Kac-Moody Lie Algebras Research Articles

$\mathfrak{sl}_{2}$ Triples Whose Nilpositive Elements Are in a Space Which Is Spanned by the Real Root Vectors in Rank 2 Symmetric Hyperbolic Kac–Moody Lie Algebras

In analogy with the theory of nilpotent orbits in finite-dimensional semisimple Lie algebras, it is known that the principal \mathfrak{sl}_{2} subalgebras can be constructed in hyperbolic Kac–Moody Lie algebras. We obtained a series of \mathfrak{sl}_{2} subalgebras in rank 2 symmetric hyperbolic Kac–Moody Lie algebras by extending the aforementioned construction. We present this result and also discuss \mathfrak{sl}_{2} modules obtained by the action of the \mathfrak{sl}_{2} subalgebras on the original Lie algebras.

Twisted tensor products of quantum affine vertex algebras and coproducts

Let g be a symmetrizable Kac-Moody Lie algebra, and let Vgˆ,ħℓ, Lgˆ,ħℓ be the quantum affine vertex algebras constructed in [11]. For any complex numbers ℓ and ℓ′, we present an ħ-adic quantum vertex algebra homomorphism Δ from Vgˆ,ħℓ+ℓ′ to the twisted tensor product ħ-adic quantum vertex algebra Vgˆ,ħℓ⊗ˆVgˆ,ħℓ′. In addition, if both ℓ and ℓ′ are positive integers, we show that Δ induces an ħ-adic quantum vertex algebra homomorphism from Lgˆ,ħℓ+ℓ′ to the twisted tensor product ħ-adic quantum vertex algebra Lgˆ,ħℓ⊗ˆLgˆ,ħℓ′. Moreover, we prove the coassociativity of Δ.

Representations of the cyclotomic oriented Brauer-Clifford supercategory

Let k be an algebraically closed field with characteristic p different from 2. We generalize the notion of a weakly triangular decomposition in [7] to the super case called a super weakly triangular decomposition. We show that the underlying even category of locally finite-dimensional left A-supermodules is an upper finite fully stratified category in the sense of [6, Definition 3.34] if the superalgebra A admits an upper finite super weakly triangular decomposition. In particular, when A is the locally unital superalgebra associated with the cyclotomic oriented Brauer-Clifford supercategory in [1], the Grothendieck group of the category of left A-supermodules admitting finite standard flags has a g-module structure that is isomorphic to the tensor product of an integrable lowest weight and an integrable highest weight g-module, where g is the complex Kac-Moody Lie algebra of type A2ℓ(2) (resp., B∞) if p=2ℓ+1 (resp., p=0).

Representations of toroidal and full toroidal Lie algebras over polynomial algebras

Toroidal Lie algebras are n variable generalizations of affine Kac-Moody Lie algebras. Full toroidal Lie algebra is the semidirect product of derived Lie algebra of toroidal Lie algebra and Witt algebra, also it can be thought of n-variable generalization of Affine-Virasoro algebras. Let h̃ be a Cartan subalgebra of a toroidal Lie algebra as well as full toroidal Lie algebra without containing the zero-degree central elements. In this paper, we classify the module structure on U(h̃) for all toroidal Lie algebras as well as full toroidal Lie algebras which are free U(h̃)-modules of rank 1. These modules exist only for type Al(l ≥ 1), Cl(l ≥ 2) toroidal Lie algebras and the same is true for full toroidal Lie algebras. Also, we determined the irreducibility condition for these classes of modules for both the Lie algebras.

Fiberwise criteria for twisted forms of algebraic structures

We provide a criterion for certain algebraic objects over Jacobson schemes to be forms of each other based on their behaviour at closed fibres. This criterion permits to answer a question related to current and affine Kac-Moody Lie algebras that I. Burban had asked the authors.

The [formula omitted]-symmetric down-up algebra

In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra A. The algebra A is associative, noncommutative, and infinite-dimensional. It is defined by two generators A,B and two relations called the down-up relations. In the present paper, we introduce the Z3-symmetric down-up algebra A. We define A by generators and relations. There are three generators A,B,C and any two of these satisfy the down-up relations. We describe how A is related to some familiar algebras in the literature, such as the Weyl algebra, the Lie algebras sl2 and sl3, the sl3 loop algebra, the Kac-Moody Lie algebra A2(1), the q-Weyl algebra, the quantized enveloping algebra Uq(sl2), and the quantized enveloping algebra Uq(A2(1)). We give some open problems and conjectures.

Components of $$V(\rho ) \otimes V(\rho )$$ and Dominant Weight Polyhedra for Affine Kac–Moody Lie Algebras

AbstractKostant asked the following question: Let $$\mathfrak {g}$$ g be a simple Lie algebra over the complex numbers. Let $$\lambda $$ λ be a dominant integral weight. Then, $$V(\lambda )$$ V ( λ ) is a component of $$V(\rho )\otimes V(\rho )$$ V ( ρ ) ⊗ V ( ρ ) if and only if $$\lambda \le 2 \rho $$ λ ≤ 2 ρ under the usual Bruhat–Chevalley order on the set of weights. In an earlier work with R. Chirivi and A. Maffei, the second author gave an affirmative answer to this question up to a saturation factor. The aim of the current work is to extend this result to untwisted affine Kac–Moody Lie algebra $$\mathfrak {g}$$ g associated with any simple Lie algebra $$\mathring{\mathfrak {g}}$$ g ˚ (up to a saturation factor). In fact, we prove the result for affine $$sl_n$$ s l n without any saturation factor. Our proof requires some additional techniques including the Goddard–Kent–Olive construction and study of the characteristic cone of non-compact polyhedra.

Trigonometric Lie algebras, affine Kac-Moody Lie algebras, and equivariant quasi modules for vertex algebras

In this paper, we study a family of infinite-dimensional Lie algebras XˆS, where X stands for the type: A,B,C,D, and S is an abelian group, which generalize the A,B,C,D series of trigonometric Lie algebras. Among the main results, we identify XˆS with what are called the covariant algebras of the affine Lie algebra LSˆ with respect to some automorphism groups, where LS is an explicitly defined associative algebra viewed as a Lie algebra. We then show that restricted XˆS-modules of level ℓ naturally correspond to equivariant quasi modules for affine vertex algebras related to LS. Furthermore, for any finite cyclic group S, we completely determine the structures of these four families of Lie algebras, showing that they are essentially affine Kac-Moody Lie algebras of certain types.

Chevalley involutions for Lie tori and extended affine Lie algebras

In finite-dimensional simple Lie algebras and affine Kac-Moody Lie algebras, Chevalley involutions are crucial ingredients of the modular theory. Towards establishing the modular theory for extended affine Lie algebras, we investigate the existence of “Chevalley involutions” for Lie tori and extended affine Lie algebras. We first discuss how to lift a Chevalley involution from the centerless core which is characterized to be a centerless Lie torus to the core and then to the entire extended affine Lie algebra. We then prove by a type-dependent argument the existence of Chevalley involutions for centerless Lie tori.

Classification of simple locally finite modules over the affine-Virasoro algebras

In [12], the authors classified a class of simple modules over untwised affine Kac-Moody Lie algebras, on these modules each weight vector of the positive parts of affine Kac-Moody Lie algebras acts locally finitely. In this paper, for all affine-Virasoro algebras we also classify all simple modules with the similar property. We determine that there are precisely three classes of simple modules on which each weight vector of the positive part of any affine-Virasoro algebra acts locally finitely: simple highest weight or Whittaker Virasoro algebra modules, simple highest weight or Whittaker affine Lie algebra modules, and simple highest weight or Whittaker affine-Virasoro algebra modules which are neither simple Virasoro algebra modules nor simple affine Lie algebra modules. We also obtain three equivalent conditions to characterize such simple modules over affine-Virasoro algebras.

Weak Faces of Highest Weight Modules and Root Systems

Chari and Greenstein (Adv. Math. 220(4), 1193–1221, 2009) introduced certain combinatorial subsets of the roots of a finite-dimensional simple Lie algebra $\mathfrak {g}$ , which were important in studying Kirillov–Reshetikhin modules over $U_{q}(\widehat {\mathfrak {g}})$ and their specializations. Later, Khare (J. Algebra. 455, 32–76, 2016) studied these subsets for large classes of highest weight $\mathfrak {g}$ -modules (in finite type), under the name of weak- $\mathbb {A}$ -faces (for a subgroup $\mathbb {A}$ of $(\mathbb {R},+)$ ), and more generally, ({2};{1,2})-closed subsets. These notions extend and unify the faces of Weyl polytopes as well as the aforementioned combinatorial subsets. In this paper, we consider these “discrete” notions for an arbitrary Kac–Moody Lie algebra $\mathfrak {g}$ , in three prominent settings: (a) the weights of an arbitrary highest weight $\mathfrak {g}$ -module V; (b) the convex hull of the weights of V; (c) the weights of the adjoint representation. For (a) (respectively, (b)) for all highest weight $\mathfrak {g}$ -modules V, we show that the weak- $\mathbb {A}$ -faces and ({2};{1,2})-closed subsets agree, and equal the sets of weights on the exposed faces (respectively, equal the exposed faces) of the convex hull of weights conv(wtV ). This completes the partial progress of Khare in finite type, and is novel in infinite type. Our proofs are type-free and self-contained. For (c) involving the root system, we similarly achieve complete classifications. For the weights of $\text {ad} \mathfrak {g}$ for any indecomposable Kac–Moody $\mathfrak {g}$ , we show that the weak- $\mathbb {A}$ -faces and ({2};{1,2})-closed subsets agree, and equal the Weyl group translates of the sets of weights in certain “standard parabolic faces” (which also holds for highest weight modules). This was proved by Chari and her coauthors for Δ ⊔{0} in finite type, but is novel for other types.

Affine Kac-Moody groups and Lie algebras in the language of SGA3

In infinite-dimensional Lie theory, the affine Kac-Moody Lie algebras and groups play a distinguished role due to their many applications to various areas of mathematics and physics. Underlying these infinite-dimensional objects there are closely related group schemes and Lie algebras of finite type over Laurent polynomial rings. The language of SGA3 is perfectly suited to describe such objects. The purpose of this short article is to provide a natural description of the affine Kac-Moody groups and Lie algebras using this language.

Conversations with Flaschka: Kac–Moody groups and Verblunsky coefficients

In this paper two items are discussed which in one way or another originated from conversations with Hermann Flaschka and his students. The first is an application of the Toda lattice to the question of whether there exists a complex Lie group with an exponential map associated to an indefinite type Kac–Moody Lie algebra. The second concerns a new example of the Verblunsky correspondence.

Finite Dimensional Simple Modules over Some GIM Lie Algebras

GIM Lie algebras are the generalizations of Kac–Moody Lie algebras. However, the structures of GIM Lie algebras are more complex than the latter, so they have encountered many new difficulties to study their representation theory. In this paper, we classify all finite dimensional simple modules over the GIM Lie algebra Qn+1(2,1) as well as those over Θ2n+1.

Root components for tensor product of affine Kac-Moody Lie algebra modules

Let g \mathfrak {g} be an affine Kac-Moody Lie algebra and let λ , μ \lambda , \mu be two dominant integral weights for g \mathfrak {g} . We prove that under some mild restriction, for any positive root β \beta , V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) contains V ( λ + μ − β ) V(\lambda +\mu -\beta ) as a component, where V ( λ ) V(\lambda ) denotes the integrable highest weight (irreducible) g \mathfrak {g} -module with highest weight λ \lambda . This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) . Then, we prove the corresponding geometric results including the higher cohomology vanishing on the G \mathcal {G} -Schubert varieties in the product partial flag variety G / P × G / P \mathcal {G}/\mathcal {P}\times \mathcal {G}/\mathcal {P} with coefficients in certain sheaves coming from the ideal sheaves of G \mathcal {G} -sub-Schubert varieties. This allows us to prove the surjectivity of the Gaussian map.

Integrable semi-discretisation of the Drinfel’d–Sokolov hierarchies

We propose a novel semi-discrete Kadomtsev–Petviashvili equation with two discrete and one continuous independent variables, which is integrable in the sense of having the standard and adjoint Lax pairs, from the direct linearisation framework. By performing reductions on the semi-discrete Kadomtsev–Petviashvili equation, new semi-discrete versions of the Drinfel’d–Sokolov hierarchies associated with Kac–Moody Lie algebras , , and are successfully constructed. A Lax pair involving the fraction of graded matrices is also found for each of the semi-discrete Drinfel’d–Sokolov equations. Furthermore, the direct linearisation construction guarantees the existence of exact solutions of all the semi-discrete equations discussed in the paper, providing another insight into their integrability in addition to the analysis of Lax pairs.

Representations of Involutory Subalgebras of Affine Kac–Moody Algebras

We consider the subalgebras of split real, non-twisted affine Kac–Moody Lie algebras that are fixed by the Cartan–Chevalley involution. These infinite-dimensional Lie algebras are not of Kac–Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of {mathbb {N}}-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity and work out a detailed example in the case of the affine extension of {mathfrak {e}}_8.

Subalgebra generated by ad-locally nilpotent elements of Borcherds Generalized Kac-Moody Lie algebras

We determine the Lie subalgebra gnil of a Borcherds symmetrizable generalized Kac-Moody Lie algebra g generated by ad-locally nilpotent elements and show that it is ‘essentially’ the same as the Levi subalgebra of g with its simple roots precisely the real simple roots of g.

Space, Matter and Interactions in a Quantum Early Universe Part I: Kac–Moody and Borcherds Algebras

We introduce a quantum model for the universe at its early stages, formulating a mechanism for the expansion of space and matter from a quantum initial condition, with particle interactions and creation driven by algebraic extensions of the Kac–Moody Lie algebra e9. We investigate Kac–Moody and Borcherds algebras, and we propose a generalization that meets further requirements that we regard as fundamental in quantum gravity.

On the tensor semigroup of affine Kac-Moody lie algebras

The support of the tensor product decomposition of integrable irreducible highest weight representations of a symmetrizable Kac-Moody Lie algebra g \mathfrak {g} defines a semigroup of triples of weights. Namely, given λ \lambda in the set P + P_+ of dominant integral weights, V ( λ ) V(\lambda ) denotes the irreducible representation of g \mathfrak {g} with highest weight λ \lambda . We are interested in the tensor semigroup Γ N ( g ) ≔ { ( λ 1 , λ 2 , μ ) ∈ P + 3 | V ( μ ) ⊂ V ( λ 1 ) ⊗ V ( λ 2 ) } , \begin{equation*} \Gamma _{\mathbb {N}}(\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+}^3\,|\, V(\mu )\subset V(\lambda _1)\otimes V(\lambda _2)\}, \end{equation*} and in the tensor cone Γ ( g ) \Gamma (\mathfrak {g}) it generates: Γ ( g ) ≔ { ( λ 1 , λ 2 , μ ) ∈ P + , Q 3 | ∃ N ≥ 1 V ( N μ ) ⊂ V ( N λ 1 ) ⊗ V ( N λ 2 ) } . \begin{equation*} \Gamma (\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+,{\mathbb {Q}}}^3\,|\,\exists N\geq 1 \quad V(N\mu )\subset V(N\lambda _1)\otimes V(N\lambda _2)\}. \end{equation*} Here, P + , Q P_{+,{\mathbb {Q}}} denotes the rational convex cone generated by P + P_+ . In the special case when g \mathfrak {g} is a finite-dimensional semisimple Lie algebra, the tensor semigroup is known to be finitely generated and hence the tensor cone to be convex polyhedral. Moreover, the cone Γ ( g ) \Gamma (\mathfrak {g}) is described in Belkale and Kumar [Invent. Math. 166 (2006), pp. 185–228] by an explicit finite list of inequalities. In general, Γ ( g ) \Gamma (\mathfrak {g}) is neither polyhedral, nor closed. In this article we describe the closure of Γ ( g ) \Gamma (\mathfrak {g}) by an explicit countable family of linear inequalities for any untwisted affine Lie algebra, which is the most important class of infinite-dimensional Kac-Moody algebra. This solves a Brown-Kumar’s conjecture in this case (see Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936]). The difference between the tensor cone and the tensor semigroup is measured by the saturation factors. Namely, a positive integer d d is called a saturation factor, if V ( N λ 1 ) ⊗ V ( N λ 2 ) V(N\lambda _1)\otimes V(N\lambda _2) contains V ( N μ ) V(N\mu ) for some positive integer N N then V ( d λ 1 ) ⊗ V ( d λ 2 ) V(d\lambda _1)\otimes V(d\lambda _2) contains V ( d μ ) V(d\mu ) , assuming that μ − λ 1 − λ 2 \mu -\lambda _1-\lambda _2 belongs to the root lattice. For g = s l n \mathfrak {g}={\mathfrak {sl}}_n , the famous Knutson-Tao theorem asserts that d = 1 d=1 is a saturation factor (see Knutson and Tao [J. Amer. Math. Soc. 12 (1999), pp. 1055–1090]). More generally, for any simple Lie algebra, explicit saturation factors are known. In the Kac-Moody case, Γ N ( g ) \Gamma _{\mathbb {N}}(\mathfrak {g}) is not necessarily finitely generated and hence the existence of such a factor is unclear a priori. Here, we obtain explicit saturation factors for any affine Kac-Moody Lie algebra. For example, in type A ~ n \tilde A_n , we prove that any integer d ≥ 2 d\geq 2 is a saturation factor, generalizing the case A ~ 1 \tilde A_1 shown in Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936].