Affine Kac-Moody Algebras Research Articles

The linkage principle for restricted critical level representations of affine Kac–Moody algebras

AbstractWe study the restricted category 𝒪 for an affine Kac–Moody algebra at the critical level. In particular, we prove the first part of the Feigin–Frenkel conjecture: the linkage principle for restricted Verma modules. Moreover, we prove a version of the Bernstein–Gelfand–Gelfand-reciprocity principle and we determine the block decomposition of the restricted category 𝒪. For the proofs, we need a deformed version of the classical structures, so we mostly work in a relative setting.

The t-analog of the basic string function for twisted affine Kac–Moody algebras

We study Lusztigʼs t-analog of weight multiplicities associated to level one representations of twisted affine Kac–Moody algebras. An explicit closed form expression is obtained for the corresponding t-string function using constant term identities of Macdonald and Cherednik. The closed form involves the generalized exponents of the graded pieces of the twisted affine algebra, considered as modules for the underlying finite dimensional simple Lie algebra. This extends previous work on level 1 t-string functions for the untwisted simply-laced affine Kac–Moody algebras.

On the restricted Verma modules at the critical level

We study the restricted Verma modules of an affine Kac-Moody algebra at the critical level with special emphasis on their Jordan-Holder multiplicities. The Feigin-Frenkel conjecture gives a formula for these multiplicities that involves the periodic Kazhdan-Lusztig polynomials. We prove this conjecture for all subgeneric blocks and for the case of anti-dominant simple subquotients.

The t-analog of the level one string function for twisted affine Kac–Moody algebras

We study Lusztigʼs t-analog of weight multiplicities, or affine Kostka–Foulkes polynomials, associated to level one representations of twisted affine Kac–Moody algebras. We obtain an explicit closed form expression for the unique t-string function, using constant term identities of Macdonald and Cherednik. This extends previous work on t-string functions for the untwisted simply-laced affine Kac–Moody algebras.

Finite order automorphisms and real forms of affine Kac-Moody algebras in the smooth and algebraic category

Let $\mathfrak{g}$ be a real or complex (finite dimensional) simple Lie algebra and $\sigma\in\mathrm{Aut}\mathfrak{g}$. The authors study automorphisms of the twisted loop algebra $L(\mathfrak{g},\sigma)$ of smooth $\sigma$-periodic maps from $\mathbb{R}$ to $\mathfrak{g}$ as well as of the ``smooth'' affine Kac-Moody algebra $\hat L(\mathfrak{g},\sigma)$, which is a $2$-dimensional extension of $L(\mathfrak{g},\sigma)$. It turns out that these automorphisms which either preserve or reverse the orientation of loops, and are correspondingly called to be of first and second kind, can be described essentially by curves of automorphisms of $\mathfrak{g}$. If the order of the automorphisms is finite, then the corresponding curves in $\mathrm{Aut}\mathfrak{g}$ allow us to define certain invariants and these turn out to parametrize the conjugacy classes of the automorphisms. If their order is $2$ the authors carry this out in detail and deduce a complete classification of involutions and real forms (which correspond to conjugate linear involutions) of smooth affine Kac-Moody algebras. The resulting classification can be seen as an extension of Cartan's classification of symmetric spaces, i.e. of involutions on $\mathfrak{g}$. If $\mathfrak{g}$ is compact, then conjugate linear extensions of involutions from $\hat L(\mathfrak{g},\sigma)$ to conjugate linear involutions on $\hat L(\mathfrak{g}_{\mathbb{C}},\sigma_{\mathbb{C}})$ yield a bijection between their conjugacy classes and this gives existence and uniqueness of Cartan decompositions of real forms of complex smooth affine Kac-Moody algebras. The authors show that their methods work equally well also in the algebraic case where the loops are assumed to have finite Fourier expansions.

Tame Quivers and Affine Enveloping Algebras

Let g be an affine Kac-Moody algebra with symmetric Cartan datum, n + be the maximal nilpotent subalgebra of g. By the Hall algebra approach, we construct integral bases of the Z-form of the enveloping algebra U(n + ). In particular, the representation theory of tame quivers is essentially used in this paper.

Modular branching rules for projective representations of symmetric groups and lowering operators for the supergroup 𝑄(𝑛)

There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects projective representation theory of the symmetric group and representation theory of the algebraic supergroup $Q(n)$ via appropriate Schur (super)algebras and Schur functors. The second approach follows the work of Grojnowski for classical affine and cyclotomic Hecke algebras and connects projective representation theory of symmetric groups in characteristic $p$ to the crystal graph of the basic module of the twisted affine Kac-Moody algebra of type $A_{p-1}^{(2)}$. The goal of this work is to connect the two approaches mentioned above and to obtain new branching results for projective representations of symmetric groups.

Extremal projectors for contragredient Lie (super)symmetries (short review)

A brief review of the extremal projectors for contragredient Lie (super)symmetries (finite-dimensional simple Lie algebras, basic classical Lie superalgebras, infinite-dimensional affine Kac-Moody algebras and superalgebras, as well as their quantum q analogs) is given. Some bibliographic comments on the applications of extremal projectors are presented.

A Brylinski filtration for affine Kac–Moody algebras

Braverman and Finkelberg have recently proposed a conjectural analogue of the geometric Satake isomorphism for untwisted affine Kac–Moody groups. As part of their model, they conjecture that (at dominant weights) Lusztig's q-analog of weight multiplicity is equal to the Poincare series of the principal nilpotent filtration of the weight space, as occurs in the finite-dimensional case. We show that the conjectured equality holds for all affine Kac–Moody algebras if the principal nilpotent filtration is replaced by the principal Heisenberg filtration. The main body of the proof is a Lie algebra cohomology vanishing result. We also give an example to show that the Poincare series of the principal nilpotent filtration is not always equal to the q-analog of weight multiplicity. Finally, we give some partial results for indefinite Kac–Moody algebras.

Krichever-Novikov algebras, their representations and applications in geometry and mathematical physics

1. Krichever–Novikov algebras and their place in the theory of Lie algebras, geometry and topology of moduli spaces, the theory of integrable systems, and conformal quantum field theory. In the 1960s–1980s, the theory of infinite-dimensional Lie algebras experienced a period of fierce growth related to the appearance of affine Kac–Moody algebras and Virasoro algebras, which combine direct inheritance from the classical theory of semisimple Lie algebras with successful applications. In this period, two important disciplines of mathematical physics were formed, the theory of integrable systems and conformal field theory, characterized by the deep penetration of methods of the theory of Riemann surfaces into theoretical physics. The discovery of the role of Kac–Moody and Virasoro algebras as the algebras of formally gauge and conformal infinitesimal symmetries of these theories and an important tool for integrating nonlinear equations has united both objects. The passage from formal types of symmetries to more geometric ones required considering current algebras and vector fields on Riemann surfaces. In the late 1980s, several works on this topic appeared, which were mainly related to elliptic curves, but not only to them. In 1987, in relation to the study of solitons and conformal field theory, I. M. Krichever and S. P. Novikov introduced the central extensions of current algebras and algebras of vector fields on Riemann surfaces (with complex structure and marked points), which later acquired the name Krichever–Novikov algebras [1–3]. These algebras naturally generalize affine (untwisted) Kac– Moody algebras and Virasoro algebras, respectively. Among other algebras of currents and vector fields on Riemann surfaces the Krichever–Novikov algebras are distinguished by the important property of carrying an almost graded structure, which is weaker than grading but stronger than filtration and leads to many important results; in particular, this makes it possible to consider analogues of highest weight representations. As applied to central extensions, being almost graded is equivalent to the locality property of the corresponding cocycles; moreover, the cocycles are uniquely determined by this property in themost important cases. Affine (untwisted) Kac–Moody algebras andVirasoro algebras are special cases of Krichever–Novikov algebras; from which point of view, they correspond to the Riemann sphere with two marked points. Krichever–Novikov algebras are related in many ways to fundamental problems of geometry, analysis, and mathematical physics. Being introduced as algebras of Fourier–Laurent series on a Riemann surface, they are a part of harmonic analysis. The representation theory of Krichever–Novikov algebras is closely related to the theory of holomorphic bundles on Riemann surfaces. In particular, holomorphic bundles play the key role in parameterizing fermion representations. The this class of representations is most important for the theory among those known at present; it was introduced by the author in [4]. The construction of fermion representations also demonstrates another important relationship, namely, to integrable systems: this construction uses essentially objects of the theory of commutative rings of difference operators constructed in [5–7].

Generalized Castelnuovo–Mumford regularity for affine Kac–Moody algebras

The graded modules over noncommutative algebras often have minimal free resolutions of infinite length, resulting in infinite Castelnuovo–Mumford regularity. In Kang et al. (2010) [6], we introduced a generalized notion of Castelnuovo–Mumford regularity to overcome this difficulty. In this paper, we compute the generalized Castelnuovo–Mumford regularity for integrable highest weight representations of all affine Kac–Moody algebras. It is shown that the generalized regularity depends only on the type and rank of algebras and the level of representations.

Twisted fermionic and bosonic representations for a class of BC-graded Lie algebras

In this paper, we study the fermionic and bosonic representations for a class of BC-graded Lie algebras coordinatized by skew Laurent polynomial rings. This generalizes the fermionic and bosonic constructions for the affine Kac-Moody algebras of type AN(2).

$U_q({\\widehat{\\mathcal{G}}})$ HEISENBERG FAMILIES

We explicitly describe Heisenberg families of elements in an arbitrary grading subspaces of the quantized universal enveloping algebra of an affine Kac–Moody algebra in the Drinfeld formulation.

On black attractors in 8D and heterotic/type IIA duality

Motivated by the study of black attractors in 8D supergravity with 16 supersymmetries, we use the field theory approach and 8D supersymmetry with non trivial central charges to shed light on the exact duality between heterotic string on T^2 and type IIA on real connected and compact surfaces {\Sigma}2. We investigate the two constraints that should be obeyed by {\Sigma}2 and give their solutions in terms of intersecting 2-cycles as well their classification using Dynkin diagrams of affine Kac-Moody algebras. It is shown as well that the moduli space of these dual theories is given by SO(1,1)x((SO(2,r+2))/(SO(2)xSO(r+2))) where r stands for the rank of the gauge symmetry G_{r} of the 10D heterotic string on T^2. The remarkable cases r=-2,-1,0 as well as other features are also investigated.

On [formula omitted]-cofiniteness of parafermion vertex operator algebras

It is proved that any locally regular vertex operator algebra is C 2 -cofinite and the regularity of parafermion vertex operator algebras associated to integrable highest weight modules for affine Kac–Moody algebra A 1 ( 1 ) implies the C 2 -cofiniteness of parafermion vertex operator algebras associated to integrable highest weight modules for any affine Kac–Moody algebra. In particular, the parafermion vertex operator algebra associated to an integrable highest weight module of small level for any affine Kac–Moody algebra is C 2 -cofinite and has only finitely many irreducible modules. Also, the parafermion vertex operator algebras with level 1 are determined explicitly.

Subcategories of fixed points of mutations in root categories with type à n−1

For any n ⩾ 3, let R(n) denote the root category of finite-dimensional nilpotent representations of cyclic quiver with n vertices. In the present paper, we prove that R(n − 1) is triangle-equivalent to the subcategory of fixed points of certain left (or right) mutation in R(n). As an application, it is shown that the affine Kac-Moody algebra of type A n−2 is isomorphic to a Lie subalgebra of the Kac-Moody algebra of type A n−1.

The Structure of Parafermion Vertex Operator Algebras: General Case

The structure of the parafermion vertex operator algebra associated to an integrable highest weight module for any affine Kac-Moody algebra is studied. In particular, a set of generators for this algebra has been determined.

Irreducible Representations for the Baby Tits–Kantor–Koecher Algebra

The baby Tits–Kantor–Koecher (TKK) algebra constructed from the smallest (nonlattice) semilattice is related to the “smallest” extended affine Lie algebras other than the finite dimensional simple Lie algebras and the affine Kac–Moody algebras. In this article, we classify the finite dimensional irreducible representations for the baby TKK algebra. It turns out that such representations can be lifted from modules of direct sums of finitely many copies of the simple Lie algebra sp 4(ℂ).

Irregular Wakimoto modules and the Casimir connection

We study some non-highest weight modules over an affine Kac-Moody algebra at non-critical level. Roughly speaking, these modules are non-commutative localizations of some non-highest weight "vacuum" modules. Using free field realization, we embed some rings of differential operators in endomorphism rings of our modules. These rings of differential operators act on a localization of the space of coinvariants of any module over the Kac-Moody algebra with respect to a certain level subalgebra. In a particular case this action is identified with the Casimir connection.

Soliton equations, vertex operators, and simple singularities

We prove the equivalence of two hierarchies of soliton equations associated to a simply-laced finite Dynkin diagram. The first was defined by Kac and Wakimoto (Proc. Symp. Pure Math. 48:138–177, 1989) using the principal realization of the basic representations of the corresponding affine Kac–Moody algebra. The second was defined in Givental and Milanov (The Breadth of Symplectic and Poisson Geometry, Progress in Mathematics, vol. 232, pp. 173–201, Birkhäuser, Basel, 2005) using the Frobenius structure on the local ring of the corresponding simple singularity. We also obtain a deformation of the principal realization of the basic representation over the space of miniversal deformations of the corresponding singularity. As a by-product, we compute the operator product expansions of pairs of vertex operators defined in terms of Picard–Lefschetz periods for more general singularities. Thus, we establish a surprising link between twisted vertex operators and deformation theory of singularities.