rank case.We have: which is precisely the formula from Definition 2.3.1.Hence we see that this is indeed the finite-rank definition rewritten in an element-free way.Now we would like to show how we can construct some of the objects of the category ReppH t,k pνqq as ultraproducts.
Remark 3.3.2.Below we will denote by t n , k n the elements of F pn such that ś F t n " t and ś F k n " k under the fixed isomorphism of ś F F pn » C. We will use the similar notation for all other parameters of algebras used in the paper.
Lemma 3.3.3.Suppose M n is a sequence of objects of Rep pn pH tn,kn pν n qq such that their (restricted) ultraproduct as objects of Rep pn pS νn q lies in INDpReppS ν qq.Suppose x n and y n are the maps which define the action of generators of the corresponding Cherednik algebra on M n .Then p ś C,r F M n , ś F x n , ś F y n q defines an object of ReppH t,k pνqq.
Proof.It's easy to see that the data p ś C,r F M n , ś F x n , ś F y n q is well defined.Since x n and y n satisfy the same conditions in finite rank and complex rank it follows that by Loś's theorem this is indeed an object of ReppH t,k pνqq.

Now we would like to construct an interpolation of the functors Ind
H tn,kn pνnq Sν n .It is possible to construct the full functor as ultraproduct directly, but this functor would a priori have ś F Rep pn pH tn,kn pν n qq as its target category, so we would need to explain why the functor really gives us objects of ReppH t,k pνqq.Instead we will construct this functor directly, which will also show that it agrees with the ultraproduct functor when applied to objects of ReppS ν q.
The idea is, following the PBW theorem, to think about "H t,k pνq" as "the direct sum À i,jě0 S i ph ˚q b S j phq b CrS ν s" and take the tensor product with V P ReppS ν q "over CrS ν s".Construction 3.3.4.For an object V P ReppS ν q, consider an ind-object I V " ' i,jě0 I i,j , where I i,j " S i ph ˚q b S j phq b V , and maps x V : h ˚b I V Ñ I V and y : h b I V Ñ I V , which are defined as follows.
First, note that S i`1 phq is isomorphic to a direct summand of h b S i phq, let's denote the corresponding inclusion and projection as ι i`1,y and π i`1,y respectively.The same is true for h ˚, the corresponding morphisms are ι i`1,x and π i`1,x .Now define px V q| I i,j : h ˚b I i,j Ñ I i`1,j to be equal to π i`1,x b 1 for all i, j.Also define py V q| I 0,j : hbI 0,j Ñ I 0,j`1 as π j`1,y b1.And lastly we define py V q| I i,j : hbI i,j Ñ I i,j`1 'I i´1,j by induction in i as: " px b 1q ˝p1 b y b 1q ˝pσ b 1q `t ¨ev h b 1 ´k ¨pev h b 1q ˝pΩ I i´1,j ´Ωh,I i´1,j q ‰ ˝p1bι i,x b1q.
Now we would like to show that this defines an object of ReppH t,k pνqq.Indeed: Lemma 3.3.5.In the notations of Construction 3.3.4,the triple pI V , x V , y V q defines an object of ReppH t,k pνqq.
Proof.Indeed, the first two formulas of Definition 3.3.1 are satisfied by the properties of symmetric powers, and we defined the action of y V by induction in such a way that the third equation is also satisfied.Another way to see that is to note that in the finite rank case this construction amounts to H tn,kn pν n qb Sν n V n , and so by Loś's theorem, we do get a correct structure of an "H t,k pνq-module".Now we need to construct the action of the induction functor on morphisms.Construction 3.3.6.In the notation of Construction 3.3.4,given a morphism φ : V Ñ U, define a morphism I φ : I V Ñ I U in the following way: pI φ q| S i ph ˚qbS j phqbV :" 1 b 1 b φ.
Proof.This is easy to see both straight from the definition, or by the ultraproduct argument, since in finite rank this defines an actual H tn,kn pν n q-module morphism.

Now we can define the actual functor:
Definition 3.3.8.Define a functor Ind H t,k pνq Sν : ReppS ν q Ñ ReppH t,k pνqq in the following way.On objects it takes V to the triple pI V , x V , y V q from Construction 3.3.4.And on morphisms it takes φ : V Ñ U to I φ from Construction 3.3.6.This is a well defined functor by Lemmas 3.3.5 and 3.3.7.

The next Corollary follows by construction and the above lemmas:
Corollary 3.3.9.For any object V P ReppS ν q such that V " ś F V n we have: where the filtration on Ind H tn,kn pνnq Sν n V n is obtained from the filtration of H tn,kn pν n q given by degpx i q " degpy i q " 1 and degpσ ij q " 0.
Remark 3.3.10.All of the constructions of the present section work for Rep ext pH t,k pνqq in the same fashion.

Symplectic reflection algebras in complex rank
In this section we will briefly generalize the results of the previous section to the context of symplectic reflection algebras.As in Section 3.2.3,we will work for transcendental ν for simplicity.Also as in that section, we fix a finite group Γ Ă SLp2, Qq.
Below we will define the category ReppH t,k,c pν, Γqq following the lines of Definition 3.3.1.To do this, we need to find the analog of V in Definition 2.4.1.
Proof.Indeed as S n -modules, each pQ 2 q n " h n ' h n , hence their ultraproduct is given by h 2 as an object of ReppS ν q.Thus by Proposition 3.2.16 it follows that it is also an object of ReppS ν ˙Γν q.The symplectic pairing is given by the ultraproduct of symplectic pairings.
We will denote this object by V and call the fundamental representation of "S ν ˙Γν ".Also V carries a natural symplectic pairing ω.Now we are ready to define the category itself.
Definition 3.3.12.Consider t, k, c C , T C as in Definition 2.4.1 with k " C. Let ν P C be a transcendental number.The objects of the category ReppH t,k,c pν, Γqq are given by pairs pM, yq, where M is an object of ReppS ν ˙Γν q and y is a map: such that the following holds: where Ω is an endomorphism from Definition 3.2.7 and Ω C is the endomorphism obtained in a similar way as the ultraproduct of endomorphisms of the identity functor arising from the sum of elements of the group belonging to the conjugacy class C. The morphisms are given by morphisms in ReppS ν ˙Γν q which commute with y.
In a fashion similar to the discussion after Definition 3.3.1 one can see that this definition is the same as in finite rank, written in an element free way.Thus for the same reasons one obtains the following statement, which generalizes Proposition 3.2.16 and Lemma 3.3.3.Proposition 3.3.13.Suppose M n are H tn,kn,cn pn, Γq-modules whose ultraproduct ś C,r F M n is a well defined object of INDpReppS ν qq.Suppose y n denotes the corresponding map pF pn q, so the conclusion follows from Corollary 3.3.9.
Note that we can define a filtration on H t,k pνqe by objects of ReppS ν q using the construction in Definition 2.1.9.Indeed, assign degpxq :" 1 and degpyq :" 1, i.e., we take F m H t,k pS ν qe to be equal to ř m i"0 S i ph ˚q b S k´i phq b C.This agrees with the filtration by Rep pn pS νn q-modules of H tn,cn pν n qe given by degpx i q " degpy i q " 1. 10Notice that the same assignment of degrees defines a grading of H t,k pνqe (and respectively H tn,cn pν n qe) by S ν -modules (S νn -modules).Hence we have a corollary: Note that this is an actual algebra over C, since it is given by a vector space of morphisms.
Also note that we can rewrite this as: End ReppH t,k pνqq pH t,k pνqeq " Hom INDpReppSν qq pC, H t,k pνqeq.
So this algebra is given by the direct sum of all trivial representations of S ν in H t,k pνqe.
Via this observation we can trivially restrict the grading of H t,k pνqe to the grading on D t,k,ν .Note that by Remark 2.3.4 in finite rank this construction gives us the spherical subalgebra B tn,kn pν n q.The spherical subalgebras inherit the gradings in a similar fashion.
To finish this section we would like to relate these algebras.
Proposition 4.1.5.The algebra D t,k,ν is given by the restricted ultraproduct of the spherical subalgebras ś r F B tn,kn pν n q with respect to the filtrations mentioned in the discussion after Lemma 4.1.2.Proof.Indeed, by the definition of the DDC-algebra we have: where the restricted ultraproduct is taken with respect to the filtrations on H tn,kn pν n qe.
Hence we can conclude that: B tn,kn pν n q, as required.
Remark 4.1.6.These results suggest that the family of algebras B tn,kn pν n q should fall into the class covered by Example 2.5.21.This is indeed the case and will be proved in Appendix A. This shows that we could have constructed D t,k,ν via the restricted ultraproduct without using Deligne categories.However, the construction via Deligne categories is more conceptual and has a number of advantages.For example, it allows one to easily define a large family of representations of D t,k,ν .Indeed, if M is an H t,k pνq-module (see [EA14] for a description of some of them), then Hom ReppSν q pC, Mq has a natural structure of a D t,k,ν -module.Admittedly, these modules are also constructible as ultraproducts (as, by definition, is everything obtained from Deligne categories), but their direct construction via Deligne categories is more transparent.
We will also need the same algebra defined in Rep ext pH t,k pνqq, and will denote it also by r D t,k,ν (note that in this case ν is not a number, but a variable).Clearly, the analog of Proposition 4.1.5also holds for this algebra.

A basis of the deformed double current algebra of type A
In this section we will construct a basis of D t,k,ν .Note that in this section t, k are arbitrary elements of k and n is any integer.
In order to do this we will start by working with the spherical subalgebras in finite rank.One question which is worthwhile to ask is: can we introduce a basis of filtered components of these algebras which stabilizes for large n? Indeed, this should be possible since their restricted ultraproduct lies in INDpReppS ν qq.
We will construct such a basis in the following way.
Definition 4.1.7.Define elements T r,q,n P B t,k pnq (over k) for r, q P Z ě0 , r `q " L using the formula ÿ r,qě0, r`q"L T r,q,n u r v q r!q! :" where u, v are formal variables.
These elements are well defined if charpkq " 0 or ą L.
Next we need to define certain combinations of these elements.
Definition 4.1.8.Denote by m a collection of non-negative integers m r,q for all r, q P Z such that r `q ą 0, all but finitely many of them zero.Denote |m| :" ř r,qě0,r`qą0 m r,q and wpmq :" ř r,qě0,r`qą0 pr `qqm r,q .Define elements T n pmq P B t,k pnq, with |m| " m, by the formula ÿ m:|m|"m T n pmq ź r,qě0 z mr,q r,q m r,q !" ´řr,qě0,r`qą0 z r,q T r,q,n ¯m m! .
Here z r,q are once again formal variables and if we work in positive characteristic, we assume that wpmq ă charpkq.
We clarify these definitions by writing these elements more explicitly.Define apr, q, jq for 1 ď j ď r `q to be apr, q, jq " x for 1 ď j ď r and apr, q, jq " y for r `1 ď j ď r `q.Then T r,q,n " 1 pr `qq! n ÿ i"1 ÿ σPS r`q ˜r`q ź j"1 apr, q, σpjqq i ¸e, where the product in ś r`q j"1 is taken from left to right (i.e., apr, q, σp1qq i apr, q, σp2qq i . . .).In other words, this element consists of sums of all possible shuffles of r copies of x i and q copies of y i .Similarly, T n pmq is proportional to the sum of all possible shuffles of m r,q copies of T r,q,n .
Let us see what happens with these elements under the "leading term" map: gr L : F L B t,k pnq Ñ F L B t,k pnq{F L´1 B t,k pnq » krx 1 , . . ., x n , y 1 , . . ., y n s Sn L .We calculate: gr r`q pT r,q,n q " Definition 4.1.9.Denote by P r,q,n the symmetric polynomial ř n i"1 x r i y q i .So we can further conclude that: gr wpmq pT n pmqq " ź r,qě0,r`qą0 P mr,q r,q,n .
From this we can conclude the following.
Lemma 4.1.10.For L ď n and charpkq " 0 or large compared to n, the vector space F L B t,k pnq{F L´1 B t,k pnq has a basis tT n pmq|wpmq " Lu.
Proof.Indeed, from invariant theory we know that krx 1 , . . ., x n , y 1 , . . ., y n s Sn L " pkrP r,q,n s r,qě0,0ăr`q q L .So a basis in krx 1 , . . ., x n , y 1 , . . ., y n s Sn L is given by products ś r,q,0ăr`q P mr,q r,q,n for m r,q such that ř r,qě0,r`qą0 m r,q pr `qq " L. But this is exactly the basis in question up to multiplication by non-zero constants.Now it follows that T n pmq form a basis of the corresponding filtered component.So we have a corollary.
Corollary 4.1.11.For L ď n and charpkq " 0 or large compared to n the vector space F L B t,k pnq has a basis given by tT n pmq|wpmq ď Lu.This tells us that F L B t,k pnq indeed stabilizes as n Ñ 8. Now we would like to construct similar elements in D t,k,ν .Notice that we can think about T n pmq as a map from k to H t,k pnqe.The image of this map lies within the filtered component of degree wpmq.Thus the ultraproduct ś F T νn pmq gives us a well-defined map from C to H t,k pνqe.
Remark 4.1.12.From this point on t, c, ν P C are the same elements as in the rest of the paper.Definition 4.1.13.By T pmq denote the element of D t,k,ν given by ś F T νn pmq.Remark 4.1.14.We can also write down these maps explicitly as follows.
First we send C to h b h ˚b C via the co-evaluation map.After that using maps pw q : h Ñ h bq and pw r : h ˚Ñ h ˚br (ultraproducts of the standard maps x i Ñ x i b¨¨¨bx i ), we send the target object of the previous map to h bq b h ˚br b C. Then we send this object to the Perm r,q phq b C, where Perm r,q phq is given by the direct sum of all possible permutations of tensor products of q copies h and r copies of h ˚.At last, we act on this object via the map, which we denote appl, sending any object Y 1 b ¨¨¨b Y r`q b C (where Y i " h or h ˚) to H t,k pνqe using the maps x and y applied starting from right.To sum up, we have: Here perm r,q is the average of all the permutaions.It's easy to see that this is the same as the ultraproduct of T r,q,νn .One can then obtain the maps T pmq by multiplication of these maps.
Using the last result we can conclude that the maps T pmq are a basis of D t,k,ν .
Proposition 4.1.15.The elements T pmq for all choices of m constitute a basis of D t,k,ν .
Proof.Indeed, from Proposition 4.1.5we know that F L D t,k,ν " ś F F L B tn,kn pν n q.But for almost all n (i.e., ν n ą L) by Corollary 4.1.11we know that the basis of F L B tn,kn pν n q is given by T νn pmq with wpmq ď L. Since D t,k,ν " ř Lě0 F L D t,k,ν it follows that T pmq constitute the basis of the whole algebra.
In a similar fashion we have a parallel proposition: Proposition 4.1.16.The elements T pmq for all choices of m constitute a basis of the Cpνq-algebra r D t,k,ν .

Deformed double current algebra of type A with central parameter
For our convenience we would like to make ν into a central element and consider our DDC-algebra over C. In order to do this, we will need the following result: Lemma 4.1.17.The structure constants of the basis T pmq P r D t,k,ν depend polynomially on ν.
Proof.This follows from the fact that the only way ν can appear in the product of two basis elements, is if in the corresponding finite rank basis vectors we encounter an empty sum ř νn i"1 , each of which contributes a multiple of ν n .For details see Appendix A.
From this we can conclude that a Crνs-lattice À m CrνsT pmq Ă r D t,k,ν inherits the structure of algebra from r D t,k,ν .Now we can define the following algebra:

The Lie algebra po
To give a presentation of D ext 1,k by generators and relations, we will have to start with the Lie algebra po of polynomials on the symplectic plane.Later it will turn out that the DDC-algebra is a flat filtered deformation of Uppoq.Definition 4.2.1.By po denote the Lie algebra over k which is krp, qs as a vector space, with the bracket defined by: rq k p l , q m p n s " plm ´nkqq k`m´1 p l`n´1 .
We will denote the element 1 P krp, qs by K.
In other words, this Lie algebra is given by the standard Poisson bracket on krp, qs determined by tp, qu " 1.
This algebra admits the following grading: Definition 4.2.2.Endow the Lie algebra po with a grading given by degpq k p l q " k `l ´2.
In this grading the bracket has degree 0.
Note that p´q 2 2 , pq, p 2 2 q constitutes an sl 2 -triple.Hence we conclude that po 0 » sl 2 .This endows po with a structure of an sl 2 -module.It is easy to see that po i is isomorphic to the simple highest weight module V i`2 of highest weight i `2.Definition 4.2.3.Denote by n the Lie subalgebra of po given by n " À ią0 po i .As an sl 2 -module we have:

A presentation of po by generators and relations.
To find a presentation of po by generators and relations, it is enough to find the corresponding presentation of n.The rest will follow easily.This was done in [VdHP91] using a computer calculation of the cohomology spaces of n to obtain a minimal set of generators and relations.We will reproduce this result below.We will also present a direct proof of this result in Appendix B.
First, it's easy to find the generators: Definition 4.2.4.The Lie algbera n is generated by n 1 .
Proof.Indeed, this easily follows by induction from the formulas p k q l " r p k`1 q l´2 k`1 , q 3 3 s for l ě 2, p k q " r p k k , pq 2 2 s and p k " r p k´1 k´1 , p 2 qs.So it follows that the algebra n is a quotient of the free Lie algebra Lpn 1 q, where n 1 » V 3 .The Lie algebra Lpn 1 q has a grading determined by degpn 1 q " 1.
To describe the relations in a language of sl 2 -modules we will first have to introduce a few definitions.Definition 4.2.5.Fix an isomorpism of n 1 with V 3 with the highest weight vector specified as c 1 " q 3 6 .Consider Λ 2 n 1 " Lpn 1 q 2 .As sl 2 -modules we have Λ 2 n 1 » V 4 ' V 0 .Denote the submodule of Λ 2 n 1 isomorophic to V 0 by φ 1 and the submodule isomorphic to V 4 by φ 2 .Fix an isomorphism of φ 1 with V 0 with the highest weight vector specified as c 1 ^c4 ´c2 ^c3 , where c i " f i´1 c 1 .Fix an isomorphism of φ 2 with V 4 with the highest weight vector specified as d 1 " c 2 ^c1 .
Consider φ 2 bn 1 Ă Lpn 1 q 3 .We have Denote the submodule isomorphic to V 1 by ψ 1 , the submodule isomorphic to V 3 by ψ 2 , the submodule isomorphic to V 5 by ψ 3 and submodule isomorphic to V 7 by ψ 4 .Fix an isomorphism of ψ 1 with V 1 with the highest weight vector specified as ´4d Consider ^2φ 2 Ă Lpn 1 q 4 .We have ^2φ 2 " V 6 ' V 2 .Denote the submodule isomorphic to V 2 by χ 1 .Fix an isomorphism of χ 1 with V 2 with the highest weight vector specified as 3d 3 ^d2 ´2d 4 ^d1 .
We have the following proposition.Proposition 4.2.6.The Lie algebra n is isomorphic to the quotient of the free Lie algebra Lpn 1 q by the ideal generated by the sl 2 -modules φ 1 , ψ 4 , ψ 1 and χ 1 .This is a minimal set of relations.
Proof.As stated in the beginning of this section, one can find a proof of this result by a computer computation in [VdHP91].See Appendix B for a more direct proof.Now we can move to the description of the whole algebra.First let us introduce the notation for the remaining part of po: Definition 4.2.7.Denote by b the Lie subalgebra of po given by po ´2 ' po ´1 ' po 0 .We have po " b ' n.
We will also need a little more notation: Definition 4.2.8.Fix an isomorphism of b 0 with sl 2 given by e Þ Ñ b 1 " ´q2 2 and f Þ Ñ b 3 " p 2 2 .Fix an isomorphism of b ´1 with V 1 with the highest weight vector specified as a 1 " q.Fix an isomorphism of b ´2 with V 0 with the highest weight vector specified as K.
Consider the free Lie algebra Lpb ' n 1 q.Consider Λ 2 b ´1 Ă Lpb ' n 1 q 2 , we have Λ 2 b ´1 » V 0 .Fix an isomorphism of Λ 2 b ´1 with V 0 with the highest weight vector specified as a 1 ^a2 . Consider Denote the submodule isomorphic to V 2 by α 1 and the submodule isomorphic to V 4 by α 2 .Fix an isomorphism of α 1 with V 2 with the highest weight vector specified as c 2 b a 1 ´2c 1 b a 2 .Proposition 4.2.9.The Lie algebra po is generated by b ' n 1 with the following set of relations: where we use the isomorphisms from Definitions 4.2.5 and 4.2.8.And by λX » µY for two sl 2 -submodules of Lpb ' n 1 q with two fixed isomorphisms with V j and two numbers λ, µ we mean that we take the quotient by the image of the map Proof.This easily follows from Proposition 4.2.6.Indeed, the first line of relations ensures that the subalgebra generated by b is indeed b, the third line ensures that the subalgebra generated by n 1 is isomorphic to n.The second line fixes the adjoint action of b on n 1 making sure that nothing more is generated.
One can also give a more explicit presentation, without using the language of sl 2modules.
Proof.In order to get this presentation from the one given in Proposition 4.2.9, to start with, we need to throw out some of the generators.Indeed, in the formulation we threw out the generator corresponding to h in the sl 2 -triple of b 0 and we have only taken one generator from the whole of n 1 -the highest-weight vector r.This is obviously enough, since we can generate the whole of sl 2 using e and f , and then generate the rest of n 1 by the action of b 0 on r.Now, it's easy to see that the first line of the relations in Proposition 4.2.9 transforms into the first two lines of relations (4.2.10) and the second line of the relations in Proposition 4.2.9 transforms into the third line of the relations (4.2.10).We only need to keep the highest-weight vectors of the third line of the relations in Proposition 4.2.9, since the rest of the relations can be generated by the action of b 0 .These four highest-weight vectors are given in the last lines of relations (4.2.10) in the same order as the corresponding sl 2 -modules in Proposition 4.2.9.
For the details of these calculations see Appendix B.

Flat filtered deformations of U ppoq
In the beginning of Section 4.2.1 we've mentioned that D ext 1,k is going to be isomorphic to a flat filtered deformation of Uppoq.For this reason in this section we will formulate a result on flat filtered deformations of Uppoq obtained via computer calculations and then present a known flat filtered deformation of Uppoq.
Using computer calculation one can arrive at the following proposition about the deformations of Uppoq.Again, before we can formulate the relations in terms of sl 2modules we need to introduce some notations: Definition 4.2.12.Consider a free associative algebra T pb ' n 1 q.Denote the subspace S 2 b ´1 Ă T pb ' n 1 q 2 isomorphic to V 2 as sl 2 -module by β 1 .Fix an isomorphism of S 2 b ´1 with V 2 with the highest weight vector specified by a 2 1 .Also for any sl 2 -submodule γ Ă T pb ' n 1 q, denote by K i γ the submodule γ b b bi ´2.If γ had a fixed isomorphism with V j with the highest weight vector specified by v γ , fix an isomorphism of γ b b bi ´2 with V j with the highest weight vector specified by v γ b K bi .
We are ready to state the main result of the section.
Proposition 4.2.13.Suppose U is a flat filtered deformation of Uppoq as an associative algebra (up to an automorphism), such that Upbq is still a subalgebra of U, and the action of Upbq on b ' n 1 is not deformed.Then U is isomorphic to A s 1 ,s 2 defined below for some values of s 1 and s 2 .The algebra A s 1 ,s 2 is generated by b' n 1 with the set of relations given by the first two lines of Proposition 4.2.9 and the following relations, which substitute the last line in Proposition 4.2.9: where s 1 , s 2 P CrKs, "»" means the same thing as in Proposition 4.2.9, and all the submodules of Lpb ' n ´1q are interpreted as submodules of T pb ' n ´1q via the map Lpb ' n ´1q Ñ T pb ' n ´1q which sends the elements of the free Lie algebra into the corresponding commutators in the free associative algebra.
Proof.First of all note that our requirement on the type of deformation effectively means that we consider such deformations of relations in Proposition 4.2.9 which change only the last four relations, augmenting them by some lower order terms.The outline of the computer calculation used is as follows.
Given a family of putative flat filtered deformations of a finitely graded algebra, the subscheme over which it is flat is cut out by the condition that for any linear combination of the deformed relations, the leading degree term is in the undeformed ideal.Just as in the commutative setting, there is a notion of Gröbner bases for noncommutative algebras, and one could in principle check flatness by computing the Gröbner bases of both the original and the deformed ideal and verifying that the leading terms agree.Unfortunately (since basic questions about noncommutative algebras are undecidable), the Gröbner basis is in general infinite, so the algorithm that produces such a basis will not terminate.However, we can still produce a subset of the equations satisfied on the flat locus via this approach, by simply stopping the calculation at some arbitrary point.In the case of interest, we do this by computing all S-polynomials of pairs of the deformed relations (noting that in the noncommutative case two relations may have more than one S-polynomial) and reducing them modulo the deformed relations.This gives us out a new collection of relations, and any such relation that vanishes in Uppoq must vanish on the flat deformation, so gives an equation for each of its coefficients.After using these equations to eliminate parameters, we find that some of the relations become independent of the parameters, and thus we may reduce mod those relations.The resulting set contains 12 relations of degree 15 that span a 10-dimensional space of relations on Uppoq, and thus gives two new relations vanishing on Uppoq, allowing us to eliminate all but two parameters as required.
Remark 4.2.14.Note that we can specialize the central element K to a number, which will give a 3-parameter flat family of algebras A s 1 ,s 2 ,K , with s 1 , s 2 , K P C.These parameters have degrees 4, 6, ´2, respectively; alternatively, we may view this deformation as one with four deformation parameters s 1 , s 2 , s 1 1 " s 1 K, s 1 2 " s 2 K of degrees 4, 6, 2, 4, respectively, which are constrained by the relation s 1 s 1 2 " s 2 s 1 1 ; i.e., deformations are parametrized by a quadratic cone in C 4 .Also, we see that up to rescaling there are only two essential parameters, s 1 " s 1 K 2 and s 2 " s 2 K 3 .
As before, this presentation can be formulated more explicitly as follows: Proposition 4.2.15.The algebra A s 1 ,s 2 is generated by the same generators as po and the same set of relations as in Proposition 4.2.10, with the last four relations deformed as follows: 4rad 3 f prq, ad 2 r pf qs ´3rad 2 f prq, ad f ad 2 r pf qs `2rad f prq, ad 2 f ad 2 r pf qs ´rr, ad 3 f ad 2 r pf qs " 15s 1 q, 3rad 2 f ad 2 r pf q, ad f ad 2 r pf qs ´2rad 3 f ad 2 r pf q, ad 2 r pf qs " 3pp30s 1 `14s 2 Kqe `7s 2 q 2 q, where s 1 , s 2 P CrKs.
Proof.This is easy to see following the proof of Proposition 4.2.10.Remark 4.2.16.We can also rewrite the above relations (Proposition 4.2.15) using the set of generators of Remark 4.2.11.Indeed, the algebra A s 1 ,s 2 is generated by the same set of generators as po in Remark 4.2.11(i.e., p, f, r) and the same set of relations as in Remark 4.2.11, with the last four (degrees 2, 3, 4) deformed as follows: ad 3 r pf q " 0, 4rad 3 f prq, ad 2 r pf qs ´3rad 2 f prq, ad f ad 2 r pf qs `2rad f prq, ad 2 f ad 2 r pf qs ´rr, ad 3 f ad 2 r pf qs " 15s 1 ad 2 p prq, 3rad 2 f ad 2 r pf q, ad f ad 2 r pf qs ´2rad 3 f ad 2 r pf q, ad 2 r pf qs " 3p7s 2 ad 2 p prq 2 ´p30s 1 `14s 2 Kqad p prqq, where K " ad 3 p prq and s 1 , s 2 P CrKs.Below we will show that the universal enveloping algebra of the Lie algebra Crx, Bs gives us an example of such a deformation.This result is well-known, see [FF80].
Definition 4.2.17.Denote by Crx, Bs the Lie algebra of polynomial differential operators, with a Lie bracket given by the commutator.
Consider a grading on Crx, Bs given by degpx k B l q " k `l ´2.We have a decomposition Crx, Bs " À i"´2 Crx, Bs i .It's easy to see that with this grading the Lie bracket decreases filtration degree at least by 2 and preserves degree modulo 2: r, s : Crx, Bs i b Crx, Bs j Ñ Crx, Bs i`j ' Crx, Bs i`j´2 ' . . . .Indeed, when we compute the commutator we use the identity rB, xs " 1 at least once, and each time it decreases the grading by 2. Lemma 4.2.18.The associated graded Lie algebra of Crx, Bs is isomorphic to po.
Proof.Writing down the commutator of basis elements, we have: So by taking the associated graded of Crx, Bs and denoting the image of x by q and the image of B by p, we end up with po.

And we have the following corollary:
Corollary 4.2.19.Crx, Bs is a non-trivial flat filtered deformation of po as a Lie algebra.
Proof.The flatness follows from Lemma 4.2.18 and the fact that the graded dimensions of the two Lie algebras are the same.
The fact that this deformation is non-trivial (which is not hard to check directly) is known as the van Hove-Groenewold's theorem in quantum mechanics, which says that classical infinitesimal symmetries deform nontrivially under quantization.See Theorem 13.13 in [Hal13].Now from Proposition 4.2.13 it follows that UpCrx, Bsq must be isomorphic to A s 1 ,s 2 for some choice of s 1 and s 2 .Let us now compute these parameters.
Proof.From Proposition 4.2.13 we know that UpCrx, Bsq » A s 1 ,s 2 .Since this deformation actually comes from the Lie algebra deformation, we can conclude that s 2 must be equal to zero.Now we can consider the Lie algebra a s 1 given by the generators and relations of Proposition 4.2.15 with s 2 " 0. So we know that Crx, Bs » a s 1 .Let's denote this isomorphism by ε : a s 1 Ñ Crx, Bs.Since ε is determined up to a constant, we can set the image of K under ε to be εpKq " 1.Now since a s 1 is a deformation of grpCrx, Bsq, we know that εpqq " x `. . ., εppq " B `. . ., εpeq " ´x2 2 `. . ., εpf q " B 2 2 `. . .and εprq " x 3 6 `. . ., where ". . ." stand for the lower order terms.Also note that since the commutator is deformed in degrees starting with ´2, it follows that the lower order terms also can appear only starting with degrees ´2.Hence εpqq " x and εppq " B. Suppose εpeq " ´x2 2 `c1 and εpf q " B 2 2 `c2 , it follows that rεpeq, εpf qs " xB `1 2 .Now by calculating rrεpeq, εpf qs, εpeqs " rxB, ´x2 2 s " ´x2 , we conclude that c 1 must be equal to 0. The same holds true for c 2 .Now suppose εprq " x 3 6 `d1 x `d2 B. Now rεpqq, εprqs " ´d2 , hence d 2 " 0. And rεppq, εprqs " x 2 2 `d1 , hence d 1 " 0. So we know the images of the commutators.Now it's enough to calculate one of the relations.
Remark 4.2.22.Of course we could have proved that Crx, Bs is isomorphic to a 1 without using computer computation and Proposition 4.2.13.Indeed, one just needs to check that 1, x, B, ´x2 2 , B 2 2 and x 3 6 satisfy the required relations, which is easy to do.

The deformed double current algebra of type A as a flat filtered deformation of U ppoq
Here we would like to show that the generic choice of parameters s 1 and s 2 can give us the algebra D ext 1,k .Below we will need to compute things in D ext 1,k .Since this algebra is defined as a certain lattice in the ultraproduct, we need to understand how we can do this.The following definition provides us with a method.Definition 4.2.23.Suppose Y P D ext 1,k is given by Y " f ptT pmquqK i , where f is a noncommutative polynomial with coefficients in C. By construction we know that Y | K"ν " ś F f n ptT νn pmquqν i n , where f n are non-commutative polynomials with coefficients in Q, such that ś F f n " f .As a shorthand notation we will write Y ∽ f n ptT νn pmquν i n q, where we will consider the r.h.s. for large enough n.
With this tool we are ready to continue: Proposition 4.2.24.The algebra D ext 1,k is a flat filtered deformation of Uppoq.Proof.Indeed, we know that the basis in this algebra is given by T pmqK i .Also recall the natural filtration we considered in the previous section (so that T pmqK i P pD ext 1,k q wpmq ).Since by Lemma 4.1.10we know that grB 1,kn pν n q " QrP r,q,νn s r,qě0,0ăr`q in sufficiently low degrees, where the associated graded is taken with respect to the filtration discussed after Lemma 4.1.2,it follows that grD ext 1,k " p ź F QrP r,q,νn s r,qě0,0ăr`qďνn q| ν"K " CrP r,q s r,qě0 , where P r,q " gr r`q pT r,q q and P 0,0 " gr 0 pKq.Now the bracket r, s acts as follows: r, s : pD ext 1,k q n b pD ext 1,k q m Ñ pD ext 1,k q m`n´2 ' pD ext 1,k q m`n´4 ' . . ., where we consider the grading of the algebra as a vector space.Indeed, this follows from the fact that rT pmq, T pnqs ∽ rT νn pmq, T νn pnqs, and to calculate the latter expression we need to use the commutator rx i , y j s at least once, which, each time we use it, lowers the degree by 2. We would like to calculate the leading term of the commutator.To calculate gr wpmq`wpnq´2 prT pmq, T pnqsq it is enough to compute it via ∽, commuting freely elements within T νn pmq and leaving only the highest term in the commutator of rx i , y j s " δ ij `. . . .So: gr wpmq`wpnq´2 prT pmq, T pnqsq ∽ « ź r,qě0,r`qą0 P mr,q r,q,νn , ź r,qě0,r`qą0 P nr,q r,q,νn ff " " ź r,qě0,r`qą0 P mr,q`nr,q r,q,νn ÿ r 1 ,r 2 ,q 1 ,q 2 m r 1 ,q 1 n r 2 ,q 2 P r 1 ,q 1 ,νn P r 2 ,q 2 ,νn rP r 1 ,q 1 ,νn , P r 2 ,q 2 ,νn s.

But now:
rP r 1 ,q 1 ,νn , P r 2 ,q 2 ,νn s " νn ÿ i,j"1 rx r 1 i y q 1 i , x r 2 j y q 2 j s " " pq 1 r 2 ´q2 r 1 qP r 1 `r2 ´1,q 1 `q2 ´1,νn , where we use P 0,0,νn to denote ν n .These formulas show us that grD ext 1,k is isomorphic to a deformation of Uppoq after identification of T i,j with p i q j .So it follows that D ext 1,k is a deformation of Uppoq.Moreover it is a flat filtered deformation, by virtue of the fact that T pmqK i constitute a basis of D ext 1,k .
Since we know all possible flat filtered deformations of Uppoq, it follows that D ext 1,k is isomorphic to A s 1 ,s 2 for some choice of constants.We would also like to calculate the exact correspondence.Proposition 4.2.25.The DDC-algebra D ext 1,k is isomorphic to A s 1 ,s 2 with s 1 " 1 `kpk `1qp1 ´Kq and s 2 " kpk `1q.
Proof.We know that D ext 1,k » A s 1 ,s 2 for some s 1 , s 2 P CrKs.Denote this isomorphism by It is enough to calculate s 1 , s 2 via evaluating one of the commutators.We will largely follow the steps of the proof of Proposition 4.2.20.
Oftentimes it will be easier for us to work with f n ptT νn pmquq as π ´1 n pf n ptπ n pT νn pmqquqq.In this case we will use another shorthand notation X ∽ β f n ptπ n pT νn pmqquqν i n , where we will consider the r.h.s for a large enough n.
Thus we conclude γ 1 " 0. A similar calculation results in γ 2 " 0. Now we can write βprq " T 3,0 6 `δ1 T 1,0 `δ2 T 0,1 for δ i P CrKs (we only need to add elements of the lower degrees which have the same parity).Let's calculate rβprq, βpqqs and rβprq, βppqs.To do this, we need to calculate rT 3,0 , T 1,0 s and rT 0,3 , T 0,1 s.The first one is obviously zero.So we have rβprq, βpa 1 qs " δ 2 rT 0,1 , T 1,0 s " δ 2 K.But this commutator should be zero.Hence δ 2 " 0. Now for the other one: rT 0,1,n , T 3,0,n s " ÿ i,j rB i , x 3 j s " 3T 2,0,n , Thus rβprq, βppqs " r T 3,0 6 `δ1 T 1,0 , T 0,1 s " T 2,0 2 ´δ1 K " ´βpeq ´δ1 K. Hence δ 1 " 0. Thus we have successfully calculated the images of all the generators.Now we need to calculate the image of 3rad 2 f ad 2 r pf q, ad f ad 2 r pf qs ´2rad 3 f ad 2 r pf q, ad 2 r pf qs.Indeed, this is the only relation where both s 1 and s 2 are present.We calculate: Similarly we can compute the results of the action of powers of ad f .The differential operator part is quite straightforward, but we will write down the part depending on c in more detail.Denoting X " ad 2 r pf q and κ " kpk `1q, we have: Now, transforming the last sum, we have: So in total we have: The next one is The second commutator in this formula amounts to: and the third one: So the original expression amounts to: So, now we can finally compute the image of the relation: The part coming from the first two commutators is just the r.h.s. of the relation when k " 0. It is equal to: as we would expect since in this case s 1 " 1, s 2 " 0. The third commutator gives: and the forth one: If we put together the formulas for the third and forth commutators in the original expression, we get: ∽ β 3κp´16b 1 pK ´1q `7pa 2 1 `2b 1 qq.Thus we see that: 3rad 2 f pXq, ad f pXqs ´2rad 3 f pXq, Xs ∽ β 3p2p15 ´κp8K ´15qqb 1 `7κa 2 1 q " " 3pp30p1 `κp1 ´Kqq `14κKqb 1 `7κa 2 1 q.And we can conclude that s 1 " 1 `kpk `1qp1 ´Kq and s 2 " kpk `1q.
Remark 4.2.27.Note that instead of using the computer calculation from Proposition 4.2.13,we could have defined the map on generators by the same formula as β and checked that it satisfies the remaining relations.This is easy to do, in fact the relation we have checked is the most complicated one.
Remark 4.2.28.One can think about the isomorphism of Proposition 4.2.25 in the following way.For the Lie algebra Crx, Bs there exists a standard map: UpCrx, Bsq Ñ S n Crx, Bs " DiffpC n q Sn .One can deform this map to arrive at the map: with s 1 " 1 `kpk `1qp1 ´nq and s 2 " kpk `1q.These maps are given by the formulas in the polynomial representation of the Cherednik algebra, which we used in the proof of Proposition 4.2.25.The isomorphism β can be thought of as a certain ultraproduct of these maps.
Remark 4.2.30.Note that via Proposition 4.1.20we can also easily obtain the presentation by generators and relations of DDC-algebras D 1,k,ν .

The Galois symmetry
Recall that the algebra D 1,k,ν is the quotient of D ext 1,k .Indeed, by Proposition 4.1.20we have D 1,k,ν " D ext 1,k,ν {pK ´νq.I.e., it is an algebra where the central parameter K became a scalar.Now we can see that the equations for s 1 and s 2 in Proposition 4.2.25 can be written in terms of the essential parameters s 1 " s 1 ν 2 and s 2 " s 2 ν 3 as follows: It is easy to check that these equations are invariant under the symmetry This implies Proposition 4.3.1.We have an isomorphism of filtered algebras . This proposition also follows from the results of [CEE09], Sections 8,9, see also [EGL15], Section 6, which establish similar symmetries for spherical Cherednik algebras of finite rank.There is also an obvious symmetry g 2 pk, νq " p´k ´1, νq.It is easy to see that g 2 1 " g 2 2 " 1, pg 1 g 2 q 3 " 1, so g 1 , g 2 generate a copy of the group S 3 .In fact, this S 3symmetry comes from permuting the parameters q 1 , q 2 , q 3 in the toroidal quantum group ([Mik07]), which can be degenerated to D 1,k,ν .
Moreover, this group is also the Galois group of the system of equations (4).Namely, we have s 1 `s2 " p1 `k `k2 qν 2 , s 2 " kpk `1qν 3 , so p1 `k `k2 q 3 " uk 2 pk `1q 2 , where u :" . Dividing this by k 3 , we get where ζ :" k `1 k `1.The group S 3 just mentioned is the Galois group of this cubic equation over Cpuq.Namely, Cpζq is a non-normal cubic extension of Cpuq, and Cpkq is the corresponding splitting field (a quadratic extension of Cpζq).
5 Deformed double current algebras for arbitrary Γ

The case of general Γ
In this section we will repeat the construction of Section 4.1 for the DDCA corresponding to arbitrary Γ.Here again for brevity we consider only the case of transcendental ν.Since the construction is literally the same upon changing ReppS ν q to ReppS ν ˙Γν q, we will go over it rather quickly.
First we start with a definition.

The deformed double current algebra of type B
In this section we would like to sketch some results on the presentation of the DDCA in type B by generators and relations akin to the discussion for type A in Section 4.2.Most of the results of this section were obtained through a computer computation.
First of all note that we can obtain the DDCA of type B by taking Γ " Z{2.We saw that D ext t,k was a deformation of Uppoq.It turns out that a similar statement holds for type B. Definition 5.2.2.By po `denote the Lie subaglebra of po given by the linear combinations of even degree monomials.I.e., po `" po Z{2 , where Z{2 acts on po by p Þ Ñ ´p and q Þ Ñ ´q.This Lie algebra has an even grading restricted from the grading of po, and this grading is also a grading by sl 2 -modules under the adjoint action of po 0 .
It's now easy to see, by similar arguments, that whereas the ultraproduct of type A algebras eH t,k pnqe which are isomorphic to Qrx 1 , . . ., x n , y 1 , . . ., y n s Sn as vector spaces is a deformation of Uppoq, the ultraproduct of type B algebras eH t,k,c pnqe which are isomorphic as vector spaces to Qrx 1 , . . ., x n , y 1 , . . ., y n s Sn˙pZ{2Zq n is a deformation of Uppo `q.
Now one can also provide a presentation of po `similar to Proposition 4.2.9.To state such a result we need to give a few definitions.
Definition 5.2.3.Denote by b the Lie subalgebra of po `given by po `2 ' po 0 .The Lie subalgebra n is given by po 2 ' po 4 ' . . . .So po `" b ' n. 12We will need a little more notation: Remark 5.2.9.Notice that in the same way as Crx, Bs is a flat filtered deformation of po, Feigin's Lie algebra glpλq :" Upsl 2 q{pC " λ 2 ´1 2 q (where C :" ef `f e `h2 2 is the Casimir) introduced in [Fei88] is a flat filtered deformation of po `.More precisely we have Upglpλqq » D 1,0,λ´1 2 ,ν (for any ν, as this algebra does not depend of ν); indeed, it is easy to see looking at the relations that the deformation D 1,0,λ´1 2 ,ν arises from the most general deformation of po `as a (filtered) Lie algebra.These relations are given in [GL96], at the beginning of Table 3.1.For more information about deformations of po `and glpλq see [PvdH96].
Note that the parameters s 1 , s 2 , s 3 in Proposition 5.2.8 are not independent: we have s 1 ´s2 " 5ps 3 `1q.
It is easy to see that u, v are invariant under the symmetry13 h 1 pk, λq :" ˆ1 k , λ k ˙.
Thus, we obtain the following proposition.
Remark 5.2.13.We see that when we interpolate the spherical Cherednik algebras eH 1,k,c pS n qe of type B into the DDCA D 1,k,c,ν , we lose one parameter (unlike the case of type A).Let us explain why such a loss of parameter is inevitable and to be expected a priori.
To this end, note that for generic k, c the algebra eH 1,k,c pnqe is simple and therefore has no nonzero finite dimensional representations.
On the other hand, let po `:" po `{C.We claim that any filtered deformation of Sppo `q necessarily has a 1-dimensional representation.
Indeed, let A be such a deformation.Let us show that the augmentation homomorphism ε : Sppo 0 q Ñ C lifts to a 1-dimensional representation of A. By definition, A has generators a " pa ij q with i `j ą 0 even (i, j ě 0) of filtration degree i `j (namely, lifts of p i q j ) and has defining relations ra ij , a kl s " P ijkl paq, where P ijkl is a noncommutative polynomial of degree ď i `j `k `l ´2 whose part of degree exactly i `j `k `l ´2 is pjk ´ilqa i`k´1,j`l´1 .In particular, setting ´a20 {2 " e, a 11 " h, a 02 {2 " f, we get rh, es " 2e `c1 , rh, f s " ´2f `c2 , re, f s " h `c3 , for some constants c i P C.These constants give a 2-cocycle on sl 2 , which must be a coboundary since H 2 psl 2 q " 0, so by shifting e, f, h by constants we can make sure that c i " 0. Thus A contains sl 2 and we can write all the relations sl 2 -equivariantly.Thus we can assume that a ij with i `j " 2m span the representation V 2m .So ra ij , a kl s belongs to the representation V 2m b V 2n if i `j " 2m, k `l " 2n, m ‰ n, and to Λ 2 V 2m if m " n.These representations don't contain C, so the polynomial P ijkl can be chosen without constant term.Thus we have a 1-dimensional A-module in which all a ij act by 0, as claimed.Note that this argument (and the statement itself) fails for po (type A), since Λ 2 V r contains C for odd r.
This loss of parameter is similar to the one for Deligne categories: the interpolation of the category RepGLpm|nq is the Deligne category RepGL ν with ν " m ´n.So the interpolation procedure forgets m, n and remembers only the difference m ´n.
A Appendix: On structure constants of the deformed double current algebra of type A As was promised in the proof of Lemma 4.1.17,in this Appendix we will prove that the structure constants of B t,k pnq depend polynomially on n, making this sequence of algebras fit Example 2.5.21.So, what we want to show is that T n pm 1 q ¨Tn pm 2 q can be written as a linear combination of T n pmq with coefficients which depend polynomially on n.This proof is due to Travis Schedler.
To start with, we will need the following definition.
It is easy to see that T r,q,n " r!q! pr `qq! ÿ a:rr`qsÑtx,yu |a ´1pxq|"r n ÿ i"1 ap1q i . . .apr `qq i .
So T r,q,n is a sum of admissible sums with l " r `q and m " 1 with coefficients which do not depend on n.
It is also easy to see that the product of admissible sums is an admissible sum (we just need to combine two pairs of functions into a single pair by concatenation).So, since T n pmq is given by the sum of products of T r,q,n with coefficients which do not depend on n, it follows that T n pmq is a sum of admissible sums with coefficients which do not depend on n.
We are now ready to prove the following proposition.
Proposition A.0.2.The product of T n pm 1 q and T n pm 2 q can be written as a sum of T n pmq with coefficients which depend polynomially on n.
Proof.By the preceding discussion T n pm 1 qT n pm 2 q is a sum of admissible sums with coefficients independent of n.So it is enough to prove that any admissible sum can be written as a sum of T n pmq with coefficients which depend on n polynomially.We will prove this by induction on l (the cardinality of the source of a in the definition of admissible sum, or the degree of admissible sum).Assume that the statement holds for admissible sums with l ă M. Suppose we are given an admissible sum A with l " M defined by functions a : rMs Ñ tx, yu and u : rMs Ñ rks.If u is not surjective, it follows that some of the summations are redundant and just give a coefficient in the form of the power of n (this is where polynomial dependence on n actually comes from).So we can reduce to the case of u being surjective.For j P rks define r j " |u ´1pjq X a ´1pxq| and q j " |u ´1pjq X a ´1pyq|, i.e., this is the number of x's and y's in the summation corresponding to i j .Define m in the following way: we set m r,q :" |tj P rks| r j " r, q j " qu|.We want to prove that there is a number α which does not depend on n such that A ´αT pmq is given by the sum of admissible sums with n-independent coefficients all of which have degree l ă M. Indeed, by the choice of m it follows that the highest orders of A and T n pmq are proportional up to some factor coming from the factorials in the definition of T n pmq, so choose α P k such that gr wpmq pAq ´α ¨gr wpmq pT n pmqq " 0.
To calculate the actual difference one would need to permute x's and y's in A to bring it to the form of T n pmq.Obviously when we permute x's and y's, an admissible sum stays admissible.So we only need to see what happens with the parts arising due to commutators.
If we commute x i 1 with y i 2 , we reduce the number of generators by 2 (so the resulting degree is less than M) and insert t ´k ř m‰i 1 s m,iq in case i 1 " i 2 or ks i 1 i 2 in case of i 1 ‰ i 2 .In either case we can commute group elements to the right and absorb them into e.What we have afterwards is a sum of sums which only differ from admissible sums by the fact that they sometimes have the condition i 1 ‰ i 2 .But since ř i 1 ‰i 2 " ř i 1 ,i 2 ´ři 1 "i 2 this reduces to the sum of admissible sums, and we are done.
has the same dimension as n 3 and π 1 is an isomorphism.Note that this also shows that the minimal set of relations must contain ψ 1 , ψ 4 .
To finish we need to consider l 4 .As before, we have a surjective map The general formulas from the induction step allow us to conclude that ξ 4 pV 8 ' V 4 q " 0. Now we need to deal with V 2 .However, as we can see from the general formulas, α 1 defined in Equation 6 becomes linearly dependent with α 2 and α 3 in degree 4. Indeed, it turns out that V 2 does not belong to the ideal generated by ψ 1 , ψ 4 and φ 0 .
We see that all we can generate by φ 0 in degree 4 is given by φ 0 b Λ 2 n 1 » V 0 ' V 4 , so it does not contain anything isomorphic to V 2 .All we can generate by ψ 4 is ψ 4 b n 1 » V 10 ' V 8 ' V 6 ' V 4 , so it does not contain anything isomorphic to V 2 .So the only chance to kill V 2 is ψ 1 b n 1 " V 4 ' V 2 .But using our calculation (and similar ones) it follows that this doesn't kill V 2 in l 3 b n 1 .
But the relation χ 1 takes care of it.So it follows both that l 4 is isomorphic to n 4 under π 1 and that the minimal set of relations must contain χ 1 .

C References
˝p1 b xq ´x ˝p1 b xq ˝pσ b 1q " 0, as a map from h ˚b h ˚b M to M; y ˝p1 b yq ´y ˝p1 b yq ˝pσ b 1q " 0, as a map from h b h b M to M; y ˝p1 b xq ´x ˝p1 b yq ˝pσ b 1q " t ¨ev h b 1 ´k ¨pev h b 1q ˝pΩ 3 ´Ω1,3 q, as a map from h b h ˚b M to M, where Ω is a central element from Definition 3.2.7,and indices indicate the spaces on which Ω acts in the tensor product h b h ˚b M.
kn pν n qe.Now we are ready to define the DDC-algebra in question.Definition 4.1.4.The algebra D t,k,ν is the endomorphism algebra End ReppH t,k pνqq pH t,k pνqeq.
Definition 4.1.18.By D ext t,k denote the algebra À m CrνsT pmq, regarded as an algebra over C. In this context we will denote the central element ν by K. We can write down a basis of D ext t,k : Proposition 4.1.19.The elements T pmqK j for all tuples m and j ě 0 constitute a C-basis of D ext t,k .Proof.This is evident from the definition.Note that trivially we also have the following result: Proposition 4.1.20.For ν P CzZ we have, D ext t,k {pK ´νq " D t,k,ν .And for ν P Cpνq we have pD ext t,k b C Cpνqq{pK ´νq " r D t,k,ν .
Definition 5.2.1.Denote D t,k,c,ν :" D t,k,c,ν pZ{2q.Here c is just a single number, since Z{2 has a single non-trivial conjugacy class.Define r D t,k,c,ν and D ext t,k,c in the same way.
Definition 2.2.4.By ReppS n ˙Γn ; kq denote the category of representations of the wreath product S n ˙Γn over k.By Rep f pS n ˙Γn ; kq denote the full subcategory of finite dimensional representations.
It's easy to see that this representation is in fact H t,k pnqe.Now the spherical subalgebra is given as follows: B t,k pnq " eH t,k pnqe " Hom Sn pk, H t,k pnqeq " End H t,k pnq pInd Definition 2.3.5.By ReppH t,k pnq; kq denote the category of (possibly infinite dimensional) representations of the rational Cherednik algebra H t,k pnq " H t,k pn, kq.Also set Rep p pH t,k pnqq " ReppH t,k pnq, F p q.
Via the isomorphism constructed in the previous paragraph this is an element of C. Notice that this element cannot satisfy any nontrivial polynomial equation over Q (indeed, the corresponding polynomial must have infinitely many roots), hence ś F n is a transcendental element of C. By an automorphism of C we can send this element into any transcendental element of C. Let us show that there exists a sequence of integers ν n and prime numbers p n such that ν n ă p n and ś F ν n " ν inside ś F F pn » C; this will be needed in what follows.
an object of ReppH t,k,c pν, Γqq.Also repeating the steps of Section 3.3.1 we can construct the induction functor.Since the construction is almost literally the same, we just state the result.Construction via the Deligne category ReppS ν q4.1.1TheconstructionInthis section we will construct the DDC-algebra of type A we are after, which we will call D t,k,ν .We will do this by taking endomorphisms of a certain object of the Deligne category.First define the following object: Definition 4.1.1.Define an object H t,k pνqe P ReppH t,k pνqq to be equal to Ind The object H t,k pνqe is isomorphic to ś C,r F H tn,kn pν n qe.Proof.Indeed, we know that H tn,kn pν n qe " Ind H tn,kn pνnq Sν n Definition 5.1.1.The object H t,k,c pν, Γqe P ReppH t,k,c pν, Γqq is defined to be equal to IndH t,k,c pν,Γq Sν ˙Γν pkq.It follows that H t,k,c pν, Γqe " ś C,rF H tn,kn,cn pn, Γqe.Note that assigning degpV q " 1 gives us the filtration on H t,k,c pν, Γqe in the same fashion as in the discussion after Lemma 4.1.2.The same filtration works in finite rank.Now we can define the DDCA itself:Definition 5.1.2.The DDC algebra D t,k,c,ν pΓq is given by: D t,k,c,ν pΓq :" End ReppH t,k,c pν,Γqq pH t,k,c pν, Γqeq " Hom ReppSν ˙Γν q pC, H t,k,c pν, Γqeq.The algebra D t,k,c,ν pΓq can be constructed as the restricted ultraproduct of spherical subalgebras ś r F B tn,kn,cn pn, Γq with respect to the filtrations mentioned after Definition 5.1.1.Remark 5.1.4.We can also do the same thing in the Deligne categories over Cpνq and obtain the algebra r D t,k,c,ν pΓq over Cpνq.Remark 5.1.5.The analogs of the results of Section 4.1.3still hold and we can also construct the algebra D ext t,k,c pΓq over C, where ν becomes a central element.Remark 5.1.6.Note that we obtain the case of type A if we set Γ " 1, the trivial group.i.e., we have D t,k,H,ν p1q " D t,k,ν .