Symmetric and skew-symmetric polynomial identities with involution for the upper triangular matrix algebras of even order (Q2669184)

Let \(UT_m\) be the algebra of the upper triangular \(m\times m\) matrices over an infinite field of characteristic different from 2. This algebra and its polynomial identities are rather important in PI theory. It should be noted that the description of the ideal of identities of \(UT_m\) is a (relatively) standard task. As a general rule, introducing an additional structure on an algebra makes studying the corresponding identities ``easier''. For example group gradings on algebras and their graded identities are of importance due to two main reasons: first they give useful and powerful insights and information about the ordinary identities, and second they are as a rule easier to handle. The same applies to identities with trace. This is not the case though with identities with involution. While it is well known that the existence of an identity with involution on an algebra implies the existence of an ordinary identity the description of the identities with involution for a given algebra is extremely hard problem. The involutions of the first kind for \(UT_m\) were described by \textit{O. M. Di Vincenzo} et al. [Adv. Appl. Math. 37, No. 4, 541--568 (2006; Zbl 1116.16029)]; in that same paper the authors described the ideals of the identities with involution in the cases \(m=2\) and 3. Even for \(m=3\) the situation becomes rather complicated, and if \(m>3\) nothing is known about the identities with involution for \(UT_m\). The paper under review considers the case when \(m=2n\). Let \(d\) be an integer and denote by \(D_k^d\) the vector subspace of \(UT_k\) consisting of all matrices that have zeros at position \((i,j)\) whenever \(j-i<d\). Denote by \(H_d\) the subalgebra of \(UT_m\) consisting of the block matrices having two blocks on the diagonal, both coming from \(UT_n\), and the off-diagonal block coming from the vector subspace \(D_n^{-d}\). The identities with involution for these types of algebras allow the author to describe all identities with involution for \(UT_m\) of the form \(f\pm f^*\) where \(*\) is the involution, and \(f\) is a product of commutators. Moreover, the relatively free algebras determined by such polynomials are described.