Related papers: Fermionic realization of toroidal Lie algebras of …

Symplectic slice theorems elucidate the local structure of symplectic manifolds carrying Hamiltonian actions of compact Lie groups. We generalize these theorems in two natural settings. The first is based on the idea that complex reductive…

Realization by linear vector fields is constructed for any Lie algebra which admits a biorthogonal system and for its any suitable representation. The embedding into Lie algebras of linear vector fields is analogous to the classical…

Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac-Moody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic…

The formulation of classical mechanics applicable to fermionic degrees of freedom is presented in mathematically rigorous terms, including a description of how the mathematical structure relates to the quantization of the theory. Canonical…

In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of…

In this paper we study a family of commutative algebras generated by two infinite sets of generators. These algebras are parametrized by Young diagrams. We explain a connection of these algebras with the fusion product of integrable…

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as…

One of the four well-known series of simple Lie algebras of Cartan type is the series of Lie algebras of Special type, which are divergence-free Lie algebras associated with polynomial algebras and the operators of taking partial…

We develop some foundations of commutative algebra, with a view towards algebraic geometry, in symmetric tensor categories. Most results establish analogues of classical theorems, in tensor categories which admit a tensor functor to some…

We employ the Poisson-Lie group of pseudo-difference operators to define lattice analogs of classical $W_m$-algebras. We then show that the so-constructed algebras coincide with the ones given by discrete Drinfeld-Sokolov type reduction.

We discuss the steps to construct Dirac operators which have arbitrary fermion offsets, gauge paths, a general structure in Dirac space and satisfy the basic symmetries (gauge symmetry, hermiticity condition, charge conjugation, hypercubic…

In this paper we construct a large class of modules for toroidal Lie superalgebras. Toroidal Lie superalgebras are universal central extension of G tensor A where G is a basic classical Lie superalgebra and A is a Laurent polynomial ring in…

For every field $F$ which has a quadratic extension $E$ we show there are non-metabelian infinite-dimensional thin graded Lie algebras all of whose homogeneous components, except the second one, have dimension $2$. We construct such Lie…

We present the most general polynomial Lie algebra generated by a second order integral of motion and one of order M, construct the Casimir operator, and show how the Jacobi identity provides the existence of a realization in terms of…

This overview paper is intended as a quick introduction to Lie algebras of vector fields. Originally introduced in the late 19th century by Sophus Lie to capture symmetries of ordinary differential equations, these algebras, or…

Iterated loop algebras are by definition obtained by repeatedly applying the loop construction, familiar from the theory of affine Kac-Moody Lie algebras, to a given base algebra. Our interest in this iterated construction is motivated by…

Contractions of Lie algebras are combined with the classical matrix method of Gel'fand to obtain matrix formulae for the Casimir operators of inhomogeneous Lie algebras. The method is presented for the inhomogeneous pseudo-unitary Lie…

The weak operator topology closed operator algebra on $L^2(R)$ generated by the one-parameter semigroups for translation, dilation and multiplication by $exp(i\lambda x), \lambda \geq 0$, is shown to be a reflexive operator algebra, in the…

In this note we explain, in terms of finite dimensional representations of Lie algebras $\mathfrak{sp}_{2\ell}\subset\mathfrak{sl}_{2\ell}$, a combinatorial coincidence of difference conditions in two constructions of combinatorial bases…

We develop the method of the hamiltonian reduction of affine Lie superalgebras to obtain explicit and general expressions both for the classical and the quantum extended superconformal algebras. By performing the gauge transformation which…