ABSTRACT
Graded polynomial identities are one of the principal tools in PI theory. On the other hand, gradings on algebras and their graded identities are interesting and important on their own. This importance was revealed by the celebrated work of Kemer (see for example [20]) on the structure theory of T-ideals that led him to the positive solution of the famous Specht problem. Graded identities provide quite a lot of information about the ordinary identities satisfied by a given algebra. It is well known that if A is
a G-graded algebra where G is a finite group then A is PI if and only if its neutral component in the grading is PI, see [4, 7]. Furthermore one may use graded identities in order to describe the concrete ordinary identities satisfied by a given algebra. In Kemer’s theory graded identities were essential in obtaining the PI equivalence of classes of verbally prime PI algebras, the so called Tensor product theorem. Later on Regev (see [26]) gave another proof of this theorem, and his proof was based once again on graded identities. Several cases of the tensor product theorem were handled in the papers [9, 13, 14, 22, 2, 3], once more using graded identities. Given an algebra (or a class of algebras) it is of importance to describe
all possible gradings on it by a fixed group. In [32] all Z2-graded simple algebras were described. In recent years the gradings on matrix algebras by finite groups were described, see for example the bibliography of [6]. Most of these result have also been transferred to the case of simple Jordan and Lie algebras, see for example the references of [29]. The algebras UTn(K) of the upper triangular matrices of order n over
a field K are of particular interest for PI theory. It is well known that the behavior of their identities determines the subvarieties of the variety of algebras generated by the matrix algebra of order two, when the base field is of characteristic 0. The identities of UTn(K) are known for every field K, see for example [31]. The corresponding T-ideal is finitely generated as a T-ideal for every field K. In particular, if K is infinite then the T-ideal of UTn(K) is generated by the polynomial [x1, x2][x3, x4] . . . [x2n−1, x2n] where [a, b] = ab − ba is the usual commutator of a and b. All Z2-gradings on UT2(K) when charK = 0 were described in [28]. Furthermore in the latter paper a variety of numerical characteristics of these graded identities were described as well. In [29] the gradings on UTn(K) by finite abelian groups were given provided thatK is algebraically closed of characteristic 0. It turns out that such gradings are isomorphic to elementary ones, that is, gradings where the matrix units eij are all homogeneous. Recall that according to [5], when G is abelian, every G-grading on Mn(K) is isomorphic to the tensor product of an elementary and a fine grading. The grading is fine if every homogeneous component is of dimension at most 1. It turns out that algebras of block-triangular matrices are quite important
in the description of extremal varieties of algebras with respect to their codimension growth, see [17, 18, 19]. Since upper triangular matrices are the simplest such algebras we believe that studying their gradings and graded identities will give new information about the behavior of such extremal algebras. When one studies Brauer groups one is led naturally to the concept of
involution ∗ on an algebra A, that is, an automorphism of order 2 of the additive group of A such that (ab)∗ = b∗a∗ for every a, b ∈ A. The involutions on central simple algebras are well understood, see for example
[21]. An involution is of the first kind if it is a linear automorphism, that is, if it preserves pointwise the centre of A, otherwise the involution is of the second kind. Concerning the algebras Mn(K), there exist two classes of nonequivalent involutions of the first kind on them. The first is generated by the usual matrix transpose t and the second is generated by the usual symplectic involution (the latter case is possible only when n is even). Identities with involution have long been studied in PI theory, see for
example [27] and its bibliography. Concerning matrix algebras little is known about the identities with involution satisfied by them The only situation where one knows the exact identities with involution for Mn(K) is n = 2, see [24, 25] for the cases of characteristic 0 and of finite field, and [8] for infinite fields of characteristic different from 2. When one studies involutions on UTn(K) a major problem is that this
algebra is rather far from being even semisimple, its radical is ample. Nevertheless one can say quite a lot about the involutions on UTn(K). Below we discuss briefly this point.
