Group gradings on the Lie and Jordan algebras of block-triangular matrices

Abstract

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.

Superinvolutions on block-triangular matrix algebras

“…Proof. Theorem 4.2 implies that U r and U s are isomorphic if and only if there exists z ∈ Z 2 such that Ξ(κ i , γ) = zΞ( κ i , γ), (10) for i = 1, . .…”

“…This implies that r = s. Now assume that (9) holds, in this case (10) implies that s 0 i0 = s 1 i0 and that i 0 is the smallest element in the set of indexes i such that s 0 i = s 1 i . Therefore condition (9) for the tuple s implies that s 0 i0 < s 1 i0 .…”

“…Algebras of upper-triangular matrices and matrix algebras are particular cases of algebras of upper block-triangular matrices. The group gradings and graded involutions for these algebras were studied in [3], [6] and [12] and the group gradings for the Lie and Jordan algebras of block-triangular matrices were classified in [10].…”

Group gradings on finite dimensional incidence algebras

“…Advancements in the classification of group gradings on non-simple algebras have also been made, as seen in the articles [2,4,7]. In particular, the complete classification of isomorphism classes of group gradings on the algebra of upper triangular matrices is given in the works [3,11].…”

Graded Lie-Rinehart Algebras

“…On the other hand, gradings appear elsewhere in the theory of Lie algebras, for example in the Cartan decomposition of a finite-dimensional complex semisimple Lie algebra (see for instance [1,6,10,15,16,19,23]). Also, graded modules have attracted the attention of many researchers in the last years (see [2,4,5,8,28,31]).…”