Kac-Moody Lie Algebras Research Articles

Cluster realizations of Weyl groups and higher Teichmüller theory

For a symmetrizable Kac–Moody Lie algebra $${\mathfrak {g}}$$ , we construct a family of weighted quivers $$Q_m({\mathfrak {g}})$$ ( $$m \ge 2$$ ) whose cluster modular group $$\Gamma _{Q_m({\mathfrak {g}})}$$ contains the Weyl group $$W({\mathfrak {g}})$$ as a subgroup. We compute explicit formulae for the corresponding cluster $${{\mathcal {A}} }$$ - and $${{\mathcal {X}} }$$ -transformations. As a result, we obtain green sequences and the cluster Donaldson–Thomas transformation for $$Q_m({\mathfrak {g}})$$ in a systematic way when $${\mathfrak {g}}$$ is of finite type. Moreover if $${\mathfrak {g}}$$ is of classical finite type with the Coxeter number h, the quiver $$Q_{kh}({\mathfrak {g}})$$ ( $$k \ge 1$$ ) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichmuller space of a once-punctured disk with 2k marked points on the boundary, up to frozen vertices. This correspondence induces the action of direct products of Weyl groups on the higher Teichmuller space of a general marked surface. We finally prove that this action coincides with the one constructed in Goncharov and Shen (Adv Math 327:225–348, 2018) from the geometrical viewpoint.

Adapted sequences and polyhedral realizations of crystal bases for highest weight modules

The polyhedral realizations for crystal bases of the integrable highest weight modules of Uq(g) have been introduced in Nakashima (1999) [13], which describe the crystal bases as sets of lattice points in the infinite Z-lattice Z∞ given by some system of linear inequalities, where g is a symmetrizable Kac-Moody Lie algebra. To construct the polyhedral realization, we need to fix an infinite sequence ι from the indices of the simple roots. If the pair (ι,λ) (λ: a dominant integral weight) satisfies the ‘ample’ condition then there are some procedure to calculate the sets of linear inequalities.In this article, we show that if ι is an adapted sequence (defined in our paper [5]) then the pair (ι, λ) satisfies the ample condition for any dominant integral weight λ in the case g is a classical Lie algebra. Furthermore, we reveal the explicit forms of the polyhedral realizations of the crystal bases B(λ) associated with arbitrary adapted sequences ι in terms of column tableaux. As an application, we will give a combinatorial description of the function εi⁎ on the crystal base B(∞).

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.

Diagram automorphisms and canonical bases for quantum affine algebras

Let Uq− be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra g, and U_q− the algebra corresponding to the orbit algebra of g obtained from a diagram automorphism σ on g. Let Bσ be the set of σ-fixed elements in the canonical basis of Uq−, and B_ the canonical basis of U_q−. Lusztig proved that there exists a canonical bijection Bσ≃B_ based on his geometric construction of canonical bases. In this paper, we prove (the signed bases version of) this fact, in the case where g is finite or affine type, in an elementary way, in the sense that we don't appeal to the geometric theory of canonical bases nor Kashiwara's theory of crystal bases. We also discuss the correspondence for PBW-bases, by using a new type of PBW-bases of Uq− obtained by Muthiah-Tingley, which is a generalization of PBW-bases constructed by Beck-Nakajima.

Drinfeld-type presentations of loop algebras

Let g \mathfrak {g} be the derived subalgebra of a Kac-Moody Lie algebra of finite-type or affine-type, let μ \mu be a diagram automorphism of g \mathfrak {g} , and let L ( g , μ ) \mathcal {L}(\mathfrak {g},\mu ) be the loop algebra of g \mathfrak {g} associated to μ \mu . In this paper, by using the vertex algebra technique, we provide a general construction of current-type presentations for the universal central extension g ^ [ μ ] \widehat {\mathfrak {g}}[\mu ] of L ( g , μ ) \mathcal {L}(\mathfrak {g},\mu ) . The construction contains the classical limit of Drinfeld’s new realization for (twisted and untwisted) quantum affine algebras [Soviet Math. Dokl. 36 (1988), pp. 212–216] and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras [Geom. Dedicata 35 (1990), pp. 283–307] as special examples. As an application, when g \mathfrak {g} is of simply-laced-type, we prove that the classical limit of the μ \mu -twisted quantum affinization of the quantum Kac-Moody algebra associated to g \mathfrak {g} introduced in [J. Math. Phys. 59 (2018), 081701] is the universal enveloping algebra of g ^ [ μ ] \widehat {\mathfrak {g}}[\mu ] .

THE PBW THEOREM FOR AFFINE YANGIANS

We prove that the Yangian associated to an untwisted symmetric affine Kac-Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed by the authors in arXiv:1407.7994 as an algebraic formalism of the cohomological Hall algebras. As a consequence, we obtain the Poincare-Birkhoff-Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given recently by Guay-Regelskis-Wendlandt.

Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian and geometric Satake equivalence

Let $G$ be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian $Gr_G$. We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with $U(\check{\mathfrak n})$ (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra $\check{\mathfrak g}$). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.

Fock Space Representation of the Circle Quantum Group

Abstract In [12] we have defined quantum groups $\mathbf{U}_{\upsilon }(\mathfrak{sl}(\mathbb{R}))$ and $\mathbf{U}_{\upsilon }(\mathfrak{sl}(S^1))$, which can be interpreted as continuum generalizations of the quantum groups of the Kac–Moody Lie algebras of finite, respectively affine type $A$. In the present paper, we define the Fock space representation $\mathcal{F}_{\mathbb{R}}$ of the quantum group $\mathbf{U}_{\upsilon }(\mathfrak{sl}(\mathbb{R}))$ as the vector space generated by real pyramids (a continuum generalization of the notion of partition). In addition, by using a variant version of the “folding procedure” of Hayashi–Misra–Miwa, we define an action of $\mathbf{U}_{\upsilon }(\mathfrak{sl}(S^1))$ on $\mathcal{F}_{\mathbb{R}}$.

Generalized Casimir operators

Let [Formula: see text] be symmetrizable Kac–Moody Lie algebra. In this paper, we describe a new class of central operators generalizing the Casimir operator. We also prove some properties of these operators and show that these operators move highest weight vectors to new highest weight vectors.

Q-Virasoro algebra and affine Kac-Moody Lie algebras

In this paper, we introduce an infinite-dimensional Lie algebra DS for any abelian group S. If S is the additive group of integers, DS reduces to the q-Virasoro algebra Dq introduced by Belov and Chaltikian in the study of lattice conformal theories. Guided by the theory of equivariant quasi modules for vertex algebras, we introduce another Lie algebra gS with S as an automorphism group and we prove that DS is isomorphic to the S-covariant algebra of the affine Lie algebra gSˆ. We then relate restricted DS-modules of level ℓ∈C to equivariant quasi modules for the vertex algebra VgSˆ(ℓ,0) associated to gSˆ with level ℓ. Furthermore, we establish an intrinsic connection between the q-Virasoro algebra Dq and affine Kac-Moody Lie algebras. More specifically, we show that if S is a finite abelian group of order 2l+1, DS is isomorphic to the affine Kac-Moody algebra of type Bl(1).

Extended affine Lie superalgebras containing Cartan subalgebras

Multiloop (super)algebras are of great importance in the literature. Affine Kac Moody Lie (super)algebras and almost all extended affine Lie algebras are realized using an affinization process on loop superalgebras and multiloop algebras respectively. An affinization process for multiloop superalgebras turns to Lie superalgebras satisfying certain properties which are the super version of extended affine Lie algebras. In this paper we systematically study this super version called extended affine Lie superalgebras. We first derive the intrinsic properties of extended affine Lie superalgebras and then study those which are simple.

Commutative post-Lie algebra structures on Kac–Moody algebras

We determine commutative post-Lie algebra structures on some infinite-dimensional Lie algebras. We show that all commutative post-Lie algebra structures on loop algebras are trivial. This extends the results for finite-dimensional perfect Lie algebras. Furthermore, we show that all commutative post-Lie algebra structures on affine Kac–Moody Lie algebras are “almost trivial”.

Simple Singular Whittaker Modules Over the Schrödinger Algebra

There are no simple singular Whittaker modules over most of important algebras, such as simple complex finite-dimensional Lie algebras, affine Kac–Moody Lie algebras, the Virasoro algebra, the Heisenberg–Virasoro algebra and the Schrodinger–Witt algebra. In this paper, however, we construct simple singular Whittaker modules over the Schrodinger algebra. Moreover, simple singular Whittaker modules over the Schrodinger algebra are classified. As a result, simple modules for the Schrodinger algebra which are locally finite over the positive part are completely classified. We also give characterizations of simple highest weight modules and simple singular Whittaker modules.

A categorical reconstruction of crystals and quantum groups at q = 0

Abstract The quantum co-ordinate algebra Aq(g) associated to a Kac–Moody Lie algebra g forms a Hopf algebra whose comodules are direct sums of finite-dimensional irreducible Uq(g) modules. In this paper, we investigate whether an analogous result is true when q=0. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this, we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over Z whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig’s quantum group at v=∞.

Braid Group Action on Affine Yangian

We study braid group actions on Yangians associated with symmetrizable Kac-Moody Lie algebras. As an application, we focus on the affine Yangian of type A and use the action to prove that the image of the evaluation map contains the diagonal Heisenberg algebra inside $\hat{\mathfrak{gl}}_N$.

Vertex Algebras and Coordinate Rings of Semi-infinite Flags

The direct sum of irreducible level one integrable representations of affine Kac-Moody Lie algebra of (affine) type $ADE$ carries a structure of $P/Q$-graded vertex operator algebra. There exists a filtration on this direct sum studied by Kato and Loktev such that the corresponding graded vector space is a direct sum of global Weyl modules. The associated graded space with respect to the dual filtration is isomorphic to the homogenous coordinate ring of semi-infinite flag variety. We describe the ring structure in terms of vertex operators and endow the homogenous coordinate ring with a structure of $P/Q$-graded vertex operator algebra. We use the vertex algebra approach to derive semi-infinite Pl\"ucker-type relations in the homogeneous coordinate ring.

Imaginary cone and reflection subgroups of Coxeter groups

The imaginary cone of a Kac-Moody Lie algebra is the convex hull of zero and the positive imaginary roots. This paper studies the imaginary cone for a class of root systems of general Coxeter groups W. It is shown that the imaginary cone of a reflection subgroup of W is contained in that of W, and that for irreducible infinite W of finite rank, the closed imaginary cone is the only non-zero, closed, pointed W-stable cone contained in the pointed cone spanned by the simple roots. For W of finite rank, various natural notions of faces of the imaginary cone are shown to coincide, the face lattice is explicitly described in terms of the lattice of facial reflection subgroups and it is shown that the Tits cone and imaginary cone are related by a duality closely analogous to the standard duality for polyhedral cones, even though neither of them is a closed cone in general. Some of these results have application, to be given in sequels to this paper, to dominance order of Coxeter groups, associated automata, and construction of modules for generic Iwahori-Hecke algebras.

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.

Representations of quantized coordinate algebras via PBW-type elements

Inspired by the work of Kuniba-Okado-Yamada, we study some tensor product representations of quantized coordinate algebras of symmetrizable Kac-Moody Lie algebras in terms of quantized enveloping algebras. As a consequence, we describe structures and properties of certain reducible representations of quantized coordinate algebras. This paper includes alternative proofs of Soibelman's tensor product theorem and Kuniba-Okado-Yamada's common structure theorem based on our direct calculation method using global bases.

Generic free resolutions and root systems

In this paper I give an explicit construction of the generic ring R_{gen} for finite free resolutions of length 3. The corresponding problem for resolutions of length 2 was solved in 1970'ies by Hochster and Huneke. The key role is played by the defect Lie algebra introduced in my old work on the subject. The defect Lie algebra turns out to be a parabolic Lie algebra in a Kac-Moody Lie algebra associated to the graph T_{p,q,r} corresponding to the format of the resolution. The ring R_{gen} is Noetherian if and only if the graph T_{p.q.r} is a Dynkin graph.