The representations of spl(2,1)—an example of representations of basic superalgebras

Abstract

The example of spl(2,1) illustrates some important properties of the representations of basic superalgebras. Typical representations are most similar to the representations of semisimple Lie algebras. We argue that nontypical representations are a degenerate type of representation. The structure of not fully reducible nontypical representations is discussed in detail. As opposed to semisimple Lie algebras, spl(2,1) also admits representations with nondiagonal Cartan subalgebra. There is a one-to-one correspondence between the representations of spl(2,1) and those of the generalized superalgebra osp(2,0,1,1). The osp(2,0,1,1) counterparts of nontypical irreducible representations of spl(2,1) are unfaithful.