Let G be an arbitrary group and let K be a field of characteristic different from 2. We classify the G-gradings on the Jordan algebra UJn of upper triangular matrices of order n over K. It turns out that there are, up to a graded isomorphism, two families of gradings: the elementary gradings (analogous to the ones in the associative case), and the so called mirror type (MT) gradings. Moreover we prove that the G-gradings on UJn are uniquely determined, up to a graded isomorphism, by the graded identities they satisfy.
Our main goal is to extend one of classical Razmyslov's Theorem saying that any two simple finite-dimensional-algebras over an algebraically closed field, satisfying the same polynomial identities, are isomorphic. We suggest a method that allows one to reduce problems about identities of algebras with additional structure to the identities of-algebras. For the convenience of the reader, we start with a full detailed proof of Razmyslov's Theorem. Then we describe our method and its consequences for the identities of graded algebras, algebras with involution, and several others. Keywords Graded algebra • Polynomial identity • Universal algebra Mathematics Subject Classification 17A42 • 08B20 • 16R50 1 Introduction: the problem and some cases In this manuscript we consider the following problem. Given two algebras A and B over the same field, suppose that A and B satisfy the same polynomial identities, is it true that A isomorphic to B? Naturally, this question stated in all its generality has counterexamples. Dedicated to Professor Ivan Shestakov on his 70th birthday.
It was proved by Valenti and Zaicev, in 2011, that, if G is an abelian group and K is an algebraically closed field of characteristic zero, then any G-grading on the algebra of upper block triangular matrices over
Let K be a field and let U T n = U T n (K) denote the associative algebra of upper triangular n × n matrices over K. The vector space of U T n can be given the structure of a Lie and of a Jordan algebra, respectively, by means of the new products: [a, b] = ab − ba, and a • b = ab + ba. We denote the corresponding Lie and Jordan algebra by U T − n and by U T + n , respectively. If G is a group, the G-gradings on U T n were described by Valenti and Zaicev (Arch Math 89(1):33-40, 2007); they proved that each grading on U T n is isomorphic to an elementary grading (that is every matrix unit is homogeneous). Also Di Vincenzo et al. (J Algebra 275(2):550-566, 2004) classified all elementary gradings on U T n. Here we study the gradings and the graded identities on U T − n and on U T +
We classify up to isomorphism all gradings by an arbitrary group G on the Lie algebras of zero-trace upper block-triangular matrices over an algebraically closed field of characteristic 0. It turns out that the support of such a grading always generates an abelian subgroup of G.Assuming that G is abelian, our technique also works to obtain the classification of G-gradings on the upper block-triangular matrices as an associative algebra, over any algebraically closed field. These gradings were originally described by A. Valenti and M. Zaicev in 2012 (assuming characteristic 0 and G finite abelian) and classified up to isomorphism by A. Borges et al. in 2018. Finally, still assuming that G is abelian, we classify G-gradings on the upper block-triangular matrices as a Jordan algebra, over an algebraically closed field of characteristic 0. It turns out that, under these assumptions, the Jordan case is equivalent to the Lie case.2010 Mathematics Subject Classification. Primary 17B70, secondary 16W50, 17C99.
Let A and B be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose A and B are graded by a semigroup S so that the graded identitical relations of A are the same as those of B. Then A is isomorphic to B as an S-graded algebra.2010 Mathematics Subject Classification. 17A42, 08B20, 16R50.
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