Affine Kac-Moody Algebras Research Articles

Teleparallel Geroch geometry

We construct the teleparallel dynamics for extended geometry where the structure algebra is (an extension of) an untwisted affine Kac-Moody algebra. This provides a geometrisation of the Geroch symmetry appearing on dimensional reduction of a gravitational theory to two dimensions. The formalism is adapted to the underlying tensor hierarchy algebra, and will serve as a stepping stone towards the geometrisation of other infinite-dimensional, e.g. hyperbolic, symmetries.

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 $v=\sqrt {|\Bbbk |}$.

Classical Whittaker modules for the classical affine Kac-Moody algebras

Inspired by Sugawara operators, we introduce a new basis for the universal non-degenerate Whittaker module of level κ over the classical affine Kac-Moody algebra of type BN(1), CN(1) or DN(1). This basis is different from the one derived from the PBW basis and would help us to understand the structure of Whittaker modules. As a result, we classify simple non-degenerate Whittaker modules for the classical affine Kac-Moody algebras whether at the critical or noncritical level.

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.

Characterization of simple smooth modules

In this paper, we characterize simple smooth modules over some infinite-dimensional Z-graded Lie algebras. More precisely, we prove that if one specific positive root element of a Z-graded Lie algebra g locally finitely acts on a simple g-module V, then V is a smooth g-module. These infinite-dimensional Z-graded Lie algebras include the Virasoro algebra, affine-Virasoro algebras, the (twisted, mirror) Heisenberg-Virasoro algebras, the planar Galilean conformal algebra, and many others. This result for untwisted affine Kac-Moody algebras holds unless we change the condition from “locally finitely” to “locally nilpotently”. We also show that these are not the case for the Heisenberg algebra.

Wild Local Structures of Automorphic Lie Algebras

We study automorphic Lie algebras using a family of evaluation maps parametrised by the representations of the associative algebra of functions. This provides a descending chain of ideals for the automorphic Lie algebra which is used to prove that it is of wild representation type. We show that the associated quotients of the automorphic Lie algebra are isomorphic to twisted truncated polynomial current algebras. When a simple Lie algebra is used in the construction, this allows us to describe the local Lie structure of the automorphic Lie algebra in terms of affine Kac-Moody algebras.

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.

Generalized imaginary Verma and Wakimoto modules

We develop a general technique of constructing new irreducible weight modules for any affine Kac-Moody algebra using the parabolic induction, in the case when the Levi factor of a parabolic subalgebra is infinite-dimensional and the central charge is nonzero. Our approach unifies and generalizes all previously known results with imposed restrictions on inducing modules. We also define generalized imaginary Wakimoto modules which provide an explicit realization for generic generalized imaginary Verma modules.

On Minimal Tilting Complexes in Highest Weight Categories

We explain the construction of minimal tilting complexes for objects of highest weight categories and we study in detail the minimal tilting complexes for standard objects and simple objects. For certain categories of representations of complex simple Lie algebras, affine Kac-Moody algebras and quantum groups at roots of unity, we relate the multiplicities of indecomposable tilting objects appearing in the terms of these complexes to the coefficients of Kazhdan-Lusztig polynomials.

Affine Lie algebra representations induced from Whittaker modules

We use induction from parabolic subalgebras with infinite-dimensional Levi factor to construct new families of irreducible representations for arbitrary affine Kac-Moody algebras. Our first construction defines a functor from the category of Whittaker modules over the Levi factor of a parabolic subalgebra to the category of modules over the affine Lie algebra. The second functor sends tensor products of a module over the affine part of the Levi factor (in particular any weight module) and of a Whittaker module over the complement Heisenberg subalgebra to the affine Lie algebra modules. Both functors preserve irreducibility when the central charge is nonzero.

Affine Kac-Moody Algebras and Tau-Functions for the Drinfeld-Sokolov Hierarchies: the Matrix-Resolvent Method

For each affine Kac-Moody algebra $X_n^{(r)}$ of rank $\ell$, $r=1,2$, or $3$, and for every choice of a vertex $c_m$, $m=0,\dots,\ell$, of the corresponding Dynkin diagram, by using the matrix-resolvent method we define a gauge-invariant tau-structure for the associated Drinfeld-Sokolov hierarchy and give explicit formulas for generating series of logarithmic derivatives of the tau-function in terms of matrix resolvents, extending the results of [Mosc. Math. J. 21 (2021), 233-270, arXiv:1610.07534] with $r=1$ and $m=0$. For the case $r=1$ and $m=0$, we verify that the above-defined tau-structure agrees with the axioms of Hamiltonian tau-symmetry in the sense of [Adv. Math. 293 (2016), 382-435, arXiv:1409.4616] and [arXiv:math.DG/0108160].

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.

Real Hamiltonian forms of affine Toda field theories: Spectral aspects

The paper is devoted to real Hamiltonian forms of 2-dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral theory of the associated Lax operators. Real Hamiltonian forms are a special type of "reductions" of Hamiltonian systems, similar to real forms of semi-simple Lie algebras. Examples of real Hamiltonian forms of affine Toda field theories related to exceptional complex untwisted affine Kac-Moody algebras are studied. Along with the associated Lax representations, we also formulate the relevant Riemann-Hilbert problems and derive the minimal sets of scattering data that determine uniquely the scattering matrices and the potentials of the Lax operators.

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.

The Brylinski Filtration for Affine Kac-Moody Algebras and Representations of ${\mathscr{W}}$-algebras

The Brylinski Filtration for Affine Kac-Moody Algebras and Representations of ${\mathscr{W}}$-algebras

Characters, coadjoint orbits and Duistermaat-Heckman integrals

The asymptotics of characters χkλ(exp⁡(h/k)) of irreducible representations of a compact Lie group G for large values of the scaling factor k are given by Duistermaat-Heckman (DH) integrals over coadjoint orbits of G. This phenomenon generalises to coadjoint orbits of central extensions of loop groups LGˆ and of diffeomorphisms of the circle Diffˆ(S1). We show that the asymptotics of characters of integrable modules of affine Kac-Moody algebras and of the Virasoro algebra factorize into a divergent contribution of the standard form and a convergent contribution which can be interpreted as a formal DH orbital integral.For some Virasoro modules, our results match formal DH integrals recently computed by Stanford and Witten. In this case, the k-scaling has the same origin as the one which gives rise to classical conformal blocks. Furthermore, we consider reduced spaces of Virasoro coadjoint orbits and we suggest a new invariant which replaces symplectic volume in the infinite dimensional situation. We also consider other modules of the Virasoro algebra (in particular, the modules corresponding to minimal models) and we obtain DH-type expressions which do not correspond to any Virasoro coadjoint orbits.We introduce volume functions V(x) corresponding to formal DH integrals over coadjoint orbits of the Virasoro algebra and show that they are related by the Hankel transform to spectral densities ρ(E) recently studied by Saad, Shenker and Stanford.

Generalised holonomies and K(E9)

The involutory subalgebra K( mathfrak{e} 9) of the affine Kac-Moody algebra mathfrak{e} 9 was recently shown to admit an infinite sequence of unfaithful representations of ever increasing dimensions [1]. We revisit these representations and describe their associated ideals in more detail, with particular emphasis on two chiral versions that can be constructed for each such representation. For every such unfaithful representation we show that the action of K( mathfrak{e} 9) decomposes into a direct sum of two mutually commuting (‘chiral’ and ‘anti-chiral’) parabolic algebras with Levi subalgebra mathfrak{so} (16)+ ⊕ mathfrak{so} (16)−. We also spell out the consistency conditions for uplifting such representations to unfaithful representations of K( mathfrak{e} 10). From these results it is evident that the holonomy groups so far discussed in the literature are mere shadows (in a Platonic sense) of a much larger structure.

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.