Kac-Moody Algebra Research Articles

Majid conjecture: quantum Kac-Moody algebras version

Based on the n-fold tensor product version of the generalized double-bosonization construction, we prove the Majid conjecture of the quantum Kac-Moody algebras version. Particularly, we give explicitly the double-bosonization type-crossing constructions of quantum Kac-Moody algebras for affine types $$G_2^{(1)}$$ , $$E_6^{(1)}$$ , and Tp,q,r, and in this way, we can recover generators of quantum Kac-Moody algebras with braided groups defined by R-matrices in the related braided tensor category. This gives us a better understanding for the algebra structures themselves of the quantum Kac-Moody algebras as a certain extension of module-algebras/module-coalgebras with respect to the related quantum subalgebras of finite types inside.

A quiver variety approach to root multiplicities

We present combinatorial upper bounds on dimensions of certain imaginary root spaces for symmetric Kac-Moody algebras. These come from the realization of the corresponding infinity-crystal using quiver varieties. The framework is general, but we only work out specifics in rank two. In that case we give explicit bounds. These turn out to be quite accurate, and in many cases exact, even for some fairly large roots.

Classification of classical twists of the standard Lie bialgebra structure on a loop algebra

The standard Lie bialgebra structure on an affine Kac–Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. In this paper we classify all classical twists of the induced Lie bialgebra structures in terms of Belavin–Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced bialgebra structures are defined by certain solutions of the classical Yang–Baxter equation (CYBE) with two parameters. Then, using the algebro–geometric theory of CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by Belavin and Drinfeld. The classification of twists in the case of parabolic subalgebras allows us to answer recently posed open questions regarding the so-called quasi-trigonometric solutions of CYBE.

A study of Kostant-Kumar modules via Littelmann paths

We study, by means of Littelmann's theory of paths, Kostant-Kumar modules (KK modules for short), which by definition are certain submodules of the tensor product of two irreducible integrable highest weight representations of a symmetrizable Kac-Moody algebra. Our main result is an identification of a path model for any KK module as a subset of the well known path model for the tensor product consisting of concatenations of Lakshmibai-Seshadri paths. The technical results about extremal elements in Coxeter groups that we formulate and prove en route and the technique of their proofs should be of independent interest. We also discuss the existence of PRV components and generalised PRV components in KK modules.

Rigged configurations and the ⁎-involution for generalized Kac–Moody algebras

We construct a uniform model for highest weight crystals and B(∞) for generalized Kac–Moody algebras using rigged configurations. We also show an explicit description of the ⁎-involution on rigged configurations for B(∞): that the ⁎-involution interchanges the rigging and the corigging. We do this by giving a recognition theorem for B(∞) using the ⁎-involution. As a consequence, we also characterize B(λ) as a subcrystal of B(∞) using the ⁎-involution.

$${\varvec{\pi }}$$-systems of symmetrizable Kac–Moody algebras

As part of his classification of regular semisimple subalgebras of semisimple Lie algebras, Dynkin introduced the notion of a $$\pi $$ -system. This is a subset of the set of roots such that pairwise differences of its elements are not roots. Such systems arise as simple systems of regular semisimple subalgebras. Morita and Naito generalized this notion to all symmetrizable Kac–Moody algebras. In this work, we systematically develop the theory of $$\pi $$ -systems of symmetrizable Kac–Moody algebras and establish their fundamental properties. For several Kac–Moody algebras with physical significance, we study the orbits of the Weyl group on $$\pi $$ -systems and completely determine the number of orbits. In particular, we show that there is a unique $$\pi $$ -system of type $${A}^{++}_1$$ (the Feingold–Frenkel rank 3 hyperbolic algebra) in $$E_{10}$$ (the rank 10 hyperbolic algebra) up to Weyl group action and negation.

A COMBINATORIAL STUDY OF AFFINE SCHUBERT VARIETIES IN THE AFFINE GRASSMANNIAN

Let \( {\overline{\mathrm{X}}}_{\uplambda} \) be the closure of the I-orbit \( {\overline{\mathrm{X}}}_{\uplambda} \) in the affine Grassmanian Gr of a simple algebraic group G of adjoint type, where I is the Iwahori subgroup and λ is a coweight of G. We find a simple algorithm which describes the set Ψ(λ) of all I-orbits in \( {\overline{\mathrm{X}}}_{\uplambda} \) in terms of coweights. We introduce R-operators (associated to positive roots) on the coweight lattice of G, which exactly describe the closure relation of I-orbits. These operators satisfy Braid relations generically on the coweight lattice. We also establish a duality between the set Ψ(λ) and the weight system of the level one affine Demazure module of \( {}^L\tilde{\mathfrak{g}} \) indexed by λ, where \( {}^L\tilde{\mathfrak{g}} \) is the affine Kac–Moody algebra dual to the affine Kac–Moody Lie algebra \( \tilde{\mathfrak{g}} \) associated to the Lie algebra \( \mathfrak{g} \) of G.

Higher Kac–Moody algebras and symmetries of holomorphic field theories

We introduce a higher dimensional generalization of the affine Kac-Moody algebra using the language of factorization algebras. In particular, on any complex manifold there is a factorization algebra of currents associated to any Lie algebra. We classify local cocycles of these current algebras, and compare them to central extensions of higher affine algebras recently proposed by Faonte-Hennion-Kapranov. A central goal of this paper is to witness higher Kac-Moody algebras as symmetries of a class of holomorphic quantum field theories. In particular, we prove a generalization of the free field realization of an affine Kac-Moody algebra and also develop the theory of q-characters for this class of algebras in terms of factorization homology. Finally, we exhibit the large N behavior of higher Kac-Moody algebras and their relationship to symmetries of non-commutative field theories.

English

In this paper, we study a special class of indefinite Kac-Moody algebras. Based on the study of hyperbolic Kac-Moody algebras, we give the definition of type Kac-Moody algebras and study some properties of this special type Kac-Moody algebras.

Casimir elements and Sugawara operators for Takiff algebras

For every simple Lie algebra g, we consider the associated Takiff algebra gℓ defined as the truncated polynomial current Lie algebra with coefficients in g. We use a matrix presentation of gℓ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra U(gℓ). A similar matrix presentation for the affine Kac–Moody algebra ĝℓ is then used to prove an analog of the Feigin–Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. The proof relies on an explicit construction of a complete set of Segal–Sugawara vectors for the Lie algebra gℓ.

The trascendence of Kac-Moody algebras

With the main objective that it can be consulted by all researchers, mainly young people, interested in the study of the Kac-Moody algebras and their applications, this paper aims to present the past and present state-of-the-art developments and results on these algebras at various levels of analysis and fields of application in different disciplines. Indeed, lot of references in the literature on these algebras are analyzed and a study of the most relevant topics regarding them is shown.

On mKdV equations related to Kac-Moody algebras $A_5^{(1)}$ and $A_5^{(2)}$

On mKdV equations related to Kac-Moody algebras $A_5^{(1)}$ and $A_5^{(2)}$

Hecke-Langlands Duality and Witten’s Gravitational Moonshine

The purpose of this research is to give a dual description of conformal blocks of d=2 rational CFT (conformal field theory) in terms of Hecke eigenforms and eigensheaves. In particular, partition functions, conformal characters and lattice theta functions may be reconstructed from the action of Hecke operators. This method can be applied to: 1) rings of integers of Galois number fields equipped with the trace (or anti-trace) form; 2) root lattices of affine Kac-Moody algebras and WZW-models; 3) minimal models of Belavin-Polyakov-Zamolodchikov and related d=2 spin-chain/lattice models; 4) vertex algebras of Leech and Niemeier lattices and others. We also use the original Witten’s idea to construct the 3-dimensional quantum gravity as the AdS/CFT-dual of c=24 Monster vertex algebra of Frenkel-Lepowsky- Meurman. Concerning the geometric Langlands duality, we use results of Beilinson-Drinfeld, Frenkel-Ben-Zvi, Gukov-Kapustin-Witten and many others (cf. references). The main new result in this paper is the construction of number-theoretical lattice vertex superalgebras in Section 5 and applications to conformal field theories and quantum gravity.

Keys and Demazure crystals for Kac–Moody algebras

The Key map is an important tool in the determination of the Demazure crystals associated to Kac–Moody algebras. In finite type A, it can be computed in the tableau realization of crystals by a simple combinatorial procedure due to Lascoux and Schützenberger. We show that this procedure is a part of a more general construction holding in the Kac–Moody case that we illustrate in finite types and affine type A. In affine type A, we introduce higher level generalizations of core partitions which are expected to play an important role in the representation theory of Ariki–Koike algebras.

Rewriting modulo isotopies in Khovanov-Lauda-Rouquier's categorification of quantum groups

We study a presentation of the Khovanov - Lauda - Rouquier's candidate 2-categorification of a quantum group using algebraic rewriting methods. We use a computational approach based on rewriting modulo the isotopy axioms of its pivotal structure to compute a family of linear bases for all the vector spaces of 2-cells of this 2-category. We show that these bases correspond to Khovanov and Lauda's conjectured generating sets, proving the non-degeneracy of their diagrammatic calculus. This implies that this 2-category is a categorification of Lusztig's idempotented and integral quantum group Uq(g) associated with a symmetrizable simply-laced Kac-Moody algebra g.

Automorphic Lie algebras and corresponding integrable systems

We study automorphic Lie algebras and their applications to integrable systems. Automorphic Lie algebras are a natural generalisation of celebrated Kac-Moody algebras to the case when the group of automorphisms is not cyclic. They are infinite dimensional and almost graded. We formulate the concept of a graded isomorphism and classify sl(2,C) based automorphic Lie algebras corresponding to all finite reduction groups. We show that hierarchies of integrable systems, their Lax representations and master symmetries can be naturally formulated in terms of automorphic Lie algebras.

The Category $\mathcal O$ for Lie Algebras of Vector Fields (I): Tilting Modules and Character Formulas

In this article, we exploit the theory of graded module categories with semi-infinite character developed by Soergel (Character formulas for tilting modules over Kac–Moody algebras, Represent. Theor. 2 (1998), 432–448) to study representations of infinite-dimensional Lie algebras of vector fields W(n), S(n) and H(n) (n \geq 2) , and obtain a description of indecomposable tilting modules. The character formulas for those tilting modules are determined.

Lattice of dominant weights of affine Kac–Moody algebras

The dual space of the Cartan subalgebra in a Kac–Moody algebra has a partial ordering defined by the rule that two elements are related if and only if their difference is a non-negative or non-positive integer linear combination of simple roots. In this paper, we study the subposet formed by dominant weights in affine Kac–Moody algebras. We give a more explicit description of the covering relations in this poset. We also study the structure of basic cells in this poset of dominant weights for untwisted affine Kac–Moody algebras of type A.

Fusion rules for ℤ2-orbifolds of affine and parafermion vertex operator algebras

This paper is about the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra K(sl2,k) associated to the integrable highest weight modules for the affine Kac—Moody algebra $$A_1^{(1)}$$ is the building block of the general parafermion vertex operator K( $$K(\mathfrak{g},k)$$ ,k) for any finite-dimensional simple Lie algebra $$\mathfrak{g}$$ and any positive integer k. We first classify the irreducible modules of ℤ2-orbifold of the simple affine vertex operator algebra of type $$A_1^{(1)}$$ and determine their fusion rules. Then we study the representations of the ℤ2-orbifold of the parafermion vertex operator algebra K(sl2, k). The quantum dimensions, and more technically, fusion rules for the ℤ2-orbifold of the parafermion vertex operator algebra K(sl2, k) are completely determined.

Connectedness of Lakshmibai-Seshadri path crystals for hyperbolic Kac-Moody algebras of rank 2

Let be a hyperbolic Kac-Moody algebra of rank 2. We give a necessary and sufficient condition for the crystal graph of the Lakshmibai-Seshadri path crystal to be connected for an arbitrary integral weight λ.