We construct and study various dual pairs between finite dimensional classical Lie groups and infinite dimensional Lie algebras in some Fock representations. The infinite dimensional Lie algebras here can be either a completed infinite rank affine Lie algebra, the W1+∞ algebra or its certain Lie subalgebras. We give a formulation in the framework of vertex algebras. We also formulate several co...

We describe a Gauss decomposition for the Yangian Y (glm|n) of the general linear Lie superalgebra. This gives a connection between this Yangian and the Yangian of the classical Lie superalgebra Y (A(m− 1, n− 1)) (with m 6= n) defined and studied in papers by Stukopin, and suggests natural definitions for the Yangians Y (sln|n) and Y (A(n, n)). We also show that the coefficients of the quantum ...

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical r-matrices. A super dynamical r-matrix r satisfies the zero weight condition if [h⊗ 1 + 1⊗ h, r(λ)] = 0 for all h ∈ h, λ ∈ h∗. In this paper we explicitly quantize zero-weight super dynamical r-matrices with zero coupling constant for the Lie superalgebra gl(m,n). We also answer some quest...

Let e be an arbitrary even nilpotent element in the general linear Lie superalgebra glM|N and let We associated finite W-superalgebra. Ym|n super Yangian to glm|n. A subalgebra of Ym|n, called shifted denoted by Ym|n(σ), is defined studied. Moreover, explicit isomorphism between a quotient Ym|n(σ) established.

Let F be the underlying base field of characteristic p > 3 and denote by W and S the even parts of the finite-dimensional generalized Witt Lie superalgebra W and the special Lie superalgebra S, respectively. We first give the generator sets of the Lie algebras W and S. Using certain properties of the canonical tori of W and S, we then determine the derivation algebra of W and the derivation spa...

We study quadratic Lie conformal superalgebras associated with Novikov superalgebras. For every superalgebra $(V,\circ)$, we construct an enveloping differential Poisson $U(V)$ a derivation $d$ such that $u\circ v=ud(v)$ and ${u,v}=u\circ v-(-1)^{|u||v|}v\circ u$ for $u,v\in V$. The latter means the commutator Gelfand–Dorfman of $V$ is special. Next, prove constructed on finite-dimensional spec...

For any finitely generated abelian group Q, we reduce the problem of classification of Q-graded simple Lie algebras over an algebraically closed field of “good” characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional Q-graded simple Lie algebras over any algebraically closed field of characteristi...

We consider a simple solution of a Yang-Baxter equation on loop algebra and deduce from it a Sklyanin Poisson structure which operates continuously on a Sobolev test algebra on the Wiener space of the Lie algebra. It is very classical that the solution of the classical Yang-Baxter equation on a finite dimensional algebra gives a Poisson structure on the algebra of smooth function on the finite ...

An explicit character formula is established for any strongly generic nite-dimensional irreducible g-module, g being an arbitrary nite-dimensional complex Lie superalgebra. This character formula had been conjectured earlier by Vera Serganova and the author for any generic irreducible nite-dimensional g-module, i.e. such that its highest weight is far enough from the walls of the Weyl chambers....