Affine Kac Research Articles

AGT relations for abelian quiver gauge theories on ALE spaces

We construct level one dominant representations of the affine Kac–Moody algebra g l ̂ k on the equivariant cohomology groups of moduli spaces of rank one framed sheaves on the orbifold compactification of the minimal resolution X k of the A k − 1 toric singularity C 2 / Z k . We show that the direct sum of the fundamental classes of these moduli spaces is a Whittaker vector for g l ̂ k , which proves the AGT correspondence for pure N = 2 U ( 1 ) gauge theory on X k . We consider Carlsson–Okounkov type Ext-bundles over products of the moduli spaces and use their Euler classes to define vertex operators. Under the decomposition g l ̂ k ≃ h ⊕ s l ̂ k , these vertex operators decompose as products of bosonic exponentials associated to the Heisenberg algebra h and primary fields of s l ̂ k . We use these operators to prove the AGT correspondence for N = 2 superconformal abelian quiver gauge theories on X k .

On conjugacy of Cartan subalgebras in extended affine Lie algebras

That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter–Dynkin diagrams and indecomposable irreducible root systems, is arguably one of the most elegant results in mathematics. The definition of the root system is done by fixing a Cartan subalgebra of the given Lie algebra. The remarkable fact is that (up to isomorphism) this construction is independent of the choice of the Cartan subalgebra. The modern way of establishing this fact is by showing that all Cartan subalgebras are conjugate.For symmetrizable Kac–Moody Lie algebras, with the appropriate definition of Cartan subalgebra, conjugacy has been established by Peterson and Kac. An immediate consequence of this result is that the root systems and generalized Cartan matrices are invariants of the Kac–Moody Lie algebras. The purpose of this paper is to establish conjugacy of Cartan subalgebras for extended affine Lie algebras; a natural class of Lie algebras that generalizes the finite-dimensional simple Lie algebra and affine Kac–Moody Lie algebras.

Lie algebra extensions of current algebras on S3

An affine Kac–Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac–Moody algebras give it for two-dimensional conformal field theory.

Iwahori–Hecke algebras for p-adic loop groups

This paper is a continuation of Braverman and Kazhdan (Ann Math (2) 174(3):1603–1642, 2011) in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac–Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori–Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik’s double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald’s formula for the spherical function in the finite-dimensional case. The results of this paper have been previously announced in Braverman and Kazhdan (European Congress of Mathematics. European Mathematical Society, Zurich, 2014).

Automorphism group of parafermion vertex operator algebras

The automorphism group of parafermion vertex operator algebra associated with the irreducible highest weight module for the affine Kac–Moody algebra A1(1) was easily determined since the Virasoro primary vector of weight 3 in this parafermion vertex operator algebra is unique up to a scalar. However, it is highly nontrivial to determine the automorphism group of parafermion vertex operator algebra associated with the irreducible highest weight module for the affine Kac–Moody algebra An(1) with the rank n≥2. As the first step, in this paper, we determine the full automorphism group of parafermion vertex operator algebra associated with the irreducible highest weight module for the affine Kac–Moody algebra A2(1), which shows the idea for a complete determination for the full automorphism group of the parafermion vertex operator algebra associated with the irreducible highest weight module for any affine Kac–Moody algebra.

Twisted Vertex Operators and Unitary Lie Algebras

AbstractA representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral ℤ2–lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by–product, some fundamental representations of affine Kac–Moody Lie algebra of type A(2)n are recovered by the new method.

Relaxed singular vectors, Jack symmetric functions and fractional level [formula omitted] models

The fractional level models are (logarithmic) conformal field theories associated with affine Kac–Moody (super)algebras at certain levels k∈Q. They are particularly noteworthy because of several longstanding difficulties that have only recently been resolved. Here, Wakimoto's free field realisation is combined with the theory of Jack symmetric functions to analyse the fractional level slˆ(2) models. The first main results are explicit formulae for the singular vectors of minimal grade in relaxed Wakimoto modules. These are closely related to the minimal grade singular vectors in relaxed (parabolic) Verma modules. Further results include an explicit presentation of Zhu's algebra and an elegant new proof of the classification of simple relaxed highest weight modules over the corresponding vertex operator algebra. These results suggest that generalisations to higher rank fractional level models are now within reach.

Invariant bilinear forms of algebras given by faithfully flat descent

The existence of nondegenerate invariant bilinear forms is one of the most important tools in the study of Kac–Moody Lie algebras and extended affine Lie algebras. In practice, these forms are created, or shown to exist, either by assumption or in an ad hoc basis. The purpose of this work is to describe the nature of the space of invariant bilinear forms of certain algebras given by faithfully flat descent (which includes the affine Kac–Moody Lie algebras, as well as Azumaya algebras and multiloop algebras) within a functorial framework. This will allow us to conclude the existence, uniqueness and nature of invariant bilinear forms for many important classes of algebras.

On the stable moment graph of an affine Kac–Moody algebra

In 1980 Lusztig proved a stabilisation property of the affine Kazhdan-Lusztig polynomials. In this paper we give a categorical version of such a result using the theory of sheaves on moment graphs. This leads us to associate with any Kac-Moody algebra its stable moment graph.

Fusion Product Structure of Demazure Modules

Let 𝔤 be a finite–dimensional complex simple Lie algebra. Given a non–negative integer ℓ, we define $\mathcal {P}^{+}_{\ell }$ to be the set of dominant weights λ of 𝔤 such that ℓΛ0+λ is a dominant weight for the corresponding untwisted affine Kac–Moody algebra $\widehat {{\mathfrak {g}}}$ . For the current algebra 𝔤[t] associated to 𝔤, we show that the fusion product of an irreducible 𝔤–module V(λ) such that $\lambda \in \mathcal {P}^{+}_{\ell }$ and a finite number of special family of 𝔤–stable Demazure modules of level ℓ (considered in Fourier and Littelmann, Nagoya Math. J. 182, 171–198 (2006), Adv. Math. 211(2), 566–593 2007) again turns out to be a Demazure module. This fact is closely related with several important conjectures. We use this result to construct the 𝔤[t]–module structure of the irreducible ${\widehat {\mathfrak {g}}}$ –module V(ℓ Λ0 + λ) as a semi–infinite fusion product of finite dimensional 𝔤[t]–modules as conjectured in Fourier and Littelmann, Adv. Math. 211(2), 566–593 (2007). As a second application we give further evidence to the conjecture on the generalization of Schur positivity (see Chari, Fourier and Sagaki 2013).

A Jantzen sum formula for restricted Verma modules over affine Kac–Moody algebras at the critical level

For a restricted Verma module of an affine Kac-Moody algebra at the critical level we describe the Jantzen filtration and give an alternating sum formula which corresponds to the Jantzen sum formula of a baby Verma module over a modular Lie algebra. This also implies a new proof of the linkage principle which was already deduced by Arakawa and Fiebig.

Zhu's algebra, [formula omitted]-algebra and [formula omitted]-cofiniteness of parafermion vertex operator algebras

We study Zhu's algebra, C2-algebra and C2-cofiniteness of parafermion vertex operator algebras. We first give a detailed study of Zhu's algebra and C2-algebra of parafermion vertex operator algebras associated with the affine Kac–Moody Lie algebra slˆ2. We show that they have the same dimension and Zhu's algebra is semisimple. The classification of irreducible modules is also established. Finally, we prove that the parafermion vertex operator algebras for any finite dimensional simple Lie algebras are C2-cofinite.

Leibniz central extensions of Lie superalgebras

In this paper, we develop a general theory on Leibniz central extensions of Lie superalgebras and apply it to determine the second Leibniz cohomology groups for several classes of Lie superalgebras, including classical Lie superalgebras, Neveu–Schwarz superalgebras, differentiably simple Lie superalgebras, and affine (toroidal) Kac–Moody Lie superalgebras.

Koszul duality of affine Kac–Moody algebras and cyclotomic rational double affine Hecke algebras

We give a proof of the parabolic/singular Koszul duality for the category O of affine Kac–Moody algebras. The main new tool is a relation between moment graphs and finite codimensional affine Schubert varieties. We apply this duality to q-Schur algebras and to cyclotomic rational double affine Hecke algebras. This yields a proof of a conjecture of Chuang–Miyachi relating the level-rank duality with the Ringel–Koszul duality of cyclotomic rational double affine Hecke algebras.

Vertex-algebraic structure of principal subspaces of standard $A^{(2)}_2$-modules, I

Extending earlier work of the authors, this is the first in a series of papers devoted to the vertex-algebraic structure of principal subspaces of standard modules for twisted affine Kac–Moody algebras. In this part, we develop the necessary theory of principal subspaces for the affine Lie algebra [Formula: see text], which we expect can be extended to higher rank algebras. As a "test case", we consider the principal subspace of the basic [Formula: see text]-module and explore its structure in depth.

The extended affine Lie algebra associated with a connected non-negative unit form

Given a connected non-negative unit form we construct an extended affine Lie algebra by giving a Chevalley basis for it. We also obtain this algebra as a quotient of an algebra defined by means of generalized Serre relations by M. Barot, D. Kussin and H. Lenzing. This is done in an analogous way to the construction of the simply-laced affine Kac–Moody algebras. Thus, we obtain a family of extended affine Lie algebras of simply-laced Dynkin type and arbitrary nullity. Furthermore, there is a one-to-one correspondence between these Lie algebras and the equivalence classes of connected non-negative unit forms.

Extended affinization of invariant affine reflection algebras

The class of invariant affine reflection algebras is the most general known extension of the class of affine Kac--Moody Lie algebras, introduced in 2008. We develop a method known as ``affinization'' for the class of invariant affine reflection algebras, and show that starting from an algebra belonging to this class together with a certain finite order automorphism, and applying the so called ``affinization method'', we obtain again an invariant affine reflection algebra. This can be considered as an important step towards the realization of invariant affine reflection algebras.

Tame Fréchet submanifolds of co-Banach type

Abstract We introduce a new class of submanifolds of co-Banach type in tame Fréchet manifolds and construct tame Fréchet submanifolds as inverse images of regular values of certain tame maps. Our method furnishes an easy way to construct tame Fréchet manifolds. The results presented are key ingredients in the construction of affine Kac–Moody symmetric spaces; they have also important applications in the study of isoparametric submanifolds in tame Fréchet spaces.

A cohomological proof of Peterson–Kacʼs theorem on conjugacy of Cartan subalgebras for affine Kac–Moody Lie algebras

This paper deals with the problem of conjugacy of Cartan subalgebras for affine Kac–Moody Lie algebras. Unlike the methods used by Peterson and Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildings.

Characterization of Simple Highest Weight Modules

Abstract.We prove that for simple complex finite dimensional Lie algebras, affine Kac–Moody Lie algebras, the Virasoro algebra, and the Heisenberg–Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro algebras and for Heisenberg algebras.