Affine Kac Moody Lie Algebras Research Articles

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.

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.

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.

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.

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.

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].

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.

THE PBW THEOREM FOR AFFINE YANGIANS

We prove that the Yangian associated to an untwisted symmetric affine Kac-Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed by the authors in arXiv:1407.7994 as an algebraic formalism of the cohomological Hall algebras. As a consequence, we obtain the Poincare-Birkhoff-Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given recently by Guay-Regelskis-Wendlandt.

Q-Virasoro algebra and affine Kac-Moody Lie algebras

In this paper, we introduce an infinite-dimensional Lie algebra DS for any abelian group S. If S is the additive group of integers, DS reduces to the q-Virasoro algebra Dq introduced by Belov and Chaltikian in the study of lattice conformal theories. Guided by the theory of equivariant quasi modules for vertex algebras, we introduce another Lie algebra gS with S as an automorphism group and we prove that DS is isomorphic to the S-covariant algebra of the affine Lie algebra gSˆ. We then relate restricted DS-modules of level ℓ∈C to equivariant quasi modules for the vertex algebra VgSˆ(ℓ,0) associated to gSˆ with level ℓ. Furthermore, we establish an intrinsic connection between the q-Virasoro algebra Dq and affine Kac-Moody Lie algebras. More specifically, we show that if S is a finite abelian group of order 2l+1, DS is isomorphic to the affine Kac-Moody algebra of type Bl(1).

Extended affine Lie superalgebras containing Cartan subalgebras

Multiloop (super)algebras are of great importance in the literature. Affine Kac Moody Lie (super)algebras and almost all extended affine Lie algebras are realized using an affinization process on loop superalgebras and multiloop algebras respectively. An affinization process for multiloop superalgebras turns to Lie superalgebras satisfying certain properties which are the super version of extended affine Lie algebras. In this paper we systematically study this super version called extended affine Lie superalgebras. We first derive the intrinsic properties of extended affine Lie superalgebras and then study those which are simple.

Commutative post-Lie algebra structures on Kac–Moody algebras

We determine commutative post-Lie algebra structures on some infinite-dimensional Lie algebras. We show that all commutative post-Lie algebra structures on loop algebras are trivial. This extends the results for finite-dimensional perfect Lie algebras. Furthermore, we show that all commutative post-Lie algebra structures on affine Kac–Moody Lie algebras are “almost trivial”.

Simple Singular Whittaker Modules Over the Schrödinger Algebra

There are no simple singular Whittaker modules over most of important algebras, such as simple complex finite-dimensional Lie algebras, affine Kac–Moody Lie algebras, the Virasoro algebra, the Heisenberg–Virasoro algebra and the Schrodinger–Witt algebra. In this paper, however, we construct simple singular Whittaker modules over the Schrodinger algebra. Moreover, simple singular Whittaker modules over the Schrodinger algebra are classified. As a result, simple modules for the Schrodinger algebra which are locally finite over the positive part are completely classified. We also give characterizations of simple highest weight modules and simple singular Whittaker modules.

Vertex Algebras and Coordinate Rings of Semi-infinite Flags

The direct sum of irreducible level one integrable representations of affine Kac-Moody Lie algebra of (affine) type $ADE$ carries a structure of $P/Q$-graded vertex operator algebra. There exists a filtration on this direct sum studied by Kato and Loktev such that the corresponding graded vector space is a direct sum of global Weyl modules. The associated graded space with respect to the dual filtration is isomorphic to the homogenous coordinate ring of semi-infinite flag variety. We describe the ring structure in terms of vertex operators and endow the homogenous coordinate ring with a structure of $P/Q$-graded vertex operator algebra. We use the vertex algebra approach to derive semi-infinite Pl\"ucker-type relations in the homogeneous coordinate ring.

Algebraic proofs for shallow water bi–Hamiltonian systems for three cocycle of the semi-direct product of Kac–Moody and Virasoro Lie algebras

Abstract We prove new theorems related to the construction of the shallow water bi-Hamiltonian systems associated to the semi-direct product of Virasoro and affine Kac–Moody Lie algebras. We discuss associated Verma modules, coadjoint orbits, Casimir functions, and bi-Hamiltonian systems.

Vertex-algebraic structure of principal subspaces of the basic modules for twisted affine Lie algebras of type [formula omitted

We obtain a presentation of principal subspaces of the basic modules for the twisted affine Kac–Moody Lie algebras of type A2n−1(2), Dn(2) and E6(2). Using this presentation, we construct exact sequences among these principal subspaces, and use these exact sequences to obtain recursions satisfied by graded dimensions of the principal subspaces. Solving these recursions, we obtain the graded dimensions of the principal subspaces.

A note on the zeroth products of Frenkel–Jing operators

Let [Formula: see text] be an untwisted affine Kac–Moody Lie algebra. The top of every irreducible highest weight integrable [Formula: see text]-module is the finite-dimensional irreducible [Formula: see text]-module, where the action of the simple Lie algebra [Formula: see text] is given by zeroth products arising from the underlying vertex operator algebra theory. Motivated by this fact, we consider zeroth products of level [Formula: see text] Frenkel–Jing operators corresponding to Drinfeld realization of the quantum affine algebra [Formula: see text]. By applying these products, which originate from the quantum vertex algebra theory developed by Li, on the extension of Koyama vertex operator [Formula: see text], we obtain an infinite-dimensional vector space [Formula: see text]. Next, we introduce an associative algebra [Formula: see text], a certain quantum analogue of the universal enveloping algebra [Formula: see text], and construct some infinite-dimensional [Formula: see text]-modules [Formula: see text] corresponding to the finite-dimensional irreducible [Formula: see text]-modules [Formula: see text]. We show that the space [Formula: see text] carries a structure of an [Formula: see text]-module and, furthermore, we prove that the [Formula: see text]-module [Formula: see text] is isomorphic to the [Formula: see text]-module [Formula: see text].

Standard cocycles: Variations on themes of C. Kassel’s and R. Wilson’s

Abstract Central extensions of Lie algebras can be understood and classified by means of 2-cocycles. The Lie algebras we are interested in are “twisted forms” (defined by Galois descent) of algebras of the form 𝔤 ⊗ k R {{\mathfrak{g}}\otimes_{k}R} with 𝔤 {{\mathfrak{g}}} split finite-dimensional simple over a base field k of characteristic 0 and R a commutative unital and associative k-algebra (such algebras are ubiquitous in modern infinite-dimensional Lie theory). We introduce a special type of cocycle that we called standard. Our main result shows that any cocycle is cohomologous to a unique standard cocycle. As an application we give a precise description of the universal central extension of the twisted forms of 𝔤 ⊗ k R {{\mathfrak{g}}\otimes_{k}R} mentioned above. This yields a new proof of a classic theorem of C. Kassel [8]. For multiloop algebras, we obtain a “twisted” version of Kassel’s result (which is due to R. Wilson [21] in the case of the affine Kac–Moody Lie algebras).