Symmetrizable Kac Moody Algebra Research Articles

Coherence for Plactic Monoids via Rewriting Theory and Crystal Structures

Rewriting methods have been developed for the study of coherence for algebraic objects. This consists in starting with a convergent presentation and expliciting a family of generating confluences to obtain a coherent presentation – one with generators, generating relations, and generating relations between relations (syzygies). In this article we develop these ideas for a class of monoids which encode the representation theory of complex symmetrizable Kac-Moody algebras, called plactic monoids. The main tools for this are the crystal realization of plactic monoids due to Kashiwara, and a class of presentations compatible with a crystal structure, called crystal presentations. We show that the compatibility of the crystal structure with the presentation reduces many aspects of the study of plactic monoids via rewriting theory to computations with components of highest weight in the crystal. We thus obtain reduced versions of Newman’s Lemma and Critical Pair Lemma, which are results for verifying convergence of a presentation. Further we show that the family of generating confluences of a convergent crystal presentation is entirely determined by the components of highest weight, and show that this result applies to the plactic monoids of types An, Bn, Cn, Dn, and G2, via their convergent presentations due to Cain–Gray–Malheiro (Cain et al. J. Comb. Theory Ser. A. 162, 406–466, 6).

A note on parabolic Verma module homomorphisms over Kac-Moody algebras

We consider a general parabolic category O S over symmetrizable Kac-Moody algebras. We introduce a reduction rule of hom-spaces between parabolic Verma modules over different Kac-Moody algebras which yield some applications on parabolic Verma module homomorphisms .

2-Verma modules

Abstract We construct a categorification of parabolic Verma modules for symmetrizable Kac–Moody algebras using KLR-like diagrammatic algebras. We show that our construction arises naturally from a dg-enhancement of the cyclotomic quotients of the KLR-algebras. As a consequence, we are able to recover the usual categorification of integrable modules. We also introduce a notion of dg-2-representation for quantum Kac–Moody algebras, and in particular of parabolic 2-Verma modules.

Defining relations for quantum symmetric pair coideals of Kac–Moody type

Classical symmetric pairs consist of a symmetrizable Kac-Moody algebra $\mathfrak{g}$, together with its subalgebra of fixed points under an involutive automorphism of the second kind. Quantum group analogs of this construction, known as quantum symmetric pairs, replace the fixed point Lie subalgebras by one-sided coideal subalgebras of the quantized enveloping algebra $U_q(\mathfrak{g})$. We provide a complete presentation by generators and relations for these quantum symmetric pair coideal subalgebras. These relations are of inhomogeneous $q$-Serre type and are valid without restrictions on the generalized Cartan matrix. We draw special attention to the split case, where the quantum symmetric pair coideal subalgebras are generalized $q$-Onsager algebras.

Integral quantum cluster structures

We prove a general theorem for constructing integral quantum cluster algebras over Z[q±1/2], namely, that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over Z[q±1/2]. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that, for every symmetrizable Kac–Moody algebra g and Weyl group element w, the dual canonical form Aq(n+(w))Z[q±1] of the corresponding quantum unipotent cell has the property that Aq(n+(w))Z[q±1]⊗Z[q±1]Z[q±1/2] is isomorphic to a quantum cluster algebra over Z[q±1/2] and to the corresponding upper quantum cluster algebra over Z[q±1/2].

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.

A study of Kostant-Kumar modules via Littelmann paths

We study, by means of Littelmann's theory of paths, Kostant-Kumar modules (KK modules for short), which by definition are certain submodules of the tensor product of two irreducible integrable highest weight representations of a symmetrizable Kac-Moody algebra. Our main result is an identification of a path model for any KK module as a subset of the well known path model for the tensor product consisting of concatenations of Lakshmibai-Seshadri paths. The technical results about extremal elements in Coxeter groups that we formulate and prove en route and the technique of their proofs should be of independent interest. We also discuss the existence of PRV components and generalised PRV components in KK modules.

$${\varvec{\pi }}$$-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.

Uniqueness of Coxeter structures on Kac–Moody algebras

Let g be a symmetrizable Kac–Moody algebra, and Uħg the corresponding quantum group. We showed in [1,2] that the braided Coxeter structure on integrable, category O representations of Uħg which underlies the R-matrix actions arising from the Levi subalgebras of Uħg and the quantum Weyl group action of the generalized braid group Bg can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R-matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in [3] to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of Uħg. Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g.

Quivers with relations for symmetrizable Cartan matrices IV: crystal graphs and semicanonical functions

We generalize Lusztig's nilpotent varieties, and Kashiwara and Saito's geometric construction of crystal graphs from the symmetric to the symmetrizable case. We also construct semicanonical functions in the convolution algebras of generalized preprojective algebras. Conjecturally these functions yield semicanonical bases of the enveloping algebras of the positive part of symmetrizable Kac-Moody algebras.

Quiver varieties and crystals in symmetrizable type via modulated graphs

Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac-Moody algebra. The underlying set consists of the irreducible components of Lusztig's quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. We generalize this to symmetrizable Kac-Moody algebras by replacing Lusztig's preprojective algebra with a more general one due to Dlab and Ringel. In non-symmetric types we are forced to work over non-algebraically-closed fields.

Face functors for KLR algebras

Simple representations of KLR algebras can be used to realize the infinity crystal for the corresponding symmetrizable Kac-Moody algebra. It was recently shown that, in finite and affine types, certain sub-categories of “cuspidal” representations realize crystals for sub-Kac-Moody algebras. Here we put that observation on a firmer categorical footing by exhibiting a corresponding functor between the category of representations of the KLR algebra for the sub-Kac-Moody algebra and the category of cuspidal representations of the original KLR algebra.

A new involution for quantum loop algebras

In this article, we introduce a completion Uˆv+(Lg) of the positive half of the quantum affinization Uv+(Lg) of a symmetrizable Kac–Moody algebra g. On Uˆv+(L(g)), we define a new “bar-involution” and construct the analogue Kashiwara's operators. We conjecture that the resulting pair (Lˆ,Bˆ) is a crystal basis which provides the existence of the “canonical basis” on the (completion of the) of the positive half of the quantum affinization.

Separating Ore sets for prime ideals of quantum algebras

Brown and Goodearl stated a conjecture that provides an explicit description of the topology of the spectra of quantum algebras. The conjecture takes on a more explicit form if there exist separating Ore sets for all incident pairs of torus-invariant prime ideals of the given algebra. We prove that this is the case for the two largest classes of algebras of finite Gelfand–Kirillov dimension that fit the setting of the conjecture: the quantized coordinate rings of all simple algebraic groups and the quantum Schubert cell algebras for all symmetrizable Kac–Moody algebras.

Universal K-matrix for quantum symmetric pairs

Abstract Let 𝔤 {{\mathfrak{g}}} be a symmetrizable Kac–Moody algebra and let U q ⁢ ( 𝔤 ) {{U_{q}(\mathfrak{g})}} denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras B 𝐜 , 𝐬 {{B_{\mathbf{c},\mathbf{s}}}} of U q ⁢ ( 𝔤 ) {{U_{q}(\mathfrak{g})}} have a universal K-matrix if 𝔤 {{\mathfrak{g}}} is of finite type. By a universal K-matrix for B 𝐜 , 𝐬 {{B_{\mathbf{c},\mathbf{s}}}} we mean an element in a completion of U q ⁢ ( 𝔤 ) {{U_{q}(\mathfrak{g})}} which commutes with B 𝐜 , 𝐬 {{B_{\mathbf{c},\mathbf{s}}}} and provides solutions of the reflection equation in all integrable U q ⁢ ( 𝔤 ) {{U_{q}(\mathfrak{g})}} -modules in category 𝒪 {{\mathcal{O}}} . The construction of the universal K-matrix for B 𝐜 , 𝐬 {{B_{\mathbf{c},\mathbf{s}}}} bears significant resemblance to the construction of the universal R-matrix for U q ⁢ ( 𝔤 ) {{U_{q}(\mathfrak{g})}} . Most steps in the construction of the universal K-matrix are performed in the general Kac–Moody setting. In the late nineties T. tom Dieck and R. Häring-Oldenburg developed a program of representations of categories of ribbons in a cylinder. Our results show that quantum symmetric pairs provide a large class of examples for this program.

Mirković–Vilonen polytopes and Khovanov–Lauda–Rouquier algebras

We describe how Mirković–Vilonen (MV) polytopes arise naturally from the categorification of Lie algebras using Khovanov–Lauda–Rouquier (KLR) algebras. This gives an explicit description of the unique crystal isomorphism between simple representations of KLR algebras and MV polytopes. MV polytopes, as defined from the geometry of the affine Grassmannian, only make sense in finite type. Our construction on the other hand gives a map from the infinity crystal to polytopes for all symmetrizable Kac–Moody algebras. However, to make the map injective and have well-defined crystal operators on the image, we must in general decorate the polytopes with some extra information. We suggest that the resulting ‘KLR polytopes’ are the general-type analogues of MV polytopes. We give a combinatorial description of the resulting decorated polytopes in all affine cases, and show that this recovers the affine MV polytopes recently defined by Baumann, Kamnitzer, and the first author in symmetric affine types. We also briefly discuss the situation beyond affine type.

The bar involution for quantum symmetric pairs

We construct a bar involution for quantum symmetric pair coideal subalgebras B c , s B_{\mathbf {c},\mathbf {s}} corresponding to involutive automorphisms of the second kind of symmetrizable Kac-Moody algebras. To this end we give unified presentations of these algebras in terms of generators and relations, extending previous results by G. Letzter and the second-named author. We specify precisely the set of parameters c \mathbf {c} for which such an intrinsic bar involution exists.

Quantum symmetric Kac–Moody pairs

The present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac–Moody algebras. The resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras. They give rise to triangular decompositions, including a quantum analog of the Iwasawa decomposition, and they can be written explicitly in terms of generators and relations. Moreover, their centers and their specializations are determined. The constructions follow G. Letzter's theory of quantum symmetric pairs for semisimple Lie algebras. The main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable Kac–Moody algebras due to Kac and Wang. The resulting theory comprises various classes of examples which have previously appeared in the literature, such as q-Onsager algebras and the twisted q-Yangians introduced by Molev, Ragoucy, and Sorba.

Weyl group multiple Dirichlet series for symmetrizable Kac-Moody root systems

Weyl group multiple Dirichlet series, introduced by Brubaker, Bump, Chinta, Friedberg and Hoffstein, are expected to be Whittaker coefficients of Eisenstein series on metaplectic groups. Chinta and Gunnells constructed these multiple Dirichlet series for all the finite root systems using the method of averaging a Weyl group action on the field of rational functions. In this paper, we generalize Chinta and Gunnells’ work and construct Weyl group multiple Dirichlet series for the root systems associated with symmetrizable Kac-Moody algebras, and establish their functional equations and meromorphic continuation.

Canonical bases and quantum coordinate algebras

Some filtrations of the tensor product of a highest weight module and a lowest weight module over quantum group Uq(g) are constructed in [1] and one can use them to define a two-sided ideal of the modified quantized enveloping algebra. It is shown that the quotient algebra inherits a canonical basis from the modified quantized enveloping algebra and is dual to the quantum coordinate algebra defined by Kashiwara for a symmetrizable Kac–Moody algebra g.