A $q$-analogue of derivations on the tensor algebra and the $q$-Schur-Weyl duality

This paper presents a $q$-analogue of an extension of the tensor algebra given by the same author. This new algebra naturally contains the ordinary tensor algebra and the Iwahori-Hecke algebra type $A$ of infinite degree. Namely this algebra can be regarded as a natural mix of these two algebras. Moreover, we can consider natural"derivations"on this algebra. Using these derivations, we can easily prove the $q$-Schur-Weyl duality (the duality between the quantum enveloping algebra of the general linear Lie algebra and the Iwahori-Hecke algebra of type $A$).


Introduction
This paper presents a q-analogue of an extension of the tensor algebra given in [4]. Using this algebra, we can easily prove the q-Schur-Weyl duality (the duality between the quantum enveloping algebra U q (gl n ) and the Iwahori-Hecke algebra of type A).
First, let us recall the algebraT (V ) given in [4]. This algebraT (V ) naturally contains the ordinary tensor algebra T (V ) and the infinite symmetric group S ∞ . Moreover, we can consider natural "derivations" on this algebra, which satisfy an analogue of canonical commutation relations. This algebra and these derivations are useful to study representations on the tensor algebra. For example, we can prove the Schur-Weyl duality easily using this framework.
In this paper, we give a q-analogue of this algebraT (V ). This new algebraT (V ) naturally contains the ordinary tensor algebra T (V ) and the Iwahori-Hecke algebra H ∞ (q) of type A ∞ . Namely, we can regard thisT (V ) as a natural mix of T (V ) and H ∞ (q). We can also consider natural "derivations" on the algebraT (V ).
This research was partially supported by JSPS Grant-in-Aid for Young Scientists (B) 24740021.
These derivations are useful to describe the natural action of U q (gl n ) on V ⊗ p . Moreover, using these derivations, we can easily prove the q-Schur-Weyl duality.
Some applications ofT (V ) were given in [4]: (1) invariant theory in the tensor algebra (for example, a proof of the first fundamental theorem of invariant theory with respect to the natural action of the special linear group), and (2) application to immanants and the quantum immanants (a linear basis of the center of the universal enveloping algebra U (gl n ); see [6,7]). The author hopes that the alge-braT (V ) will be useful to study representation theory and invariant theory related to U q (gl n ).
The author would like to thank the referee for the valuable comments.

Definition ofT (V )
Let us start with the definition of the algebraT (V ) determined by a vector space V = C n . We recall that the ordinary tensor algebra is defined by Noting this, we defineT (V ) as a vector space bŷ whereT p (V ) is the following induced representation: Here, the notation is as follows. First, H p (q) is the Iwahori-Hecke algebra of type A p−1 . Namely, this is the C-algebra defined by the following generators and relations: We define H ∞ (q) as the inductive limit of the natural inclusions H 0 (q) ⊂ H 1 (q) ⊂ · · · . Next, H p (q) naturally acts on T p (V ) = V ⊗ p as follows [5]: Here, we define t ∈ End(V ⊗ V ) by where e 1 , . . . , e n mean the standard basis of V . Note that we omit the symbol "⊗." Thus, we have explained the definition ofT (V ) as a vector space. Moreover, we consider a natural algebra structure ofT (V ). Namely, for σ v 1 · · · v k ∈T k (V ) and τ w 1 · · · w l ∈T l (V ), we define their product by Here, σ and τ are elements of H ∞ (q), and v 1 , . . . , v k , w 1 , . . . , w l are vectors in V . Moreover, α is the algebra endomorphism on H ∞ (q) defined by This multiplication is well defined. With this multiplication,T (V ) becomes an associative graded algebra.
Remark. In [4], the definition ofT (V ) was based on the left action of S p on V ⊗ p . However, in this paper, we definedT (V ) using the right action of H p (q) on V ⊗ p . Actually, we can also define a similar algebra using the left action, but we employ our definition because this is compatible with the action of U q (gl(V )) (see Section 5).

The Multiplication by v ∈ V and the Derivation by v * ∈ V *
In this section, we define two series of fundamental operators onT (V ), namely the multiplications by vectors in V and the derivations by covectors in V * .
First, let R(ϕ) denote the right multiplication by ϕ ∈T (V ): This operator is obviously fundamental, and the following two cases are particularly fundamental: (1) the case that ϕ is a vector in V , and (2) the case that ϕ is an element of H ∞ (q). Indeed, the other cases can be generated by these two cases. Note that R(v) for v ∈ V ⊂T 1 (V ) raises the degree by one, and R(σ ) for Here, k i is the linear transformation on V , and g i is the linear map from V to H 2 (q) ⊗ V defined as follows: Based on this, we define R(v * ) in such a way that R : Let us check the well-definedness of the definition (3.1) of R(e * i ). For this, we consider a linear map f r : For the well-definedness of (3.1), it suffices to show that t 1 , . . . , t p−1 commute with p r =1 f r . Namely, we only have to show the following lemma: Proof. We put e J = e j 1 · · · e j p for J = ( j 1 , . . . , j p ). Let us fix I = (i 1 , . . . , i p ) and 1 ≤ s ≤ p, and put (1), it suffices to show t γ s f r (e I ) = q δ i s i s+1 f r (e I ) for r = s, s + 1. When r > s + 1, this can be deduced from the relation t We can show the the case r < s by a direct calculation.
We can also show (2) by a direct calculation.
Remark. This well-definedness means that R(v * ) commutes with the action of H ∞ (q).

Commutation Relations
For the multiplications and derivations introduced in the previous section, we have the following commutation relations.
where K i is the linear transformation onT (V ) defined by Namely, we can exchange two multiplications by vectors putting t 1 or t −1 1 ∈ H 2 (q) on the right of these two operators. Similarly, we can exchange two derivations putting t 1 or t −1 1 on the left of two operators. The most interesting one is the commutation relation between a derivation and a multiplication. This time, t 1 or t −1 1 appears in the middle of these two operators. We can regard these relations as an analogue of the canonical commutation relations.
Proof of Theorem 4.1. These relations can be checked by direct calculations except for the commutation relations between two derivations. Thus, we here prove the sixth relation, from which the fifth relation is immediate. We can prove the fourth relation similarly (actually more easily).
To show the sixth relation, it suffices to prove  These are equal, because we have t (a+b−r ) r by a calculation.
It is natural to consider the operator algebra generated by R(v), R(v * ) and R(σ ) with v ∈ V , v * ∈ V * and σ ∈ H ∞ (q). We can regard this operator algebra as an analogue of the Weyl algebras and the Clifford algebras.
The following commutation relations with K i are also fundamental:

The Natural Representation of U q (gl(V )) on V ⊗ p
We can use the operators introduced in Section 3 to study the natural representation of the quantum enveloping algebra U q (gl(V )) on V ⊗ p . First, let us recall the definition of U q (gl(V )). For V = C n , we define the Calgebra U q (gl(V )) by the following generators and relations [5]: generators: q ±ε 1 /2 , . . . , q ±ε n /2 ,ê 1 , . . . ,ê n−1 ,f 1 , . . . ,f n−1 , Here, we denote q a 1 · · · q a k simply by q a 1 +···+a k .
Next, we defineÊ i j andÊ ji ∈ U q (gl(V )) for 1 ≤ i < j ≤ n bŷ We call thisÊ i j (a) for 1 ≤ i, j ≤ n the L-operator.
We denote by π the natural representation of the quantum enveloping algebra U q (gl(V )) on V ⊗ p . This is determined by the following actions of generators [5]: Here, k 1/2 i and E i j are the linear transformations on V defined by We can use our operators to express this representation π : Proof. We can check the assertion by a direct calculation when i = j. Let us show the case i = j. We note that for i < j. Thus, it suffices to show for i < j. We can check these relations for j = i + 1 by a direct calculation. To show the other cases, we put Indeed, using Theorems 4.1 and 4.2, we see the first relation as follows: We can show the second relation similarly. Combining these, we have (5.1).
Remark. Theorem 5.1 is quite similar to the natural action of the Lie algebra gl(V ) on P(V ) the space of all polynomial functions on V . This action μ can be expressed as Here, x i means the canonical coordinate of V , and E i j means the standard basis of gl(V ). Using Theorems 4.1 and 4.2, we have the following relations: Moreover, we have the following proposition. Indeed, using Proposition 5.2, we can rewrite R(v k ) · · · R(v 1 )R(v * 1 ) · · · R(v * k ) as a sum of products of R(v)R(v * ) and K i . gl(V ))).

q-Schur-Weyl Duality
We can use our results to prove the following Jimbo duality, namely the qanalogue of the Schur-Weyl duality. This theorem was first given in [5], and several proofs have been given (see [3,8] for example).  (q)) and π(U q (gl(V ))) are mutual commutants of each other. Namely, we have Here [k] = [k] q is a q-integer, and [k]! = [k] q ! is a q-factorial: To prove this theorem, we consider the following analogue of the Euler operator: Here, we put For this E, the following relation holds: Proof. We put and moreover Thus, by Proposition 5.3, we see that f ∈ π(U q (gl(V ))).
Remark. For any group G, every map f : G → G commuting with all right translations is equal to a left translation. This fact is proved quickly as follows. Let e be the identity element of G. For any element x of G, we have f (x) = f (ex) = f (e)x, because f commutes with the right multiplication by x. Thus f is equal to the left multiplication by f (e), as we claimed. It should be noted that our proof of Theorem 6.1 is based on the same principle (the operator E plays a role of the identity element e). Theorem 6.1 holds, if and only if q satisfy [ p]! = 0 (this condition is also equivalent with the condition that H p (q) is semisimple). Indeed, when [ p]! = 0, this proof fails because there exists I such that [I ]! = 0. It is interesting that the condition [ p]! = 0 appears this way.
I hope that the algebraT (V ) and the differential operators onT (V ) will be useful to study invariant theory in quantum enveloping algebras.
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