Yuly Billig

Representations of the Twisted Heisenberg-Virasoro Algebra at Level Zero

We describe the structure of the irreducible highest weight modules for the twisted Heisen- berg-Virasoro Lie algebra at level zero. We prove that either a Verma module is irreducible or its maximal submodule is cyclic.

Principal vertex operator representations for toroidal Lie algebras

We introduce the principal vertex operator representations for the toroidal Lie algebras generalizing the construction for the affine Kac–Moody algebras. We also represent the derivations of the toroidal algebras and introduce analogs of the Sugawara operators.

A category of modules for the full toroidal Lie algebra

Toroidal Lie algebras are very natural multi-variable generalizations of affine Kac-Moody algebras. The theory of affine Lie algebras is rich and beautiful, having connections with diverse areas of mathematics and physics. Toroidal Lie algebras are also proving themselves to be useful for the applications. Frenkel, Jing and Wang [FJW] used representations of toroidal Lie algebras to construct a new form of the McKay correspondence. Inami et al., studied toroidal symmetry in the context of a 4-dimensional conformal field theory [IKUX], [IKU]. There are also applications of toroidal Lie algebras to soliton theory. Using representations of the toroidal algebras one can construct hierarchies of non-linear PDEs [B2], [ISW]. In particular, the toroidal extension of the Korteweg-de Vries hierarchy contains the Bogoyavlensky’s equation, which is not in the classical KdV hierarchy [IT]. One can use the vertex operator realizations to construct n-soliton solutions for the PDEs in these hierarchies. We hope that further development of the representation theory of toroidal Lie algebras will help to find new applications of this interesting class of algebras. The construction of a toroidal Lie algebra is totally parallel to the well-known construction of an (untwisted) affine Kac-Moody algebra [K1]. One starts with a finite-dimensional simple Lie algebra ġ and considers Fourier polynomial maps from an N + 1-dimensional torus into ġ. Setting tk = e ixk , we may identify the algebra of Fourier polynomials on a torus with the Laurent polynomial algebra R = C[t0 , t ± 1 , . . . , t ± N ], and the Lie algebra of the ġ-valued maps from a torus with the multi-loop algebra C[t±0 , t ± 1 , . . . , t ± N ] ⊗ ġ. When N = 0, this yields the usual loop algebra. Just as for the affine algebras, the next step is to build the universal central extension (R⊗ ġ)⊕K of R⊗ ġ. However unlike the affine case, the center K is infinite-dimensional when N ≥ 1. The infinite-dimensional center makes this Lie algebra highly degenerate. One can show, for example, that in an irreducible bounded weight module, most of the center should act trivially. To eliminate this degeneracy, we add the Lie algebra of vector fields on a torus, D = Der (R) to (R⊗ ġ)⊕K. The resulting algebra,

Magnetic hydrodynamics with asymmetric stress tensor

In this paper we study equations of magnetic hydrodynamics with a stress tensor. We interpret this system as the generalized Euler equation associated with an Abelian extension of the Lie algebra of vector fields with a nontrivial 2-cocycle. We use the Lie algebra approach to prove the energy conservation law and the conservation of cross-helicity.

Abelian Extensions of the Group of Diffeomorphisms of a Torus

In this Letter we construct Abelian extensions of the group of diffecomorphisms of a torus. We consider the Jacobian map, which is a crossed homomorphism from the group of diffeomorphisms into a toroidal gauge group. A pull-back under this map of an invariant central 2-cocycle on a gauge group turns out to be an Abelian cocycle on the group of diffeomorphisms. In the case of a circle we get an interpretation of the Virasoro–Bott cocycle as a pull-back of the Heisenberg cocycle. We also give an Abelian generalization of the Virasoro–Bott cocycle to the case of a manifold with a volume form.

Representations of the Lie algebra of vector fields on a torus and the chiral de Rham complex

The goal of this paper is to study the representation theory of a classical infinite-dimensional Lie algebra - the Lie algebra of vector fields on an N-dimensional torus for N > 1. The case N=1 gives a famous Virasoro algebra (or its centerless version - the Witt algebra). The algebra of vector fields has an important class of tensor modules parametrized by finite-dimensional modules of gl(N). Tensor modules can be used in turn to construct bounded irreducible modules for the vector fields on N+1-dimensional torus, which are the central objects of our study. We solve two problems regarding these bounded modules: we construct their free field realizations and determine their characters. To solve these problems we analyze the structure of the irreducible modules for the semidirect product of vector fields with the quotient of 1-forms by the differentials of functions. These modules remain irreducible when restricted to the subalgebra of vector fields, unless they belongs to the chiral de Rham complex, introduced by Malikov-Schechtman-Vaintrob.

Verma modules for Yangians

AbstractWe study the Verma modules M((μu)) over the Yangian Y

Conjugacy theorem for the strictly hyperbolic kac-moody algebras

(\mathfrak{a})