RICERCA ITALIANA     

Research program

Polynomial identities and combinatorial methods

University Co-ordinator

Università degli Studi di BARI - MATEMATICA - ()

Research Unit Leader

Onofrio Mario Di Vincenzo

Description

The research project is facing topics from the theory of associative algebras and their polynomial identities under the action of groups of (anti)-automorphisms and/or in the presence of (generalized) derivations.

A meaningful part of the research project concerns one among the main topics of the theory of algebras with polynomial identities, namely the description of the ideals of the free algebra F< X > corresponding to the polynomial identities of an assigned F-algebra. Describing such ideals is generally an extremely difficult problem. In the ordinary case they are the so-called T-ideals of F< X >, that is those ideals which are invariant under the action of any endomorphism of the free algebra.
In 1950 Specht conjectured that, if the base field is of zero characteristic, every proper T-ideal is finitely generated as T-ideal. The complete proof of this result lies on a description of the structure of the varieties of associative algebras and involves the study of the Z_2-graded polynomial identities of associative superalgebras [36],[37]. The Grassmann algebra E of and infinite-dimensional vector space is among them. Thoroughly the mentioned description the all important algebras are the matrix algebras M_n(F), M_n(E) and M_{k,t}(E). In fact, in characteristic zero, any non trivial verbally prime variety is generated by one of them. Notice that each of them is either simple and finite dimensional either isomorphic to the Grassmann envelope of a finite-dimensional simple superalgebra. More generally in zero characteristic any non trivial variety is generated by the Grassmann envelope of a finite dimensional superalgebra [37].
Within these settings, remarkably meaningful is the study of the graded polynomial identities satisfied by a generic superalgebra. More generally, polynomial identities of associative algebras graded by an arbitrary group can be considered. Notice that in case of finite abelian groups any G-grading is equivalent to the action of a group of automorphisms isomorphic to G.

The project's main goal and the proper tasks of this research unit are precisely to describe the polynomial identities of peculiar and meaningful classes of associative algebras under the action of automorphisms or anti-automorphisms groups and semigroups. Also into account will be taken cases in which algebraic structures are enriched by the presence of derivations. When (generalized) derivations do satisfy suitable and natural conditions, the algebra structure will be investigated together with its polynomial identities.

A precise goal of the research program is to study the free algebra ideals arising from the Z_2-graded polynomial identities of a superalgebra or from the *-polynomial identities of an algebra with involution. As mentioned above, these settings correspond to the case in which a group of order 2 acts as a group of automorphisms or anti-automorphisms on the algebra respectively. In both cases it is possible to focus on noteworthy subspaces, namely the components of the graded decomposition for superalgebras or those of symmetric and skew-symmetric elements for algebras with involution. More precisely, we want to determine the sets of polynomials generating these ideals as bilateral ideals or as ideals which are invariant under suitable semigroups of endomorphisms of the free associative algebra.

In order to give an efficient description of these structures the research unit is going to employ the combinatorial methods properly pertaining to the group representation theory. In fact in zero characteristic, the polynomial identities (in the non ordinary cases as well) are determined by the multi-linear ones, and the permutation groups naturally act on these spaces. Such actions are equivalent to the action of the linear groups on the spaces of multi-homogeneous polynomials. The research unit goal is to get information about the invariants characterizing these actions: the codimension sequence and its exponential growth, the cocharacter sequence and the multiplicities of their irreducible components, the Hilbert series for the relatively free algebra of the variety.

The research planning develops through the following steps: at first by further analyzing the class of algebras with involution brought up in [18](O.M. Di Vincenzo, R. La Scala, "Minimal algebras with respect to their *-exponent" Journal of Algebra, 317,2007). These are subalgebras of block triangular matrix algebras with the involution consisting on the flip about the secondary diagonal. In [18] it has been conjectured that, in zero characteristic, those algebras with involution generate exactly the varieties which are minimal with respect to the *-exponent values. We intend to enlarge the set of cases in which this conjecture has been positively verified. The next step is to investigate the structure of the ideal of *-polynomial identities of the verbally prime superalgebras M_{p,q}(E). Here M_{p,q}(E) is endowed with the natural involution induced by a pair of superinvolutions, defined on the matrix superalgebra M_{p+q}(F) with entries in the field F and on the Grassmann algebra E, respectively. In particular, we want to get the generators of the ideal of the *-polynomial identities and to describe the associated cocharacter sequence, at least in some meaningful cases. It is remarkable that a complete classification of the involutions which can be defined on these algebras is missing so far.
In a second phase the ideal structure of the graded identities of the tensor products of verbally prime superalgebras will be investigated. In particular, one among the unit's tasks is to positively solve a conjecture made by Regev and Seeman (A.Regev, T. Seeman, "Z_2-graded tensor products of p.i. algebras" Journal of Algebra 291, 2005, no. 1, 274--296) by means of the above description.

Beside to the mentioned tasks, we intend to proceed deeper into the study of functional identities (FI) holding for associative algebras.
An identity can be described in terms of ring elements together with arbitrary functions. One of the goals in the theory of FI is to determine the features of the functions involved into the formal expression of the functional identity or, in case this is too difficult, to determine the structure of the algebra admitting the assigned FI.
In order to have an adequate setting to these topics and to gain an exhaustive bibliography we address to the paper of M. Bresar "Functional identities: a survey" (Contemp. Math. 259, 93-109, 2000), to the recent book by M. Bresar, M.A. Chebotar and W.S. Martindale III, "Functional identities, Frontiers in Mathematics" (Birkhauser, 2007) or the book "Rings with generalized identities" by K.I.Beidar, W.S.Martindale III, A.V.Mikhalev (Dekker, 1996). Here we just recall that the study of FI in associative rings is strictly linked to Lie map problems, and formerly Herstein, in 1961, planned a study program about this topic, in case of prime rings. Peculiar instances of FI are those in which:
1) the involved functions are polynomial functions in non commutative indeterminates. In this case any functional identity is indeed a generalized polynomial identity. It should be noticed, however, that PI-theory and FI-theory are not that close as it could seem, and actually they are in some sense complementary: in fact the study of PI concerns rings which are "low dimension" algebras, while the study of FI-theory provides results for algebras with sufficiently large, when not infinity, dimension.
2) the involved functions are (generalized) derivations. In this case, a functional identity is a (generalized) differential identity.

We recall that an additive map F is said to be commuting on a subset A of a ring R if F(x)x-xF(X)=0 for all x in A. Analogously, it is said centralizing in A if F(x)x-xF(x) belongs to the center of R for all x in A. A classic result by E.C. Posner (Derivations in prime rings, Proc. Amer. Math. Soc. 8, 1093-1100, 1957), states that a prime ring with a centralizing derivation must be commutative. In 1993 Bresar proved that any commuting additive map F in a prime ring is of kind F(x)=ax+b for some elements a,b belonging to the centroid of the ring (Centralizing mappings and derivations in prime rings, J. Algebra 156, 385-394, 1993).

This result has been extended towards several directions. In particular we want to focus to the case when A is the subset of evaluations of an assigned polynomial f(x_1,…,x_n) in non commutative indeterminates. If R is an algebra with a derivation F and the functional polynomial [F(x),x]_k (for x in A) is central, then R satisfies some concrete polynomial identity ( T.K.Lee , Derivations with Engel conditions on polynomials, Algebra Coll. 5-1,13-24, 1998).

Later on M.A. Chebotar and P.H. Lee (On certain subgroups of prime rings with derivations, Comm. Algebra 29-7, 3083-3087, 2001) and T.L. Wong (On certain subgroups of semiprime rings with derivations, Comm. Algebra 32-5, 1961-1968, 2004) investigate the case when k=2. They prove that under some additional assumptions the additive subgroup generated by the evaluation set S of the polynomial [F(x),x]_2 (for x in R) contains a Lie ideal of R.

Our contribution is devoted to study the general case when the subset A follows from the evaluations of the polynomial f(x_1,...,x_n). We studied the annihilator and the centralizer of S, showing that both are trivial (V.De Filippis, O.M. Di Vincenzo, Posner's second theorem and an annihilator condition. Math. Pannon. 12, 2001, 69--81).
Moreover if a derivation vanishes on S then it must be the zero map on R (V. De Filippis, O.M. Di Vincenzo, Posner's second theorem, multilinear polynomials and vanishing derivations, Journal of Australian Math. Soc. 76, 2004, 357-368). A second problem is to extend the above mentioned result by T.K. Lee (1998) to the case of a generalized derivation F. We proved that, similarly, the evaluation set of the functional polynomial
[F(x),x]_k is central in R just in case R is a PI-algebra or F(x)=ax for a belonging to the centroid of R (V. De Filippis, Generalized derivations with Engel conditions on polynomials, Israel Journal of Math., in press).

As a first step, we will pay attention to the study of functional identities involving automorphisms and generalized derivations of the ring, aiming at a complete description for the latter. Subsequently we intend to analyze the structure of the subrings and subgroups generated by suitable functional polynomials which are not identities of the ring. In particular, we want to discriminate in which cases such subrings (subgroups, respectively) contain ideals (or Lie ideals, respectively) of the assigned ring.

The research unit intends to keep the scientific collaborative relations with the various research groups, both national and international, operating within these mathematical topics. We intend further, in a whole with the other units composing the national project, to work towards the organization of a workshop illustrating the gained results.