arXiv subjects

Victor G. Kac

Publications and source records attributed to Victor G. Kac.

120 records · Page 7Linked to original sources

arXiv1999

Quasifinite representations of W_{\infty}

We classify the quasifinite highest weight modules over a family of subalgebras W_{\infty}^{n} of the central extension W_{1+\infty} of the Lie algebra of differential operators on the circle consisting of operators of order \geq n. We classify the unitary quasifinite highest weight modules over W_{\infty}=W_{\infty}^{1} and realize them in terms of unitary highest weight representations of the Lie algebra of infinite matrices with finitely many non-zero diagonals.

math.QA↗

arXiv1999

Formal distribution algebras and conformal algebras

Conformal algebra is an axiomatic description of the operator product expansion (or rather its Fourier transform) of chiral fields in a conformal field theory. This is a review of recent developments in the subject.

q-alg↗

arXiv1999

Generalized Spencer Cohomology and filtered Deformations of Z-graded Lie Superalgebras

In this paper we introduce generalized Spencer cohomology for finite depth Z-graded Lie (super)algebras. We develop a method of finding filtered deformations of such Z-graded Lie (super)algebras based on this cohomology. As an application we determine all simple filtered deformations of certain Z-graded Lie superalgebras classified in [K3], thus completing the last step in the classification of simple infinite-dimensional linearly compact Lie superalgebras.

math.RT↗

arXiv1998

Cohomology of Conformal Algebras

Conformal algebra is an axiomatic description of the operator product expansion of chiral fields in conformal field theory. On the other hand, it is an adequate tool for the study of infinite-dimensional Lie algebras satisfying the locality property. The main examples of such Lie algebras are those ``based'' on the punctured complex plane, like the Virasoro algebra and loop algebras. In the present paper we develop a cohomology theory of conformal algebras with coefficients in an arbitrary module. It possesses standard properties of cohomology theories; for example, it describes extensions and deformations. We offer explicit computations for most of the important examples.

math.QA↗

arXiv1998

Quasifinite representations of classical Lie subalgebras of W_{1+infty}

We show that there are precisely two, up to conjugation, anti-involutions sigma_{\pm} of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension \hat{D}^{\pm} of the Lie subalgebra of this algebra fixed by - sigma_{\pm}, and find the unitary ones. We realize them in terms of highest weight representations of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the truncated polynomial algebra C[u] / (u^{m+1}) and its classical Lie subalgebras of B, C and D types. Character formulas for positive primitive representations of \hat{D}^{\pm} (including all the unitary ones) are obtained. We also realize a class of primitive representations of \hat{D}^{\pm} in terms of free fields and establish a number of duality results between these primitive representations and finite-dimensional irreducible representations of finite-dimensional Lie groups and supergroups. We show that the vacuum module V_c of \hat{D}^+ carries a vertex algebra structure and establish a relationship between V_c for half-integral central charge c and W-algebras.

math.QA↗

arXiv1997

The idea of locality

This is a review of recent results on conformal (super)algebras. It may be viewed as an amplification of my Wigner medal acceptance speech (given in July 1996 in Goslar, Germany) reproduced in the introduction.

q-alg↗

arXiv1996

Affine orbifolds and rational conformal field theory extensions of W_{1+infinity}

Chiral orbifold models are defined as gauge field theories with a finite gauge group $Γ$. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group $Γ$ of inner automorphisms or A (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra $A^Γ\subset A$ of local observables invariant under $Γ$. A set of positive energy $A^Γ$ modules is constructed whose characters span, under some assumptions on $Γ$, a finite dimensional unitary representation of $SL(2,Z)$. We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules. As an application we construct a family of rational conformal field theory (RCFT) extensions of $W_{1+\infty}$ that appear to provide a bridge between two approaches to the quantum Hall effect.

hep-th↗

arXiv1994

Finite dimensional representations of quantum affine algebras at roots of unity

We describe explicitly the canonical map $χ:$ Spec $\ue(\a{g})\ \rightarrow \ $Spec $\ze$, where $\ue(\a{g})$ is a quantum loop algebra at an odd root of unity $\ve$. Here $\ze$ is the center of $\ue(\a{g})$ and Spec $R$ stands for the set of all finite--dimensional irreducible representations of an algebra $R$. We show that Spec $\ze$ is a Poisson proalgebraic group which is essentially the group of points of $G$ over the regular adeles concentrated at $0$ and $\infty$. Our main result is that the image under $χ$ of Spec $\ue(\a{g})$ is the subgroup of principal adeles.

hep-th↗

arXiv1994

Integrable highest weight modules over affine superalgebras and number theory

In the first part of the paper we give the denominator identity for all simple finite-dimensional Lie super algebras $\frak g\/$ with a non-degenerate invariant bilinear form. We give also a character and (super) dimension formulas for all finite-dimensional irreducible $\frak g\/$-modules of atypicality $\leq 1\/$ . In the second part of the paper we give the denominator identity for the affine superalgebras $\hat{\frak g}\/$ associated to $\frak g\/$. Specializations of this identity give almost all old and many new formulas for the number of representations of an integer as sums of squares and sums of triangular numbers. At the end, we introduce the notion of an integrable $\hat{\frak g}\/$-module and give a classification of irreducible integrable highest weight $\hat{\frak g}\/$-modules.

hep-th↗

arXiv1993

Vertex Operator Superalgebras and Their Representations

After giving some definitions for vertex operator SUPERalgebras and their modules, we construct an associative algebra corresponding to any vertex operator superalgebra, such that the representations of the vertex operator algebra are in one-to-one correspondence with those of the corresponding associative algebra. A way is presented to decribe the fusion rules for the vertex operator superalgebras via modules of the associative algebra. The above are generalizations of Zhu's constructions for vertex operator algebras. Then we deal in detail with vertex operator superalgebras corresponding to Neveu-Schwarz algebras, super affine Kac-Moody algebras, and free fermions. We use the machinery established above to find the rationality conditions, classify the representations and compute the fusion rules. In the appendix, we present explicit formulas for singular vectors and defining relations for the integrable representations of super affine algebras. These formulas are not only crucial for the theory of the corresponding vertex operator superalgebras and their representations, but also of independent interest.

hep-th↗

arXiv1993

Quasifinite highest weight modules over the Lie algebra of differential operators on the circle

We classify positive energy representations with finite degeneracies of the Lie algebra $W_{1+\infty}\/$ and construct them in terms of representation theory of the Lie algebra $\hatgl ( \infty R_m )\/$ of infinite matrices with finite number of non-zero diagonals over the algebra $R_m = \C [ t ] / ( t^{m + 1} )\/$. The unitary ones are classified as well. Similar results are obtained for the sin-algebras.

hep-th↗

arXiv1993

Some quantum analogues of solvable Lie groups

In this paper we analyze the structure of some subalgebras of quantized enveloping algebras corresponding to unipotent and solvable subgroups of a simple Lie group G. These algebras have the non--commutative structure of iterated algebras of twisted polynomials with a derivation, an object which has often appeared in the general theory of non-commutative rings. In particular, we find maximal dimensions of their irreducible representations. Our results confirm the validity of the general philosophy that the representation theory is intimately connected to the Poisson geometry.

hep-th↗