The structure of sl(2,1)-supersymmetry: irreducible representations and primitive ideals

Abstract

We give a detailed study of the enveloping algebra of the Lie superalgebra sl(2, 1), including classification of irreducible HarishChandra modules, completeness of finite dimensional irreducible, explicit computation of center, and classification of primitive ideals.Introduction and main results. Lie superalgebras are important both in physics and in mathematics [5]. In physics, they are used e.g. to unify fermions and bosons in a unique picture (one irreducible representation of the structure) via supersymmetry. In mathematics, their enveloping algebras provide a class of very interesting noetherian algebras. Much information is known about enveloping algebras of Lie algebras (e.g., [4]), but for superalgebras there is a lot to do (see e.g.[2] for a pioneering work, and [13] for a very nice survey of results obtained up to now). Let us restrict to the simple case; then a natural distinction does appear between simple superalgebras with an enveloping algebra which is a domain and others. The first case is exactly the series osp(l, 2ή), which are also the only semi-simple simple superalgebras [8]. The simplest model of this case is h = osp(l, 2) U(h) was completely studied in [16], including explicit computation of Prim U(h). The simplest model of the second case is g = sl(2, 1), and the purpose of the present paper is a complete study of U(g). We shall give a classification of irreducible Harish-Chandra modules, a detailed computation of the center Z(g) of U(g), and a classification of Prim U(g).Let us recall known results: finite dimensional irreducible representations of g = sl(2, 1) are known [18], and also unitary irreducible are classified ([6], [7]). Moreover, finite dimensional representations provide a complete set of representations [2], but are generally not fully reducible.A fundamental result of our paper is the fact that finite dimensional irreducible provide a complete set, because of information that can be deduced on U(g). Actually, we deduce an explicit determination of the center Z(g), which shows that Z{g) is not a finitely

Centre and representations of at roots of unity

“…In section 6, we present complete sets of representations corresponding to infinite subsets of the set of continuous parameters. All the representations of these complete sets have the same dimension, unlike in the classical case [1]. Finally, in section 7, we prove the relations in the centre using our complete set of irreducible representations.…”

“…This terminology was used in [1], where the authors found complete sets of finite dimensional irreducible representations of the classical sl(2) and sl(2|1). For quantum groups at roots of unity, we shall obtain rather different results.…”

“…The explicit expression of a set of generators of the centre, together with the relations, was given in [1] in the classical case and in [3] in the quantum case.…”

Algebraic approach to q-deformed supersymmetric variants of the Hubbard model with pair hoppings

“…they satisfy the classical analogues of (20,21). The relations (14,15,16,22,23,24) are still valid as long as the indices p i are greater or equal to 2. Notice that the classical operators Q (±) p , C p and S p are not the limits as q goes to 1 of the corresponding quantum ones, but rather limits of some linear combinations of them (See [8]).…”

Hochschild Cohomology and Deformations of Clifford-Weyl Algebras

“…In Sections 7 and 8, we describe A Λ (n) and A λ (n) using the enveloping algebra U of the Lie superalgebra osp (1,2) and its primitive quotients [20]. Denoting by A λ the algebra A λ (0), one has:…”