Kac Moody Research Articles

The Formal Shift Operator on the Yangian Double

Abstract Let ${\mathfrak{g}}$ be a symmetrizable Kac–Moody algebra with associated Yangian $Y_\hbar{\mathfrak{g}}$ and Yangian double $\textrm{D}Y_\hbar{\mathfrak{g}}$. An elementary result of fundamental importance to the theory of Yangians is that, for each $c\in{\mathbb{C}}$, there is an automorphism $\tau _c$ of $Y_\hbar{\mathfrak{g}}$ corresponding to the translation $t\mapsto t+c$ of the complex plane. Replacing $c$ by a formal parameter $z$ yields the so-called formal shift homomorphism $\tau _z$ from $Y_\hbar{\mathfrak{g}}$ to the polynomial algebra $Y_\hbar{\mathfrak{g}}[z]$. We prove that $\tau _z$ uniquely extends to an algebra homomorphism $\Phi _z$ from the Yangian double $\textrm{D}Y_\hbar{\mathfrak{g}}$ into the $\hbar $-adic closure of the algebra of Laurent series in $z^{-1}$ with coefficients in the Yangian $Y_\hbar{\mathfrak{g}}$. This induces, via evaluation at any point $c\in{\mathbb{C}}^\times $, a homomorphism from $\textrm{D}Y_\hbar{\mathfrak{g}}$ into the completion of the Yangian with respect to its grading. We show that each such homomorphism gives rise to an isomorphism between completions of $\textrm{D}Y_\hbar{\mathfrak{g}}$ and $Y_\hbar{\mathfrak{g}}$ and, as a corollary, we find that the Yangian $Y_\hbar{\mathfrak{g}}$ can be realized as a degeneration of the Yangian double $\textrm{D}Y_\hbar{\mathfrak{g}}$. Using these results, we obtain a Poincaré–Birkhoff–Witt theorem for $\textrm{D}Y_\hbar{\mathfrak{g}}$ applicable when ${\mathfrak{g}}$ is of finite type or of simply laced affine type.

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.

Decompositions of principal series representations of Iwahori–Hecke algebras for Kac–Moody groups over local fields

Recently, Iwahori-Hecke algebras were associated to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we introduced principal series representations for these algebras and partially generalized Kato's irreducibility criterion. In this paper, we study how some of these representations decompose when they are reducible and deduce information on the irreducible representations of these algebras.

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.

The Bruhat order, the lookup conjecture and spiral Schubert Varieties of type $Ã_2$

Although the Bruhat order on a Weyl group is closely related to the singularities of the Schubert varieties for the corresponding Kac–Moody group, it can be difficult to use this information to prove general theorems. This paper uses the action of the affine Weyl group of type Ã2 on a Euclidean space V≅ℝ2 to study the Bruhat order on W. We believe that these methods can be used to study the Bruhat order on arbitrary affine Weyl groups. Our motivation for this study was to extend the lookup conjecture of Boe and Graham (which is a conjectural simplification of the Carrell–Peterson criterion for rational smoothness) to type Ã2. Computational evidence suggests that the only Schubert varieties in type Ã2 where the “nontrivial” case of the lookup conjecture occurs are the spiral Schubert varieties, and as a step towards the lookup conjecture, we prove it for a spiral Schubert variety X(w) of type Ã2. The proof uses descriptions we obtain of the elements x≤w and of the rationally smooth locus of X(w) in terms of the W-action on V. As a consequence we describe the maximal nonrationally smooth points of X(w). The results of this paper are used in a sequel to describe the smooth locus of X(w), which is different from the rationally smooth locus.

Rigged configurations and the ⁎-involution for generalized Kac–Moody algebras

We construct a uniform model for highest weight crystals and B(∞) for generalized Kac–Moody algebras using rigged configurations. We also show an explicit description of the ⁎-involution on rigged configurations for B(∞): that the ⁎-involution interchanges the rigging and the corigging. We do this by giving a recognition theorem for B(∞) using the ⁎-involution. As a consequence, we also characterize B(λ) as a subcrystal of B(∞) using the ⁎-involution.

$${\varvec{\pi }}$$-systems of symmetrizable Kac–Moody algebras

As part of his classification of regular semisimple subalgebras of semisimple Lie algebras, Dynkin introduced the notion of a $$\pi $$ -system. This is a subset of the set of roots such that pairwise differences of its elements are not roots. Such systems arise as simple systems of regular semisimple subalgebras. Morita and Naito generalized this notion to all symmetrizable Kac–Moody algebras. In this work, we systematically develop the theory of $$\pi $$ -systems of symmetrizable Kac–Moody algebras and establish their fundamental properties. For several Kac–Moody algebras with physical significance, we study the orbits of the Weyl group on $$\pi $$ -systems and completely determine the number of orbits. In particular, we show that there is a unique $$\pi $$ -system of type $${A}^{++}_1$$ (the Feingold–Frenkel rank 3 hyperbolic algebra) in $$E_{10}$$ (the rank 10 hyperbolic algebra) up to Weyl group action and negation.

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.

Higher Kac–Moody algebras and symmetries of holomorphic field theories

We introduce a higher dimensional generalization of the affine Kac-Moody algebra using the language of factorization algebras. In particular, on any complex manifold there is a factorization algebra of currents associated to any Lie algebra. We classify local cocycles of these current algebras, and compare them to central extensions of higher affine algebras recently proposed by Faonte-Hennion-Kapranov. A central goal of this paper is to witness higher Kac-Moody algebras as symmetries of a class of holomorphic quantum field theories. In particular, we prove a generalization of the free field realization of an affine Kac-Moody algebra and also develop the theory of q-characters for this class of algebras in terms of factorization homology. Finally, we exhibit the large N behavior of higher Kac-Moody algebras and their relationship to symmetries of non-commutative field theories.

Defining relations of quantum symmetric pair coideal subalgebras

AbstractWe explicitly determine the defining relations of all quantum symmetric pair coideal subalgebras of quantised enveloping algebras of Kac–Moody type. Our methods are based on star products on noncommutative${\mathbb N}$-graded algebras. The resulting defining relations are expressed in terms of continuousq-Hermite polynomials and a new family of deformed Chebyshev polynomials.

Keys and Demazure crystals for Kac–Moody algebras

The Key map is an important tool in the determination of the Demazure crystals associated to Kac–Moody algebras. In finite type A, it can be computed in the tableau realization of crystals by a simple combinatorial procedure due to Lascoux and Schützenberger. We show that this procedure is a part of a more general construction holding in the Kac–Moody case that we illustrate in finite types and affine type A. In affine type A, we introduce higher level generalizations of core partitions which are expected to play an important role in the representation theory of Ariki–Koike algebras.

Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations

To Nicolás Andruskiewitsch on his 60th birthday, with admiration We introduce bivariate versions of the continuous [Formula: see text]-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous [Formula: see text]-Hermite polynomials and the star product method of [S. Kolb and M. Yakimov, Symmetric pairs for Nichols algebras of diagonal type via star products, Adv. Math. 365 (2020), Article ID: 107042, 69 pp.] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac–Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials.

Etingof’s conjecture for quantized quiver varieties

We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing. We provide an exact count in the case when the quiver is of finite type or is of affine type and the framing is the coordinate vector at the extending vertex. The latter case precisely covers Etingof’s conjecture on the number of finite dimensional irreducible representations for Symplectic reflection algebras associated to wreath-product groups. We use several different techniques, the two principal ones are categorical Kac–Moody actions and wall-crossing functors.

The Category $\mathcal O$ for Lie Algebras of Vector Fields (I): Tilting Modules and Character Formulas

In this article, we exploit the theory of graded module categories with semi-infinite character developed by Soergel (Character formulas for tilting modules over Kac–Moody algebras, Represent. Theor. 2 (1998), 432–448) to study representations of infinite-dimensional Lie algebras of vector fields W(n), S(n) and H(n) (n \geq 2) , and obtain a description of indecomposable tilting modules. The character formulas for those tilting modules are determined.

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.

Commutator relations and structure constants for rank 2 Kac–Moody algebras

We completely determine the structure constants between real root vectors in a rank 2 Kac–Moody algebra g. Our description is computationally efficient, even in the rank 2 hyperbolic case where the coefficients of roots on the root lattice grow exponentially with height. Our approach is to extend Carter's method of finding structure constants from those on extraspecial pairs to the rank 2 Kac–Moody case. We also determine all commutator relations involving only real root vectors in all rank 2 Kac-Moody algebras. The generalized Cartan matrix of g is of the form H(a,b)=(2−b−a2) where a,b∈Z and ab≥4. If ab=4, then g is of affine type. If ab>4, then g is of hyperbolic type. Explicit knowledge of the root strings is needed, as well as a characterization of the pairs of real roots whose sums are real. We prove that if a and b are both greater than one, then no sum of real roots can be a real root. We determine the root strings between real roots β,γ in H(a,1), a≥5 and we determine the sets (Z≥0α+Z≥0β)∩Δre(H(a,b)). One of our tools is a characterization of the root subsystems generated by a subset of roots. We classify these subsystems in rank 2 Kac–Moody root systems. We prove that every rank two infinite root system contains an infinite family of non-isomorphic symmetric rank 2 hyperbolic root subsystems H(k,k) for certain k≥3, generated by either two short or two long simple roots. We also prove that a non-symmetric hyperbolic root systems H(a,b) with a≠b and ab>5 also contains an infinite family of non-isomorphic non-symmetric rank 2 hyperbolic root subsystems H(aℓ,bℓ), for certain positive integers ℓ.

Painlevé property, local and nonlocal symmetries, and symmetry reductions for a (2+1)-dimensional integrable KdV equation* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11975131 and 11435005) and the K C Wong Magna Fund in Ningbo University.

The Painlevé property for a (2+1)-dimensional Korteweg–de Vries (KdV) extension, the combined KP3 (Kadomtsev–Petviashvili) and KP4 (cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.

Depth and detection for Noetherian unstable algebras

For a connected Noetherian unstable algebra R R over the mod p p Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of R R , originally proved when R R is the mod p p cohomology ring of a finite group. This recovers the aforementioned results, and also proves versions of them when R R is the mod p p cohomology ring of a compact Lie group, a profinite group with Noetherian cohomology, a Kac–Moody group, a discrete group of finite virtual cohomological dimension, as well as for certain other discrete groups. More generally, our results apply to certain finitely generated unstable R R -modules. Moreover, we explain the results in the case of the p p -local compact groups of Broto, Levi, and Oliver, as well as in the modular invariant theory of finite groups.