Generic Irreducible Representations of Finite-Dimensional Lie Superalgebras

Abstract

A theory of highest weight modules over an arbitrary finite-dimensional Lie superalgebra is constructed. A necessary and sufficient condition for the finite-dimensionality of such modules is proved. Generic finite-dimensional irreducible representations are defined and an explicit character formula for such representations is written down. It is conjectured that this formula applies to any generic finite-dimensional irreducible module over any finite-dimensional Lie superalgebra. The conjecture is proved for several classes of Lie superalgebras, in particular for all solvable ones, for all simple ones, and for certain semi-simple ones.

Good gradings of basic Lie superalgebras

“…This then determines a decomposition g = n − ⊕ h ⊕ n + where n + := ⊕ α∈∆ + g α and n − := ⊕ α∈∆ − g α . Such a decomposition is called a triangular decomposition [15]. We have an induced triangular decomposition g0 = n ) for g (resp.…”

Shapovalov determinant for loop superalgebras

“…On poi(0 |n), more precisely, on the superspace of generating functions, the integral (equal to the coefficient of the leading term in the Taylor series in θ) determines a nondegenerate invariant bilinear form of the same parity as n: [20], [21] that the Cartan subalgebras of any finite-dimensional simple and certain nonsimple (such as poi, q, and their subquotients) Lie superalgebra are conjugate by inner automorphisms. We always fix a Cartan subalgebra t (e.g., for poi(0 |2n), we take t = C[ξ 1 η 1 , .…”

“…For finite-dimensional Lie algebras (and Kac-Moody algebras), the passage is achieved using elements of the (affine) Weyl group. For Lie superalgebras, such a passage is achieved using the "odd" reflections introduced first by Skornyakov and independently by Penkov and Serganova [21]. An interesting open problem is to describe systems of positive roots (or at least an algorithm for passing from one system to another and an explicit form of at least one system) for g = poi or h (perhaps it is better to take the superalgebra g = poi or h augmented by the grading operator).…”

Annihilation Theorem and Separation Theorem for basic classical Lie superalgebras

“…Triangular decompositions. Triangular decompositions of the superalgebras are defined in [PS2] as follows. A Lie subsuperalgebra h ⊂ g is called a Cartan subsuperalgebra if h is nilpotent and coincides with its centralizer in g. For the basic classical Lie superalgebras the set of Cartan subalgebras coincides with the set of Cartan subalgebras of g 0 .…”