Dynkin Diagram Automorphism Research Articles

Automorphism Groups of Deformations and Quantizations of Kleinian Singularities

AbstractIt is known that, for the algebra of functions on a Kleinian singularity, the parameter space of deformations and the parameter space of quantizations coincide. We prove that, for a Kleinian singularity of type $$\textbf{A}$$ A or $$\textbf{D}$$ D , isomorphisms between the quantizations are essentially the same as Poisson isomorphisms between deformations. In particular, the group of automorphisms of the deformation and the quantization corresponding to the same deformation parameter are isomorphic. We additionally describe the groups of automorphisms as abstract groups: for type $$\textbf{A}$$ A they have an amalgamated free product structure, for type $$\textbf{D}$$ D they are subgroups of the groups of Dynkin diagram automorphisms. For type $$\textbf{D}$$ D we also compute all the possible affine isomorphisms between deformations; this was not known before.

Pointed Hopf algebras over nonabelian groups with nonsimple standard braidings

AbstractWe construct finite‐dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We show conversely that every finite‐dimensional pointed Hopf algebra over a nonabelian group with nonsimple infinitesimal braiding of rank at least 4 is of this form. We follow the steps of the Lifting Method by Andruskiewitsch–Schneider. Our starting point is the classification of finite‐dimensional Nichols algebras over nonabelian groups by Heckenberger–Vendramin, which consist of low‐rank exceptions and large‐rank families. We prove that the large‐rank families are cocycle twists of Nichols algebras constructed by the second author as foldings of Nichols algebras of Cartan type over abelian groups by outer automorphisms. This enables us to give uniform Lie‐theoretic descriptions of the large‐rank families, prove generation in degree 1, and construct liftings. We also show that every lifting is a cocycle deformation of the corresponding coradically graded Hopf algebra using an explicit presentation by generators and relations of the Nichols algebra. On the level of tensor categories, we construct families of graded extensions of the representation category of a quantum group by a group of diagram automorphism.

Folding KLR algebras

This paper develops the theory of KLR algebras with a Dynkin diagram automorphism. This is foundational material intended to allow folding techniques in the theory of KLR algebras.

A Uniform Approach to Soliton Cellular Automata Using Rigged Configurations

For soliton cellular automata, we give a uniform description and proofs of the solitons, the scattering rule of two solitons, and the phase shift using rigged configurations in a number of special cases. In particular, we prove these properties for the soliton cellular automata using $B^{r,1}$ when $r$ is adjacent to $0$ in the Dynkin diagram or there is a Dynkin diagram automorphism sending $r$ to $0$.

Quantum dimensions and fusion products for irreducible VQσ-modules with σ2 = 1

ABSTRACTEvery isometry σ of a positive-definite even lattice Q can be lifted to an automorphism of the lattice vertex algebra VQ. An important problem in vertex algebra theory and conformal field theory is to classify the representations of the σ-invariant subalgebra of VQ, known as an orbifold. In the case when σ is an isometry of Q of order two, we have classified the irreducible modules of the orbifold vertex algebra and identified them as submodules of twisted or untwisted VQ-modules in [6]. Here we calculate their quantum dimensions and fusion products.The examples where Q is the orthogonal direct sum of two copies of the A2 root lattice and σ is the 2-cycle permutation as well as where Q is the An root lattice and σ is a Dynkin diagram automorphism are presented in detail.

Orbifolds of lattice vertex algebras under an isometry of order two

Every isometry σ of a positive-definite even lattice Q can be lifted to an automorphism of the lattice vertex algebra VQ. An important problem in vertex algebra theory and conformal field theory is to classify the representations of the σ-invariant subalgebra VQσ of VQ, known as an orbifold. In the case when σ is an isometry of Q of order two, we classify the irreducible modules of the orbifold vertex algebra VQσ and identify them as submodules of twisted or untwisted VQ-modules. The examples where Q is a root lattice and σ is a Dynkin diagram automorphism are presented in detail.

ISOMORPHIC INDUCED MODULES AND DYNKIN DIAGRAM AUTOMORPHISMS OF SEMISIMPLE LIE ALGEBRAS

AbstractConsider a simple Lie algebra $\mathfrak{g}$ and $\overline{\mathfrak{g}}$ ⊂ $\mathfrak{g}$ a Levi subalgebra. Two irreducible $\overline{\mathfrak{g}}$-modules yield isomorphic inductions to $\mathfrak{g}$ when their highest weights coincide up to conjugation by an element of the Weyl group W of $\mathfrak{g}$ which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$. In this paper, we study the converse problem: given two irreducible $\overline{\mathfrak{g}}$-modules of highest weight μ and ν whose inductions to $\mathfrak{g}$ are isomorphic, can we conclude that μ and ν are conjugate under the action of an element of W which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$? We conjecture this is true in general. We prove this conjecture in type A and, for the other root systems, in various situations providing μ and ν satisfy additional hypotheses. Our result can be interpreted as an analogue for branching coefficients of the main result of Rajan [6] on tensor product multiplicities.

Generalized spin Sutherland systems revisited

We present generalizations of the spin Sutherland systems obtained earlier by Blom and Langmann and by Polychronakos in two different ways: from SU(n) Yang–Mills theory on the cylinder and by constraining geodesic motion on the N-fold direct product of SU(n) with itself, for any N>1. Our systems are in correspondence with the Dynkin diagram automorphisms of arbitrary connected and simply connected compact simple Lie groups. We give a finite-dimensional as well as an infinite-dimensional derivation and shed light on the mechanism whereby they lead to the same classical integrable systems. The infinite-dimensional approach, based on twisted current algebras (alias Yang–Mills with twisted boundary conditions), was inspired by the derivation of the spinless Sutherland model due to Gorsky and Nekrasov. The finite-dimensional method relies on Hamiltonian reduction under twisted conjugations of N-fold direct product groups, linking the quantum mechanics of the reduced systems to representation theory similarly as was explored previously in the N=1 case.

Symmetries in Connection Preserving Deformations

We wish to show that the root lattice of B\"acklund transformations of the $q$-analogue of the third and fourth Painlev\'e equations, which is of type $(A_2+ A_1)^{(1)}$, may be expressed as a quotient of the lattice of connection preserving deformations. Furthermore, we will show various directions in the lattice of connection preserving deformations present equivalent evolution equations under suitable transformations. These transformations correspond to the Dynkin diagram automorphisms.

Quantum Folding

In the present paper we introduce a quantum analogue of the classical folding of a simply-laced Lie algebra g to the non-simply-laced algebra g^sigma along a Dynkin diagram automorphism sigma of g For each quantum folding we replace g^sigma by its Langlands dual g^sigma^v and construct a nilpotent Lie algebra n which interpolates between the nilpotnent parts of g and (g^sigma)^v, together with its quantized enveloping algebra U_q(n) and a Poisson structure on S(n). Remarkably, for the pair (g, (g^sigma)^v)=(so_{2n+2},sp_{2n}), the algebra U_q(n) admits an action of the Artin braid group Br_n and contains a new algebra of quantum n x n matrices with an adjoint action of U_q(sl_n), which generalizes the algebras constructed by K. Goodearl and M. Yakimov in [12]. The hardest case of quantum folding is, quite expectably, the pair (so_8,G_2) for which the PBW presentation of U_q(n) and the corresponding Poisson bracket on S(n) contain more than 700 terms each.

Modular Galois covers associated to symplectic resolutions of singularities

Let Y be a normal projective variety and p a morphism from X to Y, which is a projective holomorphic symplectic resolution. Namikawa proved that the Kuranishi deformation spaces Def(X) and Def(Y) are both smooth, of the same dimension, and p induces a finite branched cover f from Def(X) to Def(Y). We prove that f is Galois. We proceed to calculate the Galois group G, when X is simply connected, and its holomorphic symplectic structure is unique, up to a scalar factor. The singularity of Y is generically of ADE-type, along every codimension 2 irreducible component B of the singular locus, by Namikawa's work. The modular Galois group G is the product of Weyl groups of finite type, indexed by such irreducible components B. Each Weyl group factor W_B is that of a Dynkin diagram, obtained as a quotient of the Dynkin diagram of the singularity-type of B, by a group of Dynkin diagram automorphisms. Finally we consider generalizations of the above set-up, where Y is affine symplectic, or a Calabi-Yau threefold with a curve of ADE-singularities. We prove that the morphism f from Def(X) to Def(Y) is a Galois cover of its image. This explains the analogy between the above results and related work of Nakajima, on quiver varieties, and of Szendroi on enhanced gauge symmetries for Calabi-Yau threefolds.

Kirillov–Reshetikhin crystals for nonexceptional types

We provide combinatorial models for all Kirillov–Reshetikhin crystals of nonexceptional type, which were recently shown to exist. For types D n ( 1 ) , B n ( 1 ) , A 2 n − 1 ( 2 ) we rely on a previous construction using the Dynkin diagram automorphism which interchanges nodes 0 and 1. For type C n ( 1 ) we use a Dynkin diagram folding and for types A 2 n ( 2 ) , D n + 1 ( 2 ) a similarity construction. We also show that for types C n ( 1 ) and D n + 1 ( 2 ) the analog of the Dynkin diagram automorphism exists on the level of crystals.

On the uniqueness of promotion operators on tensor products of type A crystals

The affine Dynkin diagram of type A (1) has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type A n crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type A n crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara’s conjecture that all ‘good’ affine crystals are tensor products of Kirillov-Reshetikhin crystals.

Twisted spin Sutherland models from quantum Hamiltonian reduction

Recent general results on Hamiltonian reductions under polar group actions are applied to study some reductions of the free particle governed by the Laplace–Beltrami operator of a compact, connected, simple Lie group. The reduced systems associated with arbitrary finite-dimensional irreducible representations of the group by using the symmetry induced by twisted conjugations are described in detail. These systems generically yield integrable Sutherland-type many-body models with spin, which are called twisted spin Sutherland models if the underlying twisted conjugations are built on non-trivial Dynkin diagram automorphisms. The spectra of these models can be calculated, in principle, by solving certain Clebsch–Gordan problems, and the result is presented for the models associated with the symmetric tensorial powers of the defining representation of SU(N).

A Twining Character Formula For Demazure Modules

We prove a formula for the twining characters of certain Demazure modules, over a Borel subalgebra b of a finite dimensional complex semisimple Lie algebra g. This formula describes the twining character of the Demazure module by the ω-Demazure operator associated to an element of the Weyl group that is fixed by the Dynkin diagram automorphism ω of g. Our result is a refinement of the twining character formula for the irreducible highest weight g-modules of symmetric dominant integral highest weights, and also of the ordinary Demazure character formula.

Twining Character Formula of Kac-Wakimoto Type for Affine Lie Algebras

We prove a formula of Kac-Wakimoto type for the twining characters of irreducible highest weight modules of symmetric, noncritical, integrally dominant highest weights over affine Lie algebras. This formula describes the twining character in terms of the subgroup of the integral Weyl group consisting of elements which commute with the Dynkin diagram automorphism. The main tools in our proof are the (Jantzen) translation functor and the existence result of a certain local composition series which is stable under the Dynkin diagram automorphism.

Graded Lie Superalgebras, Supertrace Formula, and Orbit Lie Superalgebras

Let $\Gamma$ be a countable abelian semigroup and $A$ be a countable abelian group satisfying a certain finiteness condition. Suppose that a group $G$ acts on a $(\Gamma \times A)$-graded Lie superalgebra ${\frak L}=\bigoplus_{(\alpha,a) \in \Gamma\times A} {\frak L}_{(\alpha,a)}$ by Lie superalgebra automorphisms preserving the $(\Gamma\times A)$-gradation. In this paper, we show that the Euler-Poincar\'e principle yields the generalized denominator identity for ${\frak L}$ and derive a closed form formula for the supertraces $\text{str}(g|{\frak L}_{(\alpha,a)})$ for all $g\in G$,$(\alpha,a) \in \Gamma\times A$. We discuss the applications of our supertrace formula to various classes of infinite dimensional Lie superalgebras such as free Lie superalgebras and generalized Kac-Moody superalgebras. In particular, we determine the decomposition of free Lie superalgebras into a direct sum of irreducible $GL(n) \times GL(k)$-modules, and the supertraces of the Monstrous Lie superalgebras with group actions. Finally, we prove that the generalized characters of Verma modules and the irreducible highest weight modules over a generalized Kac-Moody superalgebra ${\frak g}$ corresponding to the Dynkin diagram automorphism $\sigma$ are the same as the usual characters of Verma modules and irreducible highest weight modules over the orbit Lie superalgebra $\breve{\frak g}={\frak g}(\sigma)$ determined by $\sigma$.

Zero actions and energy functions for perfect crystals

We give a combinatorial description of the action of the crystal operators \tilde e_0 , \tilde f_0 on certain perfect crystals of U'_q(C_n^{(1)}) , U'_q(D_n^{(1)}) and U'_q(D_{n+1}^{(2)}) , by means of Dynkin diagram automorphisms and Schensted column insertion. Also, for certain level 1 perfect crystals of these algebras we give a definition of a combinatorial “charge” related to the energy function on homogeneous paths in such crystals.

From Dynkin diagram symmetries to fixed point structures

Any automorphism of the Dynkin diagram of a symmetrizable Kac-Moody algebra g induces an automorphism of g and a mappingτ ω between highest weight modules of g. For a large class of such Dynkin diagram automorphisms, we can describe various aspects of these maps in terms of another Kac-Moody algebra, the “orbit Lie algebra” ğ. In particular, the generating function for the trace ofτ ω over weight spaces, which we call the “twining character” of g (with respect to the automorphism), is equal to a character of ğ. The orbit Lie algebras of untwisted affine Lie algebras turn out to be closely related to the fixed point theories that have been introduced in conformal field theory. Orbit Lie algebras and twining characters constitute a crucial step towards solving the fixed point resolution problem in conformal field theory.

Simple WZW superselection sectors

The superselection structure of simple currents of chiral Wess-Zumino-Witten theories, at arbitrary valuek ∈ ℕ of the corresponding affine Lie algebra, is described in terms of explicit localizable automorphisms of the affine algebra. These automorphisms are induced by certain Dynkin diagram automorphisms; under composition, they form an Abelian group isomorphic to the center of the relevant simply connected simple Lie group and, hence, reproduce the WZW fusion rules.