Reduction of the adjoint representation of sl(2,1): generators of primitive ideals

Abstract

IVe present a reduction of'the adjoint representation of the Lie superalgebra . s i ( % , I ) and astudy of the quotient algebra B(,,,, = LI/U(C -c ) + L L (~ -kc).ivhere c. I; are two complex numbers. lrnder some additional conditions, we prove that every irreducible infinite dimensional reprew~tation of B(,,k, is faithful. and that B1,,h) is a primitive algebra. LVe give espliciily a set of generators of primitive degenerate ideal of infinite codimension. Essentialiv \ve prove that any minimal primitive ideal of U ( s l J P . 1 ) ) is generated, as a 2-hided ideal, by its intersection with the algebra, of eo-invariants.

Cohomology of Lie superalgebras and their generalizations

“…The decomposition (B.3) of T s 2 (L, ε) has been found independently of Ref. [40]. As already noted, in this article a decomposition of U(L) into indecomposable L-submodules has been determined, and (B.3) is just the first non-trivial part of it.…”

“…Remark 5.3. A decomposition of the adjoint L-module U(L) into indecomposable submodules has been constructed in a recent paper by Benamor [40]. (We are grateful to the referee for drawing our attention to this article.)…”

Indecomposable doubling for representations of the type I Lie superalgebras sl(m/n) and osp(2/2n)

“…For the indecomposable doubling of quartet representations [8], the central case motivating the present work, it should be noted that the structure of the tensor product modules 4 2/3 ⊗ 4 4/3 and 4 0 ⊗ 4 2 , or more generally, of the family 4 y ⊗ 4 −y+2 or 4 y ⊗ 4 y , has been studied by several authors in the course of their analyses of representations of sl(2/1) [6,7,33,30] . In particular, in [30] , an explicit computation indeed yielded (co)homology dimension 1, confirming our more general result.…”

Indecomposable doubling for representations of the type I Lie superalgebras sl(m/n) and osp(2/2n)

“…For the indecomposable doubling of quartet representations [8], the central case motivating the present work, it should be noted that the structure of the tensor product modules 4 2/3 ⊗ 4 4/3 and 4 0 ⊗ 4 2 , or more generally, of the family 4 y ⊗ 4 −y+2 or 4 y ⊗ 4 y ′ , has been studied by several authors in the course of their analyses of representations of sl(2/1) [6,7,30,33]. In particular, in [30], an explicit computation indeed yielded (co)homology dimension 1, confirming our more general result.…”