It is proved that up to isomorphism there is only one (2,2)-dimensional supertorus associated to a nontrivial representation of its underlying 2-torus, and that it has nontrivial odd brackets. This supertorus is obtained by finding out first a canonical form for its Lie superalgebra, and then using Lie’s technique to represent it faithfully as supervector fields on a supermanifold. Those superv...

We give closed formulae for the q–characters of the fundamental representations of the quantum loop algebra in terms of a family of partitions satisfying some simple properties. We also give the multiplicities of the eigenvalues of the imaginary subalgebra in terms of these partitions. Introduction In this paper we study the q–characters of the fundamental finite–dimensional representations of ...

We extend the basic fact that every ideal of a finite dimensional semisimple Lie algebra has a unique complement to the case of closed ideals of prosemisimple Lie algebras. We prove that if A is a closed ideal of a prosemisimple Lie algebra L = lim ←−−Ln (n ∈ N), where the Ln are finite dimensional semisimple Lie algebras, then there exists a unique ideal B of L such that L = A ⊕ B. Mathematics...

In this paper we give a re-normalization of the supertrace on the category of representations of Lie superalgebras of type I, by a kind of modified superdimension. The genuine superdimensions and supertraces are generically zero. However, these modified superdimensions are non-zero and lead to a kind of supertrace which is non-trivial and invariant. As an application we show that this new super...

A finite dimensional Lie algebra f is Frobenius if there is a linear Frobenius functional F : f→ C such that the skew bilinear form BF defined by BF (x, y) = F ([x, y]) is non-degenerate. The principal element of f is then the unique element F̂ such that F (x) = F ([F̂ , x]); it depends on the choice of functional. However, if f is a subalgebra of a simple Lie algebra g and not an ideal of any la...

We prove that the super star product on a Poisson Lie supergroup leads to the structure of quantum superalgebra ( triangular Hopf superalgebra) on the super quantized enveloping algebra of the Lie superalgebra of the Lie supergroup and that equivalents super star products generate isomorphic quantum superalgebras.

Representations of the quantum superalgebra U q [osp(1/2)] and their relations to the basic hypergeometric functions are investigated. We first establish Clebsch-Gordan decomposition for the superalgebra U q [osp(1/2)] in which the representations having no classical counterparts are incorporated. Formulae for these Clebsch-Gordan coefficients are derived, and it is observed that they may be ex...

We find the free field construction of basic $W$-current and screening currents for deformed $W$-superalgebra ${\cal W}_{q,t}\bigl(A(M,N)\bigr)$ associated with Lie superalgebra type $A(M,N)$. Using this construction, we introduce higher $W$-currents obtain a closed set quadratic relations among them. These are independent choice Dynkin-diagrams $A(M,N)$, though not. This allows us to define by...

We classify nontrivial deformations of the standard embedding of the Lie superalgebra K(1) of contact vector fields on the (1,1)-dimensional supercircle into the Lie super-algebra of superpseudodifferential operators on the supercircle. This approach leads to the deformations of the central charge induced on K(1) by the canonical central extension of SΨDO.

We observe a correspondence between the zero modes of superconformal algebras S(2, 1) and W (4) ([8]) and the Lie superalgebras formed by classical operators appearing in the Kähler and hyper-Kähler geometry. 1 Lie superalgebras of the Hodge theory 1.1 Kähler manifolds Let M = (M2n, g, I, ω) be a compact Kähler manifold of real dimension 2n, where g is a Riemannian (Kähler) metric, I is a compl...