Affine Kac Research Articles

The Universe Inside Hall Algebras of Coherent Sheaves on Toric Resolutions

Abstract Let $\mathfrak {g}\neq \mathfrak {s}\mathfrak {o}_{8}$ be a simple Lie algebra of type $A,D,E$ with $\widehat {\mathfrak {g}}$ the corresponding affine Kac–Moody algebra and $\mathfrak {n}_{+}\subset \widehat {\mathfrak {g}}$ is the standard positive nilpotent subalgebra. Given $\mathfrak {n}_{+}$ as above, we provide an infinite collection of cyclic finite abelian subgroups of $SL_{3}(\mathbb {C})$ with the following properties. Let $G$ be any group in the collection, $Y=G\operatorname {-}\mbox {Hilb}(\mathbb {C}^{3})$, and $\Psi : D^{b}_{G}(Coh(\mathbb {C}^{3}))\rightarrow D^{b}(Coh(Y))$ the derived equivalence of Bridgeland, King, and Reid. We present an (explicitly described) subset of objects in $Coh_{G}(\mathbb {C}^{3})$, s.t. the Hall algebra generated by their images under $\Psi $ is isomorphic to $U(\mathfrak {n}_{+})$. In case the field $\Bbbk $ (in place of $\mathbb {C}$) is finite and $\mbox {char}(\Bbbk )$ is coprime with the order of $G$, we establish isomorphisms of the corresponding “counting” Ringel–Hall algebras and the specializations of quantized universal enveloping algebras $U_{v}(\mathfrak {n}_{+})$ at $v=\sqrt {|\Bbbk |}$.

Geometrical correspondence of the Miura transformation induced from affine Kac–Moody algebras

AbstractThe Miura transformation plays a crucial role in the study of integrable systems. There have been various extensions of the Miura transformation, which have been used to relate different kinds of integrable equations and to classify the bi‐Hamiltonian structures. In this paper, we are mainly concerned with the geometric aspects of the Miura transformation. The generalized Miura transformations from the mKdV‐type hierarchies to the KdV‐type hierarchies are constructed under both algebraic and geometric settings. It is shown that the Miura transformations not only relate integrable curve flows in different geometries but also induce the transition between different moving frames. Moreover, the Miura transformation gives the factorization of generating operators of constraint Gelfand–Dickey hierarchy. Other geometric formulations are also investigated.

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.

On string functions and double-sum formulas

String functions are important building blocks of characters of integrable highest modules over affine Kac--Moody algebras. Kac and Peterson computed string functions for affine Lie algebras of type $A_{1}^{(1)}$ in terms of Dedekind eta functions. We produce new relations between string functions by writing them as double-sums and then using certain symmetry relations. We evaluate the series using special double-sum formulas that express Hecke-type double-sums in terms of Appell--Lerch functions and theta functions, where we point out that Appell--Lerch functions are the building blocks of Ramanujan's classical mock theta functions.

Weight Multiplicities and Young Tableaux Through Affine Crystals

The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine Kac–Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the ( ( spin ) ) rigid tableaux, and prove that they are equinumerous to the weight multiplicities of the highest weight modules under our consideration. These new classes of Young tableaux arise from crystal basis elements for dominant maximal weights of the integrable highest weight modules over affine Kac–Moody algebras. By applying combinatorics of tableaux such as the Robinson–Schensted algorithm and new insertion schemes, and using integrals over orthogonal groups, we reveal hidden structures in the sets of weight multiplicities and obtain explicit closed formulas for the weight multiplicities. In particular we show that some special families of weight multiplicities form the Pascal, Catalan, Motzkin, Riordan and Bessel triangles.

3-Dimensional mixed BF theory and Hitchin’s integrable system

The affine Gaudin model, associated with an untwisted affine Kac–Moody algebra, is known to arise from a certain gauge fixing of 4-dimensional mixed topological–holomorphic Chern–Simons theory in the Hamiltonian framework. We show that the finite Gaudin model, associated with a finite-dimensional semisimple Lie algebra, or more generally the tamely ramified Hitchin system on an arbitrary Riemann surface, can likewise be obtained from a similar gauge fixing of 3-dimensional mixed BF theory in the Hamiltonian framework.

Folded quantum integrable models and deformed W-algebras

We propose a novel quantum integrable model for every non-simply laced simple Lie algebra \({{\mathfrak {g}}}\), which we call the folded integrable model. Its spectra correspond to solutions of the Bethe Ansatz equations obtained by folding the Bethe Ansatz equations of the standard integrable model associated with the quantum affine algebra \(U_q(\widehat{{{\mathfrak {g}}}'})\) of the simply laced Lie algebra \({{\mathfrak {g}}}'\) corresponding to \({{\mathfrak {g}}}\). Our construction is motivated by the analysis of the second classical limit of the deformed W-algebra of \({{\mathfrak {g}}}\), which we interpret as a “folding” of the Grothendieck ring of finite-dimensional representations of \(U_q(\widehat{{{\mathfrak {g}}}'})\). We conjecture, and verify in a number of cases, that the spaces of states of the folded integrable model can be identified with finite-dimensional representations of \(U_q({}^L{\widehat{{{\mathfrak {g}}}}})\), where \(^L{\widehat{{{\mathfrak {g}}}}}\) is the (twisted) affine Kac–Moody algebra Langlands dual to \({\widehat{{{\mathfrak {g}}}}}\). We discuss the analogous structures in the Gaudin model which appears in the limit \(q \rightarrow 1\). Finally, we describe a conjectural construction of the simple \({{\mathfrak {g}}}\)-crystals in terms of the folded q-characters.

Virasoro constraints for Drinfeld–Sokolov hierarchies and equations of Painlevé type

We construct a tau cover of the generalized Drinfeld–Sokolov hierarchy associated with an arbitrary affine Kac–Moody algebra with gradations s ⩽ 1 $\mathrm{s}\leqslant \mathbb {1}$ and derive its Virasoro symmetries. By imposing the Virasoro constraints we obtain solutions of the Drinfeld–Sokolov hierarchy of Witten–Kontsevich and of Brezin–Gross–Witten types, and of those characterized by certain ordinary differential equations of Painlevé type. We also show the existence of affine Weyl group actions on the space of solutions of such ordinary differential equations, which generalizes the theory of Noumi and Yamada on affine Weyl group symmetries of the Painlevé-type equations.

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.

Tensor Hierarchy Algebra Extensions of Over-Extended Kac\u2013Moody Algebras

Tensor hierarchy algebras are infinite-dimensional generalisations of Cartan-type Lie superalgebras. They are not contragredient, exhibiting an asymmetry between positive and negative levels. These superalgebras have been a focus of attention due to the fundamental rôle they play for extended geometry. In the present paper, we examine tensor hierarchy algebras which are super-extensions of over-extended (often, hyperbolic) Kac–Moody algebras. They contain novel algebraic structures. Of particular interest is the extension of a over-extended algebra by its fundamental module, an extension that contains and generalises the extension of an affine Kac–Moody algebra by a Virasoro derivation L_1. A conjecture about the complete superalgebra is formulated, relating it to the corresponding Borcherds superalgebra.

Simple Modules for Affine Vertex Algebras in the Minimal Nilpotent Orbit

Abstract We explicitly construct, in terms of Gelfand–Tsetlin tableaux, a new family of simple positive energy representations for the simple affine vertex algebra $V_k(\mathfrak{s}\mathfrak{l}_{n+1})$ in the minimal nilpotent orbit of $\mathfrak{s}\mathfrak{l}_{n+1}$. These representations are quotients of induced modules over the affine Kac–Moody algebra $\widehat{\mathfrak{s}\mathfrak{l}}_{n+1} $ and include in particular all admissible simple highest weight modules and all simple modules induced from $\mathfrak{s}\mathfrak{l}_2$. Any such simple module in the minimal nilpotent orbit has bounded weight multiplicities.

ON CLASSICAL LIMITS OF BETHE SUBALGEBRAS IN YANGIANS

The Yangian Y (𝔤) of a simple Lie algebra 𝔤 can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra of the current algebra \( U\left(\mathfrak{g}\left[t\right]\right) \) and the coordinate ring of the first congruence subgroup \( \mathcal{O}\left({G}_1\left[\left[{t}^{-1}\right]\right]\right). \) Both of these algebras are obtained from the Yangian by taking the associated graded with respect to an appropriate filtration on Y (𝔤).Bethe subalgebras B(C) in Y (𝔤) form a natural family of commutative subalgebras depending on a group element C of the adjoint group G. The images of these algebras in tensor products of fundamental representations give all integrals of the quantum XXX Heisenberg magnet chain.We describe the associated graded of Bethe subalgebras in the Yangian Y (𝔤) of a simple Lie algebra 𝔤 as subalgebras in \( U\left(\mathfrak{g}\left[t\right]\right) \) and in \( \mathcal{O}\left({G}_1\left[\left[{t}^{-1}\right]\right]\right) \) for all semisimple C ∈ G. In particular, we show that the associated graded in \( U\left(\mathfrak{g}\left[t\right]\right) \) of the Bethe subalgebra B(E) assigned to the unity element of G is the universal Gaudin subalgebra of \( U\left(\mathfrak{g}\left[t\right]\right) \) obtained from the center of the corresponding affine Kac–Moody algebra \( \hat{\mathfrak{g}} \) at the critical level. This generalizes Talalaev’s formula for generators of the universal Gaudin subalgebra to 𝔤 of any type. In particular, this shows that higher Hamiltonians of the Gaudin magnet chain can be quantized without referring to the Feigin–Frenkel center at the critical level.Using our general result on the associated graded of Bethe subalgebras, we compute some limits of Bethe subalgebras corresponding to regular semisimple C ∈ G as C goes to an irregular semisimple group element C0. We show that this limit is the product of the smaller Bethe subalgebra B(C0) and a quantum shift of argument subalgebra in the universal enveloping algebra of the centralizer of C0 in 𝔤. This generalizes the Nazarov–Olshansky solution of Vinberg’s problem on quantization of the (Mishchenko–Fomenko) shift of argument subalgebras.

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.

4D Chern\u2013Simons theory and affine Gaudin models

We relate two formalisms recently proposed for describing classical integrable field theories. The first (Costello and Yamazaki in Gauge Theory and Integrability, III, 2019) is based on the action of four-dimensional Chern–Simons theory introduced and studied by Costello, Witten and Yamazaki. The second (Costello and Yamazaki, in Gauge Theory and Integrability, III, 2017) makes use of classical generalised Gaudin models associated with untwisted affine Kac–Moody algebras.

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.

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.

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.

Positive Energy Representations of Affine Vertex Algebras

We construct new families of positive energy representations of affine vertex algebras together with their free field realizations by using localization technique. We introduce the twisting functor T_\alpha on the category of modules over affine Kac--Moody algebra \widehat{g}_\kappa of level \kappa for any positive root \alpha of g, and the Wakimoto functor from a certain category of g-modules to the category of smooth \widehat{g}_\kappa-modules. These two functors commute and the image of the Wakimoto functor consists of relaxed Wakimoto \widehat{g}_\kappa-modules. In particular, applying the twisting functor T_\alpha to the relaxed Wakimoto \widehat{g}_\kappa-module whose top degree component is isomorphic to the Verma g-module M^g_b(\lambda), we obtain the relaxed Wakimoto \widehat{g}_\kappa-module whose top degree component is isomorphic to the \alpha-Gelfand--Tsetlin g-module W^g_b(\lambda, \alpha). We show that the relaxed Verma module and relaxed Wakimoto module whose top degree components are such \alpha-Gelfand--Tsetlin modules, are isomorphic generically. This is an analogue of the result of E.Frenkel for Wakimoto modules both for critical and non-critical level. For a parabolic subalgebra p of g we construct a large family of admissible g-modules as images under the twisting functor of generalized Verma modules induced from p. In this way, we obtain new simple positive energy representations of simple affine vertex algebras.

Darboux Transforms for the $${\hat{B}}_n^{(1)}$$-Hierarchy

The $${\hat{B}}_n^{(1)}$$ -hierarchy is constructed from the standard splitting of the affine Kac–Moody algebra $${\hat{B}}_n^{(1)}$$ , the Drinfeld–Sokolov $${\hat{B}}_n^{(1)}$$ –KdV hierarchy is obtained by pushing down the $${\hat{B}}_n^{(1)}$$ -flows along certain gauge orbit to a cross section of the gauge action. In this paper, we