Differential Polynomial Identities of Upper Triangular Matrices Under the Action of the Two-Dimensional Metabelian Lie Algebra

We study the differential polynomial identities of the algebra UTm(F) under the derivation action of the two dimensional metabelian Lie algebra, obtaining a generating set of the TL-ideal they constitute. Then we determine the Sn-structure of their proper multilinear spaces and, for the minimal cases m = 2, 3, their exact differential codimension sequence.

to discover, and each of these types of identities completely determines the ordinary ones, at least in principle.
It must be said that in present days an effort towards a unifying approach among different types of identities is being pursued. It took its steps from Berele's influential paper [4], and has its homeland in the settings of Hopf-algebra actions. For instance, in the specific case of differential identities of an algebra under the derivations of a Lie algebra , the Hopf algebra involved is the universal enveloping algebra of . Even when no straight Hopf-algebra action is possible, such as for instance for -identities, one can consider a generalized Hopf-algebra action to keep on track ( [4], Remark at page 878. See also [2], Section 7). Of course this does not mean that differences cease to exist: as always happens, general theories are built upon concrete situations, so differential identities, as well as other types of identities, still maintain their characteristic features when dealing with specific problems.
In the present paper we consider the smallest nonabelian Lie algebra, namely the metabelian two-dimensional Lie algebra , acting on the algebra by derivations. We give a faithful representation of as the Lie algebra generated by two suitable inner derivations of , which turns out to be very convenient to computational tasks. Our main result consists in providing a full list (Definition 3.4) of differential polynomials, generating all differential polynomial identities of with respect to this -action (Theorem 4.11). Then we pass to study the differential multilinear spaces related to through their -structure. The key to these results is afforded by the use of proper polynomials, a powerful tool re-discovered and developed mainly by Drensky, who pointed out their extreme usefulness in several papers. More precisely, we determine the proper differential cocharacter sequence of recursively (Proposition 5.1 and Theorem 5.8) and explicitly (Corollary 5.9).
The last section of our paper is devoted to work out the small cases 2 and 3. In the former we regain the results of [11] as byproduct of our general description. On the other hand, 2 is too small to depict the effective impact of the -action on the differential polynomial identities in general cases. All its characteristic features are instead already present when 3, so this case is worth studying to get a more concrete idea of the general one.

Algebras with Derivations and Differential Polynomial Identities
Throughout the paper let denote a field. With the term -algebra we mean an associative -algebra. If is an -algebra, let denote the full algebra of -linear trasformations from to itself. A derivation on is any satisfying 1 2 1 2 1 2 for all 1 2 . We will adopt the exponential notation for derivations throughout the paper. The set of all derivations on is a vector subspace of , turned into a Lie algebra with respect the usual Lie product between maps, and is denoted Der . Notice that, since we use a right action, linear maps are composed from left to right. If is a Lie algebra over , it acts on as an algebra of derivations if is a Lie module of . Actually, denoting L the universal enveloping algebra of , this amounts to turn into a right L-module. We briefly recall that L is uniquely determined, and the Poincarè-Birkhoff-Witt Theorem provides a natural embedding of in L and an explicit basis of L starting from any linearly ordered basis B of , namely the set B of all semistandard products 1 2 (that is: 1 For a fixed Lie algebra , it is possible to define a free object for the class of -algebras on which acts as an algebra of derivations ( -algebras, for short). Let us start with a countable set of free indeterminates , and consider the tensor algebra of the vector space L. It is a free algebra generated by the simple tensors for and B. We will denote by and call it a letter; moreover, we may identify the indeterminate with the letter 1 . We will denote by the set of free generators (letters) , and by the free algebra they generate. Since , the free algebra is a subalgebra of . Actually, is free on the set in the class of -algebras, in the sense that if is any -algebra, any map can be uniquely extended to analgebra homomorphism , that is an algebra homomorphism extending and commuting with the derivation action of (and L). Thus is called the free -algebra on . Each element of can be thought as a polynomial in the noncommutative letters . If 1 we will say is a differential letter, while is an ordinary letter. When 1 , we say that the is the name of the letter . A polynomial is called ordinary if just ordinary letters occur in , otherwise it is called differential.
Now let be an -algebra. The following notions have been outlined in [4] in the general settings of Hopf algebra actions, and employed in [11] in the more specific situation of a derivation action. Let be the set constituted by the differential polynomials laying in the kernel of all -homomorphisms . Then is an ideal of invariant under all the -endomorphisms of , called the -ideal of , and its elements are called the -polynomial identities of . Notice that the ordinary -ideal of , namely the set of all (ordinary) polynomials of vanishing under all algebra homomorphisms from to , is contained in . If G is any subset of , the least -ideal of containing G is called the -ideal generated by G . One of the main tasks in studying the differential polynomial identities of an algebra is to find a generating set for . In order to describe a smaller set of differential polynomials is needed when char.
0. In fact standard Vandermonde arguments and multilinearization process allow reducing the description to listing the multilinear differential polynomials contains. Recall that a polynomial is multilinear of degree if each name among 1 occurs exactly once in any monomial of , and no other name occurs. As in [11] we denote span 1 1 B , the space of multilinear differential polynomials of degree . Moreover, let 0 . Then is the least -ideal of containing the set . In other words, is generated, as -ideal, by the multilinear -polynomials it contains. Actually, each is more than a vector space: the natural left -action for all turns into an -module, having as a submodule. The factor module is therefore an -module as well, and its module structure can be described through its -character, called the -th -cocharacter of . Though it essentially describes the structure of the non identity multilinear polynomials, the intere -structure of can be recovered by complete reducibility. The associated number sequence dim , for , is called the -codimension sequence of .
When dealing with unitary algebras, a further strong reduction is actually available. For any 1 2 define the commutators of length 2 by 1 2 . The unitary subalgebra of generated by the commutators of any length will be denoted , and we will call its elements -proper polynomials. The relation between and can be clearly described through Lie algebras: let L be the free Lie algebra generated by and let L L L be its derived ideal. Then L is also a free Lie algebra, and L is spanned by modulo L . By Witt's Theorem, is the universal enveloping algebra of L , while is the universal enveloping algebra of L . Furthermore, let us fix a linear order in such that the ordinary letters precede the differential ones. Then the semistandard commutators 1 2 (that is such that ) form a basis for L (see [1], Corollary of Proposition 8, (ii), p. 55), and can be completed to a basis for L by adding the elements of . The linear ordering on can be extended to a total order on this basis such that elements of precede any commutator. Then the semistandard polynomials , where is a semistandard monomial on and is a semistandard sequence of the L -basis, constitute a basis for by the Poincarè-Birkhoff-Witt Theorem.
We actually need just polynomials which are proper with respect to ordinary letters, that is elements of the subalgebra of generated by commutators and differential letters. Thus we may consider the differential letters as commutators of length 1, similarly to what has been made in [6], and consider normal semistandard commutators (nssc's for short) of length 1: a commutator 1 2 is normal if at most a single differential letter occurs in it, and in this case it is 1 . It has been proved in [6], Proposition 7, that the nssc's constitute an -basis for . It can be proved that is generated by theproper polynomials it contains. Even simpler, we just need multilinear proper polynomials. In fact, let for all and let be the union of these sets. Then is generated by . Although proper polynomials could be dealt with in much greater generality (see [7], section 4.3 for the basic definitions and results on proper polynomials; polynomials proper with respect to a distinct set of letters were introduced in [8], Section 2), in this paper we shall deal with multilinear polynomials only. As a consequence, the nssc's we are going to deal with are actually normal standard commutators (nsc for short), that is nssc's such that 1 2 . Both in order to keep the paper as self-contained as possible and to give a proof within our restricted assumptions, we prove is generated as -ideal by .
Proof Let be the -ideal generated by . Then . Since generates , if then there must be polynomials such that but , and we may choose one of minimal degree . It has to be a linear combination of products where is a standard monomial on a subset of the ordinary letters 1 and is a product of commutators in the remaining letters, say 1 where none among the monomials involves the letter 1 . Now let us consider theendomorphism of sending 1 in 1 and fixing all other indeterminates. Then . By the way, just the polynomials may contribute to 1 2 : indeed 1 participates in each , either occurring in a commutator of length 2 or as a 1-commutator, that is a single derived letter. In both cases, evaluates to zero. Therefore is still multilinear in the remaining letters, belongs to and has degree 1. Renaming the letters, we get an element in 1 . By minimality, it must be in . Therefore , hence 1 , so mod .
Since we may repeat the process on the minimal letter occurring among the , after finite steps we get 0 mod , a contradiction.

A Spanning Set for L n (U m )
In this paper we are interested in the only nonabelian 2-dimensional Lie algebra over . It is also called the metabelian Lie algebra of dimension two, although this term may be confusing. Indeed while in most cases the word metabelian means 2-solvable (see [1], p. 27, and [7], Remark 2.1.17, (iii) p. 22, but also, p. 117 for metabelian groups), sometimes it is intended as 2-nilpotent (see [10], Definition 1.1). It is well known (see for instance [12], p. 11) that if is a two-dimentional noncommutative Lie algebra then there exists a basis of such that . Any linear map such that is a Lie-homomorphism, and induces a derivation action of on . Notice also that is injective as soon as 0. In this case we denote the algebra under the -action, to distinguish this structure from the natural algebra structure, which we will continue to denote by . By [5], all derivations of are inner, so can be defined as soon as a couple of elements such that is chosen, where is the inner derivation defined by . This amounts to turn into a right L-module. Our real concern is, in fact, in the L-action on , in order to determine the differential identities of . From now on, we denote by the inner derivations of induced by 1 and respectively, that is 1 and for all . Then and satisfy the relations 2 2 0 0 hence . Therefore and generate a Lie subalgebra of the (Lie) algebra ofendomorphisms of isomorphic to . While the relation suffices to establish a Lie isomorphism between and the Lie subalgebra of generated by the nonzero derivations and , the other relations, namely 2 , 2 0 and (which, together with , implies 0) affect the concrete action of L over : it turns out that the relations 2 2 generate the kernel K of this action. Therefore the L-module structure of is equivalent to the (right) module structure afforded by the factor algebra L K spanned by 1 and .
Example 3.1 Assign 23 and 33 . Again , defined by and , is an injective Lie homomorphism, since . By the way, while still holding 2 0, it is no longer true that 2 , 0 and hold. These facts alter the computational behaviour of the L-action on because the generators of the kernel change. The resulting L-module is equivalent to the previous one, that is the factor algebra is isomorphic to the previous one, but the computations are more complex.
As a consequence of our choice, we get the following differential identities of : 2 2 . Therefore the only nontrivial differential indeterminates involved in modulo are just and . It holds There are further basic -identities of , which can be easily checked: The last basic -identity of , involving a product of 1 commutators, is the following: Proof Denote for 1 1 and let be any substitution in such that 1 2 0. Then its value belongs to span 1 1 2 . Hence just the 1 -entry of 1 contributes to 1 1 . By the way it is the same entry as 1 , hence The polynomials listed so far constitute the candidate set of generators of . Let us fix the notation: Definition 3.4 Let I denote the set constituted by the following differential polynomials: where all indeterminates belong to . Moreover, let denote the -ideal generated by I .
The -ideal is contained in , and we are going to prove that in fact the reverse inclusion holds as well. As a first step, we are going to determine a set of polynomials spanning modulo . The second step will be to prove their linear independence modulo , in the next section. As in [6] let us denote by B the subset of the basis of constituted by the products of nssc's, and let B , for all 1. Since the identity 1 1 , any product of normal standard commutators is in . Hence any element of can be written, modulo , as a linear combinations of products of at most 1 nsc's. Next let us prove the following There is a partial analogous of Lemma 3.5 holding for -letters: Moreover, working modulo , let us denote by the vector subspace of spanned by S .
Gluing together the parts, let S 1 1 S 1 be the set of selected ordinary polynomials, S 1 S the one of polynomials involving an -letter and, finally, let S S 1 S S . Extending the notations to the vectors subspaces they span modulo , we will denote the vector subspace of spanned by S . We can now summarize all the previous results stating Theorem 3.10 S spans modulo .
Proof Any basis element 1 of is either zero or equivalent to a suitable linear combination of elements in S modulo , so S spans modulo , by the preceding results. Proof Since S spans modulo and , the same holds modulo .

Remark 3.12
The size of the sets of polynomials selected so far depend upon the size . For instance, passing from 1 to we must add further polynomials to the sets S 1 1 and S 1 to get S 1 and S , while on the contrary S 1 S 1 1 . The codimension sequences we are going to compute later in this paper will measure exactly how these sizes vary. There are however a couple of points to notice right now: the first one is that the set S 1 constitutes a basis for , that is for the ordinary algebra of upper triangular matrices without any derivation (see [9]

The Linear Indipendence of S n (U m ) Modulo T L (U m )
In this section we are going to prove that the spanning set S is indeed linearly independent modulo . To this aim, starting from a given linear combination of a subset of S assumed to be in , we will exhibit a suitable substitution (that is: anhomomorphisms from to defined on ) vanishing on all involved vectors but one, so that it is linearly independent with the remaining ones and can be deleted, shortening selectively the list until it is empty.
The easiest cancellation is the following: , where runs in S 1 S and β . Let be the substitution sending all the letters of in 11 . Then 0. By the way, 0 for all , since 11 ker and the ordinary commutators vanish under . Since 11 1 and 1 11 11 0 it follows β 0, so 1 1 is linearly independent modulo with the remaining vectors of S .
In other words, we just proved that 1 .  obtained taking the first letter of will be called the main word of . The letter will be called the main letter of . The other letters of will be called secondary letters of . The sequence 1 of the lengths of the commutators 1 will be called the structure of .

Lemma 4.2 The set S 1 1 is a linearly independent subset of S and
So, for instance, the vector 2 1 4 7 5 3 6 S 7 3 5 has 2 7 3 and structure 3 2 2 . Notice that S 7 3 4 . Notice also that for any S the structure belongs to . Now we are going to delete all polynomials of S for 2 from the list, one by one. To make the clearest possible computations, we need an abuse of notation. For any 2 let be the map defined by . Its image is the subalgebra of of all matrices whose rows 1, . . . , are zero. It is immediate to notice that is not only an algebra isomorphism beween and , but it commutes with the -action: for any one has and, similarly, 1 . In what follows, by -substitution sending we mean the substitution sending to , abusing the notation merely in order to simplify the computations. . We shall instead employ the 3 -substitution sending 1 2 22 , 3 12 , 4 23 and 5 33 , that is evaluate the polynomial inside 3 and just then going back inside 3 .
Notice also that the embeddings provide a canonical ascending chain 2 3 , and hence the descending chain 2 3 of -ideals of . and assume that 0. Just among the letters 1 are substituted by off-diagonal elements, and diagonal matrices commute. So, by the pigeonhole principle, each must involve exactly one among the main letters 1 of . Actually, since any non-identity permutation of the factors in 12 1 causes the product to vanish, the letter occurs in for all , and it must occur in its first or second position, otherwise 0. Now notice that if a secondary letter of occurs in for then 0: exactly one among the entries of is 1 , and if 1 1 occurs in it then 0 because 1 1 1 0. Therefore all the letters of are confined within , derived or not. Hence . Therefore if then . If instead then and must involve the same letters. Since however must be an -letter occurring in , it has to be placed in first position, so . If 1 the task is done, so assume 1 and let us compare 1 and 1 . As before, no secondary letter of 1 may appear in if 1, so 1 1 and, if the inequality is strict, this means . If instead 1 1 then 1 and 1 involve exactly the same letters, and 1 is placed in the first or second position of 1 . By the way, since the sets of letters occurring in 1 and 1 are the same, they have the same minimum, and this letter must be placed in position 2 both in 1 and in 1 . Hence 1 must be in the first position of 1 , hence 1 1 . These arguments apply to the remaining commutators of , so either or . In the latter case the previous arguments show that then for all , hence .
Then we get a simple algorithm to delete all the vectors in S : for increasing 's, take any element in S of maximum structure 1 and consider the associated -substitution . Then is the only element of S not vanishing under , hence it is linearly independent with all the vectors not yet cancelled. Then we may delete from the list and repeat the process until no vector of S we have to stop: this procedure does not work with elements of S 1 1 S 1 , essentially because 1 1 and the last main letter of a vector may be either ordinary or differential without affecting the evaluation. Therefore so far we obtained As a consequence, we get  The set is not just a vector subspace of : as already mentioned in Section 2, the natural action of the symmetric group defined by for all 1 , and 1 , turns into a left -module. In this section we are going to describe its -structure.
Recall that a complete set of representatives for the isomorphism classes of irreducible -modules is in a bijective correspondence with the set of partitions of . If is one of them, we shall abuse the notation and use the same symbol to mean a (fixed) irreducible -module in the isomorphism class indexed by . We shall denote 1 the partition with parts 1 2 .
At first, let us study the -proper cocharacters of 2 , that is the -structure of 2 . As already remarked it holds S 3 S 3 1 , and we will denote 3 the vector space spanned by this set. In the same fashion define 2 and 1 , hence 3 2 1 . While the whole vector space is an -module, just the last summand is an -submodule of it, and we are going to determine its structure first.
A finer partition of S 1 is indexed by the structures 1 , that is by the 1 -tuples 1 1 such that 1 1, 2 for other , and 1 1 . The explicit decomposition of is the following: The partitions involved in the first summation are (with 1) and 1 1 1 . By the Young-Pieri formula, the multiplicities for both are 1. The Young-Pieri formula works with the second summand, as well. The partitions involved in its decomposition are and (with 1), with multiplicities and 1 respectively. Hence the total multiplicities follow immediately.
Also the differential codimensions can be computed from the proper ones (see [11] We have so far considered a faithful derivation action of on 2 ; now we are going to consider the unfaithful ones.
The trivial action has of course little to offer: it amounts to consider the ordinary algebra 2 . Formally, its differential polynomial identities are generated by the differential polynomials and together with the ordinary polynomial 1 1 2 2 . Thus the differential cocharacters and codimensions of 2 differ just formally from the ordinary ones.
The remaining unfaithful actions are therefore those having the commutator Lie-ideal as kernel. So let 2 be the -algebra 1 and, for 2, 2 1 1 . endowed with a derivation action of the (commutative) Lie subalgebra of . Denoting by 2 the ideal of the differential identities of 2 under the -derivation action, the formal differences between 2 and 2 consist solely in the presence or not of the polynomial among the generators. In [11] the -action has been considered, so is missing.

The proper differential codimension sequence is
To complete the picture in the case 2, in the same spirit we may consider the action of the Lie subalgebra on 2 , and investigate the corresponding algebra 2 . We continue to denote 2 the ideal of differential identities of 2 , to avoid misunderstanding. The description of 2 is not given in [11], but can be easily recovered and very similar to the one of 2 .
Proposition 6.6 The differential polynomial identities 1 and, for 2, 2 1 1 . Then, with exactly the same reasoning as for , we get

The proper differential codimension sequence is
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