Using Howe duality we compute explicitly Kostant-type homology groups for a wide class of representations of the infinite-dimensional Lie superalgebra b gl∞|∞ and its classical subalgebras at positive integral levels. We also obtain Kostant-type homology formulas for the Lie algebra b gl∞ at negative integral levels. We further construct resolutions in terms of generalized Verma modules for the...

Generalized quantum statistics such as para-statistics is usually characterized by certain triple relations. In the case of para-Fermi statistics these relations can be associated with the orthogonal Lie algebra Bn = so(2n + 1); in the case of paraBose statistics they are associated with the Lie superalgebra B(0|n) = osp(1|2n). In a previous paper, a mathematical definition of “a generalized qu...

The classification of the finite-dimensional restricted simple Lie algebras over an algebraically closed field F of prime characteristic p > 7 is announced. Such an algebra is either of classical type (an analogue over F of a finite-dimensional simple Lie algebra over the complex numbers) or of Cartan type (an analogue over F of one of the infinite Lie algebras of Cartan over the complex numbers).

We investigate cohomological support varieties for finite-dimensional Lie superalgebras defined over fields of odd characteristic. Verifying a conjecture from our previous work, we show the variety supermodule can be realized as an explicit subset nullcone underlying superalgebra. also is zero if and only finite projective dimension. As consequence, obtain positive characteristic version theore...

A theory of highest weight modules over an arbitrary nite-dimensional Lie su-peralgebra is constructed. A necessary and suucient condition for the nite-dimensionality of such modules is proved. Generic nite-dimensional irreducible representations are deened and an explicit character formula for such representations is written down. It is conjectured that this formula applies to any generic nite...

Let Γ be a countable abelian semigroup and A be a countable abelian group satisfying a certain finiteness condition. Suppose that a group G acts on a (Γ × A)-graded Lie superalgebra L = ⊕ (α,a)∈Γ×A L(α,a) by Lie superalgebra automorphisms preserving the (Γ × A)-gradation. In this paper, we show that the Euler-Poincaré principle yields the generalized denominator identity for L and derive a clos...

We describe the Lie bialgebra structure on the Lie superalgebra sl(2, 1) related to an r−matrix that cannot be obtained by a Belavin-Drinfeld type construction. This structure makes sl(2, 1) into the Drinfeld double of a four-dimensional subalgebra. It is well-known that non-degenerate r−matrices (describing quasitriangular Lie bialgebra structures) on simple Lie algebras are classified by Bela...

A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance, in the case of the orthosymplectic Lie superalgebra osp(1, 2), Pinczon gave in [Pi] an example of a Verma module whose annihilator is not gene...

Soon after parastatistics has been introduced [1], it was discovered that it has a deep algebraic structure. It turned out that any n pairs of parafermion operators generate the simple Lie algebra so(2n + 1) [2, 3], and n pairs of paraboson creation and annihilation operators b 1 , . . . , bn generate a Lie superalgebra [4], isomorphic to one of the basic classical Lie superalgebras in the clas...