Graded involutions on block-triangular matrix algebras

Superinvolutions on block-triangular matrix algebras

“…We relate the superinvolutions on algebras of upper block-triangular matrices to certain anti-automorphisms on these algebras. The classification of these anti-automorphisms is reduced to the classification of antiautomorphisms on Z 2 × Z-graded matrix algebras, analogous results are obtained in [6] to classify the involutions on graded algebras of upper block-triangular matrices. The results on the classification of anti-automorphisms on graded matrix algebras that will be needed in this paper are presented in Section 3 and the necessary results on graded automorphisms and anti-automorphisms of algebras of upper blocktriangular matrices are presented in Section 4.…”

“…The main result of [6] implies that for the isomorphism ψ in the previous corollary there exists a matrix Ψ ∈ U T (p 1 , . .…”

“…, n. The map ψ : e i,j → e σ(i),σ(j) is an isomorphism from U(κ, γ) to U(κ ′ , γ ′ ) that has the desired property. Now we present the results on isomorphisms and automorphisms of gradings on algebras of upper block-triangular matrices that were obtained in [6] and that will be needed in the next section.…”

“…Remark 4.8. Theorem 4.7 implies that if U admits a graded antiautomorphism, then, up to an isomorphism, we can assume that U admits a stable involution, see [6,Remark 28].…”

“…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].…”