We present explicit free field representations for the N = 4 doubly extended superconformal algebra, Ãγ. This algebra generalizes and contains all previous N = 4 superconformal algebras. We have found Ãγ to be obtained by hamiltonian reduction of the Lie superalgebra D(2|1;α). In addition, screening operators are explicitly given and the associated singular vectors identified. We use this to pr...

A well-known Peterson’s theorem says that the number of abelian ideals in a Borel subalgebra of a rank-r finite dimensional simple Lie algebra is exactly 2r . In this paper, we determine the dimensional distribution of abelian ideals in a Borel subalgebra of finite dimensional simple Lie algebras, which is a refinement of the Peterson’s theorem capturing more Lie algebra invariants.

We describe an algorithm for computing automorphism groups and testing isomorphisms of finite dimensional Lie algebras over finite fields. The algorithm is particularly effective for simple or almost simple Lie algebras. We show how it can be used in a computer search for new small-dimensional simple Lie algebras over the field with two elements.

Abstract We introduce a new approach to the study of finite-dimensional representations quantum group affine Lie superalgebra $ \textrm {L}{\mathfrak {g}\mathfrak {l}}_{M|N}=\mathbb {C}[t,t^{-1}]\otimes \mathfrak {l}_{M|N}$ ($M\neq N$). explain how {l}}_{M|N}$ are directly related those algebra type $A$, using an exact monoidal functor called truncation. This can be viewed as analogue super dua...

We study of the category O for the queer Lie superalgebra q(n), and the corresponding block decomposition induced by infinitesimal central characters. In particular, we show that the so-called typical blocks correspond to standardly stratified algebras, in the sense of Cline, Parshall and Scott. By standard arguments for Lie algebras, modified to the superalgebra situation, we prove that these ...

A method is developed to construct irreducible representations(irreps) of the quantum supergroup U q (C(n + 1)) in a systematic fashion. It is shown that every finite dimensional irrep of this quantum supergroup at generic q is a deformation of a finite dimensional irrep of its underlying Lie superalgebra C(n + 1), and is essentially uniquely characterized by a highest weight. The character of ...

In this paper, we introduce the Harish-Chandra homomorphism for quantum superalgebra $$\mathrm {U}_q({\mathfrak {g}})$$ associated with a simple basic Lie $${\mathfrak {g}}$$ and give an explicit description of its image. We use it to prove that center is isomorphic subring ring $$J({\mathfrak exponential super-invariants in sense Sergeev Veselov, establishing type theorem . As byproduct, obtai...

Given a finite dimensional representation of a semisimple Lie algebra there are two ways of constructing link invariants: 1) quantum group invariants using the R-matrix, 2) the Kontsevich universal link invariant followed by the Lie algebra based weight system. Le and Murakami showed that these two link invariants are the same. These constructions can be generalized to some classes of Lie super...