Bernstein Superalgebras and Supermodules

Abstract

Bernstein superalgebras are introduced and irreducible Bernstein supermodules are classified.

Representations of exceptional simple alternative superalgebras of characteristic 3

“…1) B = B (1,2), where B 0 = F · 1, B 1 = F · x + F · y, with 1 being the unit of B and xy = −yx = 1, x 2 = y 2 = 0. 2) B = B (4,2), where B 0 = M 2 (F ), B 1 = F · m 1 + F · m 2 is the 2-dimensional irreducible Cayley bimodule over B 0 ; that is, B 0 acts on B 1 by e ij · m k = δ ik m j , i, j, k ∈ {1, 2}, (1) m · a = a · m, (2) where a ∈ B 0 , m ∈ B 1 , a → a is the symplectic involution in B 0 = M 2 (F ).…”

“…2) B = B (4,2), where B 0 = M 2 (F ), B 1 = F · m 1 + F · m 2 is the 2-dimensional irreducible Cayley bimodule over B 0 ; that is, B 0 acts on B 1 by e ij · m k = δ ik m j , i, j, k ∈ {1, 2}, (1) m · a = a · m, (2) where a ∈ B 0 , m ∈ B 1 , a → a is the symplectic involution in B 0 = M 2 (F ). The odd multiplication on B 1 is defined by 3) The twisted superalgebra of vector type B = B (E, D, γ).…”

“…In this work, we study birepresentations of B (1,2) and of B (4,2). First, we classify the irreducible superbimodules over these superalgebras.…”

“…First, we classify the irreducible superbimodules over these superalgebras. It occurs that, besides a certain two-parametric series of bimodules V (λ, µ) over B (1,2), all the other unital irreducible superbimodules for these superalgebras are regular or opposite to them. As a corollary, we prove that every unital B (4,2)-superbimodule is completely reducible.…”

“…It was shown in [5] that both B (1,2) and B (4,2) admit J-admissible superinvolutions; that is, superinvolutions with symmetric elements in the nucleus. This was used in [5] for constructing new simple exceptional Jordan superalgebras of characteristic 3 as 3 × 3 Hermitian matrices over B (1,2) and B (4,2).…”