On [A,A]/[A,[A,A]][A,A]/[A,[A,A]] and on a WnW_n-action on the consecutive commutators of free associative algebra

We consider the lower central filtration of the free associative algebra $A_n$ with $n$ generators as a Lie algebra. We consider the associated graded Lie algebra. It is shown that this Lie algebra has a huge center which belongs to the cyclic words, and on the quotient Lie algebra by the center there acts the Lie algebra $W_n$ of polynomial vector fields on $\mathbb{C}^n$. We compute the space $[A_n,A_n]/[A_n,[A_n,A_n]]$ and show that it is isomorphic to the space $\Omega^2_{closed}(\mathbb{C}^n) \oplus \Omega^4_{closed}(\mathbb{C}^n) \oplus \Omega^6_{closed}(\mathbb{C}^n) \oplus ...$.Comment: 18 pages, 4 eps Figures, v2: minor corrections are mad

Zhu's algebras, C2C_2-algebras and abelian radicals

This paper consists of three parts. In the first part we prove that Zhu's and $C_2$-algebras in type $A$ have the same dimensions. In the second part we compute the graded decomposition of the $C_2$-algebras in type $A$, thus proving the Gaberdiel-Gannon's conjecture. Our main tool is the theory of abelian radicals, which we develop in the third part.Comment: 19 page

Riemann-Roch-Hirzebruch theorem and Topological Quantum Mechanics

In the present paper we discuss an independent on the Grothendieck-Sato isomorphism approach to the Riemann-Roch-Hirzebruch formula for an arbitrary differential operator. Instead of the Grothendieck-Sato isomorphism, we use the Topological Quantum Mechanics (more or less equivalent to the well-known constructions with the Massey operations from [KS], [P], [Me]). The statement that the Massey operations can "produce" the integral in some set-up, has an independent from the RRH theorem interest. We finish the paper by some open questions arising when the main construction is applied to the cyclic homology (instead of the Hochschild homology).Comment: 24 pages, no figures, LaTe

Generalized Drinfeld realization of quantum superalgebras and Uq(osp^(1,2))U_q(\hat {\frak osp}(1,2))

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g(z)=g(z^{-1})^{-1}.$ In particular, we present the Drinfeld realization of $U_q(\hat {\frak osp}(1,2))$ and its Serre relations.Comment: 14 pages, 2 figures, Ams-latex. Dedicated to Moshe Flato. Correct typos. Acknowledgments adde

Difference equations of quantum current operators and quantum parafermion construction

For the current realization of the affine quantum groups, a simple comultiplication for the quantum current operators was given by Drinfeld. With this comultiplication, we prove that, for the integrable modules of $U_q(\hat {\frak sl}(2))$ of level $k+1$, $x^\pm(z)x^\pm(zq^{\pm 2}) \cdot\cdot\cdot x^\pm(zq^{\pm 2k})$ are vertex operators satisfying certain q-difference equations, and we derive the quantum parafermions of $U_q(\hat {\frak sl}(2))$.Comment: 15 pages Amslate

Integrals of Motion and Quantum Groups

A homological construction of integrals of motion of the classical and quantum Toda field theories is given. Using this construction, we identify the integrals of motion with cohomology classes of certain complexes, which are modeled on the BGG resolutions of the associated Lie algebras and their quantum deformations. This way we prove that all classical integrals of motion can be quantized. For the Toda field theories associated to finite-dimensional Lie algebras, the algebra of integrals of motions is the corresponding W-algebra. For affine Toda field theories this algebra is a commutative subalgebra of a W-algebra; it consists of quantum KdV hamiltonians.Comment: 71 pages (final version, to appear in Lect. Notes in Math, vol. 1620

Extended vertex operator algebras and monomial bases

We present a vertex operator algebra which is an extension of the level $k$ vertex operator algebra for the $\hat{sl}_2$ conformal field theory. We construct monomial basis of its irreducible representations.Comment: To appear in the Festschrift in honor of Prof. McGuire published by World Scientific PC. In the present version several corrections are mad

Coinvariants of nilpotent subalgebras of the Virasoro algebra and partition identities

We prove that the dimensions of coinvariants of certain nilpotent subalgebras of the Virasoro algebra do not change under deformation in the case of irreducible representations of (2,2r+1) minimal models. We derive a combinatorial description of these representations and the Gordon identities from this result.Comment: 9 pages, amslatex; for non-amslatex users the .dvi file is available via anonymous ftp from math.harvard.edu/pub:coinv.dv

Functional equations in algebra

We study flat deformations of quotients of a polynomial algebra in a class of graded commutative associative algebras. Functional equations and their solutions in terms of theta functions play important role in these studies. An analog of this theory in a fermionic case is also briefly discussed.Comment: 22 pages, Late

Integrable Hierarchies and Wakimoto Modules

In our earlier papers we proposed a new approach to integrable hierarchies of soliton equations and their quantum deformations. We have applied this approach to the Toda field theories and the generalized KdV and modified KdV (mKdV) hierarchies. In this paper we apply our approach to the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and its generalizations. In particular, we show that the free field (Wakimoto) realization of an affine algebra naturally appears in the context of the generalized AKNS hierarchies. This is analogous to the appearance of the free field (quantum Miura) realization of a W-algebra in the context of the generalized KdV equations. As an application, we give here a new proof of the existence of the Wakimoto realization.Comment: 36 pages, Latex2