Affine Kac Research Articles

An admissible level $$\widehat{\mathfrak {osp}} \left( 1 \big \vert 2 \right) $$-model: modular transformations and the Verlinde formula

The modular properties of the simple vertex operator superalgebra associated with the affine Kac–Moody superalgebraosp(1|2) at level −5 4 are investigated. After classifying the relaxed highest-weight modules over this vertex operator superalgebra, the characters and supercharacters of the simple weight modules are computed and their modular transforms are determined. This leads to a complete list of the Grothendieck fusion rules by way of a continuous superalgebraic analog of the Verlinde formula. All Grothendieck fusion coefficients are observed to be non-negative integers. These results indicate that the extension to general admissible levels will follow using the same methodology once the classification of relaxed highest-weight modules is completed.

Representations of E7, equivalent combinatorics and algebraic varieties

In our earlier work on a new functor from [Formula: see text]-Mod to [Formula: see text]-Mod, we found a one-parameter ([Formula: see text]) family of inhomogeneous first-order differential operator representations of the simple Lie algebra of type [Formula: see text] in [Formula: see text] variables. Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector [Formula: see text], we prove that the space forms an irreducible [Formula: see text]-module for any [Formula: see text] if [Formula: see text] is not on an explicitly given projective algebraic variety. Certain equivalent combinatorial properties of the basic oscillator representation of [Formula: see text] over its 27-dimensional module play key roles in our proof. Our result can also be used to study free bosonic field irreducible representations of the corresponding affine Kac–Moody algebra.

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.

PRESENTATIONS OF AFFINE KAC–MOODY GROUPS

How many generators and relations does $\text{SL}\,_{n}(\mathbb{F}_{q}[t,t^{-1}])$ need? In this paper we exhibit its explicit presentation with $9$ generators and $44$ relations. We investigate presentations of affine Kac–Moody groups over finite fields. Our goal is to derive finite presentations, independent of the field and with as few generators and relations as we can achieve. It turns out that any simply connected affine Kac–Moody group over a finite field has a presentation with at most 11 generators and 70 relations. We describe these presentations explicitly type by type. As a consequence, we derive explicit presentations of Chevalley groups $G(\mathbb{F}_{q}[t,t^{-1}])$ and explicit profinite presentations of profinite Chevalley groups $G(\mathbb{F}_{q}[[t]])$.

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 family of representations of the affine Lie superalgebra [formula omitted

In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra glm|n(C). The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac–Moody Lie superalgebra glm|nˆ(C) on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter μ, and we also obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter μ is nonzero.

The next 16 higher spin currents and three-point functions in the large mathcal{N}=4 holography

By using the known operator product expansions (OPEs) between the lowest 16 higher spin currents of spins (1, frac{3}{2}, frac{3}{2}, frac{3}{2}, frac{3}{2}, 2,2,2,2,2,2, frac{5}{2}, frac{5}{2}, frac{5}{2}, frac{5}{2}, 3) in an extension of the large mathcal{N}=4 linear superconformal algebra, one determines the OPEs between the lowest 16 higher spin currents in an extension of the large mathcal{N}=4 nonlinear superconformal algebra for generic N and k. The Wolf space coset contains the group G =mathrm{SU}(N+2) and the affine Kac–Moody spin 1 current has the level k. The next 16 higher spin currents of spins (2,frac{5}{2}, frac{5}{2}, frac{5}{2}, frac{5}{2}, 3,3,3,3,3,3, frac{7}{2}, frac{7}{2}, frac{7}{2}, frac{7}{2},4) arise in the above OPEs. The most general lowest higher spin 2 current in this multiplet can be determined in terms of affine Kac–Moody spin frac{1}{2}, 1 currents. By careful analysis of the zero mode (higher spin) eigenvalue equations, the three-point functions of bosonic higher spin 2, 3, 4 currents with two scalars are obtained for finite N and k. Furthermore, we also analyze the three-point functions of bosonic higher spin 2, 3, 4 currents in the extension of the large mathcal{N}=4 linear superconformal algebra. It turns out that the three-point functions of higher spin 2, 3 currents in the two cases are equal to each other at finite N and k. Under the large (N, k) ’t Hooft limit, the two descriptions for the three-point functions of higher spin 4 current coincide with each other. The higher spin extension of SO(4) Knizhnik Bershadsky algebra is described.

Structure of parabolically induced modules for affine Kac–Moody algebras

The main result of the paper establishes the irreducibility of a large family of nonzero central charge induced modules over Affine Lie algebras for any non standard parabolic subalgebra. It generalizes all previously known partial results and provides a construction of many new irreducible modules.

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].

A remark on boundary level admissible representations

We point out that it is immediate by our character formula that in the case of a boundary level the characters of admissible representations of affine Kac–Moody algebras and the corresponding W-algebras decompose in products in terms of the Jacobi form ϑ11(τ,z).

A combinatorial formula expressing periodic R-polynomials

In 1980, Lusztig introduced the periodic Kazhdan–Lusztig polynomials, which are conjectured to have important information about the characters of irreducible modules of a reductive group over a field of positive characteristic, and also about those of an affine Kac–Moody algebra at the critical level. The periodic Kazhdan–Lusztig polynomials can be computed by using another family of polynomials, called the periodic R-polynomials. In this paper, we prove a (closed) combinatorial formula expressing periodic R-polynomials in terms of the “doubled” Bruhat graph associated to a finite Weyl group and a finite root system.

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).

Affine Weyl group multiple Dirichlet series: type

We define a multiple Dirichlet series whose group of functional equations is the Weyl group of the affine Kac–Moody root system $\widetilde{A}_{n}$, generalizing the theory of multiple Dirichlet series for finite Weyl groups. The construction is over the rational function field $\mathbb{F}_{q}(t)$, and is based upon four natural axioms from algebraic geometry. We prove that the four axioms yield a unique series with meromorphic continuation to the largest possible domain and the desired infinite group of symmetries.

Tau functions and Virasoro symmetries for Drinfeld–Sokolov hierarchies

For each Drinfeld–Sokolov integrable hierarchy associated to affine Kac–Moody algebra, we obtain a uniform construction of tau function by using tau-symmetric Hamiltonian densities, moreover, we represent its Virasoro symmetries as linear/nonlinear actions on the tau function. The relations between the tau function constructed in this paper and those defined for particular cases of Drinfeld–Sokolov hierarchies in the literature are clarified. We also show that, whenever the affine Kac–Moody algebra is simply-laced or twisted, the tau function of the Drinfeld–Sokolov hierarchy coincides with the solution of the corresponding Kac–Wakimoto hierarchy constructed from the principal vertex operator realization of the affine algebra.

Admissible positive systems of affine Kac–Moody Lie algebras: The twisted cases

We introduce admissible positive systems and Hermitian real forms of affine twisted Kac–Moody Lie algebras, and show that a real form has admissible positive system if and only if it is Hermitian. We use the Vogan diagrams to classify the Hermitian real forms, and show that their symmetric spaces carry complex structures. The affine non-twisted Kac–Moody Lie algebras have been treated in an earlier work, and this article deals with the twisted cases.

A new characterization of Kac–Moody–Malcev superalgebras

In the past two decades there has been great attention to Lie (super)algebras, which are extensions of affine Kac–Moody Lie (super)algebras, in certain typical or axiomatic approaches. These Lie (super)algebras have been mostly studied under variations of the name “extended affine Lie (super)algebras”. We show that certain classes of Malcev (super)algebras also can be put in this framework. This in particular allows us to provide new examples of Malcev (super)algebras which extend the known Kac–Moody Malcev (super)algebras.

Presentation of affine Kac–Moody groups over rings

Tits has defined Steinberg groups and Kac-Moody groups for any root system and any commutative ring R. We establish a Curtis-Tits-style presentation for the Steinberg group St of any rank > 2 irreducible affine root system, for any R. Namely, St is the direct limit of the Steinberg groups coming from the 1- and 2-node subdiagrams of the Dynkin diagram. This leads to a completely explicit presentation. Using this we show that St is finitely presented if the rank is > 3 and R is finitely generated as a ring, or if the rank is 3 and R is finitely generated as a module over a subring generated by finitely many units. Similar results hold for the corresponding Kac-Moody groups when R is a Dedekind domain of arithmetic type.

A framework of Rogers–Ramanujan identities and their arithmetic properties

The two Rogers-Ramanujan $q$-series \[ \sum_{n=0}^{\infty}\frac{q^{n(n+\sigma)}}{(1-q)\cdots (1-q^n)}, \] where $\sigma=0,1$, play many roles in mathematics and physics. By the Rogers-Ramanujan identities, they are essentially modular functions. Their quotient, the Rogers-Ramanujan continued fraction, has the special property that its singular values are algebraic integral units. We find a framework which extends the Rogers-Ramanujan identities to doubly-infinite families of $q$-series identities. If $a\in\{1,2\}$ and $m,n\geq 1$, then we have \[ \sum_{\substack{\lambda \lambda_1\leq m}} q^{a|\lambda|} P_{2\lambda}(1,q,q^2,\dots;q^n) =\textrm{"infinite product modular function"}, \] where the $P_{\lambda}(x_1,x_2,\dots;q)$ are Hall-Littlewood polynomials. These $q$-series are specialized characters of affine Kac--Moody algebras. Generalizing the Rogers-Ramanujan continued fraction, we prove in the case of $\textrm{A}_{2n}^{(2)}$ that the relevant $q$-series quotients are integral units.

JOSEPH IDEALS AND LISSE MINIMAL -ALGEBRAS

We consider a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac–Moody algebras and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closures. As an application we obtain a new family of lisse ($C_{2}$-cofinite)$W$-algebras that are not coming from admissible representations of affine Kac–Moody algebras.

Congruence Subgroups of Lattices in Rank 2 Kac–Moody Groups over Finite Fields

Let G be a rank 2 complete affine or hyperbolic simply-connected Kac–Moody group over a finite field k. Then G is locally compact and totally disconnected. Let B− = HU− be the negative minimal parabolic subgroup of G, where H is the analog of a diagonal subgroup and U− is generated by all negative real root groups. Let w1 and w2 be the generators of the Weyl group. Let be the negative standard parabolic subgroup of G corresponding to w1. It is known that the subgroups U−, B− and , are nonuniform lattice subgroups of G. Here we construct an infinite sequence of congruence subgroups of as natural generalizations of the corresponding notions for lattices in Lie groups. We also show that the group U− contains analogous congruence subgroups. Our technique involves determining graphs of groups presentations for U−, B−, and with the fundamental apartment of the Bruhat–Tits tree X a quotient graph for U− and for B− on X. When k = 𝔽q and q = 2s, the graph of groups for has the the positive half of the fundamental apartment as quotient graph. We explicitly construct the graphs of groups for the principal (level 1) congruence subgroup of and the analogous subgroups of U− giving generalized amalgam presentations for them.