Manin associated to a quadratic algebra (quantum space) the quantum matrix group of its automorphisms. This Talk aims to demonstrate that Manin's construction can be extended for quantum spaces which are non-quadratic homogeneous algebras. Here given a regular Artin-Schelter algebra of dimension 3 we construct the quantum group of its symmetries, i.e., the Hopf algebra of its automorphisms. For quadratic Artin-Schelter algebras these quantum groups are contained in the the classification of the GL(3) quantum matrix groups due to Ewen and Ogievetsky. For cubic Artin-Schelter algebras we obtain new quantum groups which are automorphisms of cubic quantum spaces.
We investigate the general structure of the automorphism group and the Lie algebra of derivations of a finitely generated vertex operator algebra. The automorphism group is isomorphic to an algebraic group. Under natural assumptions, the derivation algebra has an invariant bilinear form and the ideal of inner derivations is nonsingular.
We give a complete description of the full automorphism group of a lattice vertex operator algebra, determine the twisted Zhu's algebra for the automorphism lifted from the -1 isometry of the lattice and classify the corresponding irreducible twisted modules through the twisted Zhu algebra.
The derivation algebras, automorphism groups and second cohomology groups of the generalized loop Schrödinger-Virasoro algebras are completely determined in this paper.
Let g be a simple complex finite dimensional Lie algebra and let U_q^+(g) be the positive part of the quantum enveloping algebra of g. If g is of type A_2, the group of algebra automorphisms of U_q^+(g) is a semidirect product of a 2-dimensional torus and the group of automorphisms of the Dynkin diagram; any algebra automorphism is an automorphism of braided Hopf algebra, and preserves the standard grading [AD1]. This intriguing smallness of the group of algebra automorphisms raises questions about the extent of these phenomena. We discuss some of them in the present paper. We introduce the no…
Herein we study conformal vectors of a Z-graded vertex algebra of (strong) CFT type. We prove that the full vertex algebra automorphism group transitively acts on the set of the conformal vectors of strong CFT type if the vertex algebra is simple. The statement is equivalent to the uniqueness of self-dual vertex operator algebra structures of a simple vertex algebra. As an application, we show that the full vertex algebra automorphism group of a simple vertex operator algebra of strong CFT type uniquely decomposes into the product of certain two subgroups and the vertex operator algebra automo…
Graded quasi-commutative skew PBW extensions are isomorphic to graded iterated Ore extensions of endomorphism type, whence graded quasi-commutative skew PBW extensions with coefficients in AS-regular algebras are skew Calabi-Yau and the Nakayama automorphism exists for these extensions. With this in mind, in this paper we give a description of Nakayama automorphism for these non-commutative algebras using the Nakayama automorphism of the ring of the coefficients.
We determine the automorphism groups of the cyclic orbifold vertex operator algebras associated with coinvariant lattices of isometries of the Leech lattice in the conjugacy classes $4C,6E,6G,8E$ and $10F$. As a consequence, we have determined the automorphism groups of all the $10$ vertex operator algebras in [Hö], which are useful to analyze holomorphic vertex operator algebras of central charge $24$.
Recently, Willwacher showed that the Grothendieck-Teichmuller group GRT acts by L-infinity-automorphisms on the Schouten algebra of polyvector fields T_poly(R^d) on affine space R^d. In this article, we prove that a large class of L-infinity-automorphisms on the Schouten algebra, including Willwacher's, can be globalized. That is, given an L-infinity-automorphism of T_poly(R^d) and a general smooth manifold M with the choice of a torsion-free connection, we give an explicit construction of an L-infinity-automorphism of the Schouten algebra T_poly(M) on the manifold M, depending on the chosen c…
We continue the study of twisted automorphisms of Hopf algebras started in "Twisted automorphisms of Hopf algebras". In this paper we concentrate on the group algebra case. We describe the group of twisted automorphisms of the group algebra of a group of order coprime to 6. The description turns out to be very similar to the one for the universal enveloping algebra given in "Twisted automorphisms of Hopf algebras".
We determine the group of algebra automorphisms for the two-parameter quantized enveloping algebra $\V$. As an application, we prove that the group of Hopf algebra automorphisms for $\V$ is isomorphic to a torus of rank two.
In this paper, we completely determine the group of algebra automorphisms for the two-parameter Hopf algebra ${\check U}_{r,s}^{\geq 0}({\mathfrak sl_{3}})$. As a result, the group of Hopf algebra automorphisms is determined for $\V$ as well. We further characterize all the derivations of the subalgebra $U^{+}_{r,s}({\mathfrak sl_{3})}$, and calculate its first degree Hochschild cohomology group.
We describe the groups of automorphisms of two generated free braided associative algebras with involutive diagonal braidings over a field of characteristic $\neq 2$. Depending on the form of the diagonal involutive braiding, five different automorphism groups arise as automorphism groups of two generated free braided associative algebras. This list covers all three automorphism groups that arise in the case of quantum planes \cite{AC}.
The conjugacy classes of the involutive automorphisms of the affine Kac-Moody algebras \Cli\ for $\ell\geq 2$ are determined using the matrix formulation of automorphisms of an affine Kac-Moody algebra.
Given an algebraic Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we canonically associate to it a Lie algebra $\mathfrak{g}_{\infty}$ defined over $\mathbb{C}_{\infty}$-the reduction of $\mathbb{C}$ mod infinitely large prime, and show that for a class of Lie algebras $\mathfrak{g}_{\infty}$ is an invariant of the derived category of $\mathfrak{g}$-modules. We give two applications of this construction. First, we show that the bounded derived category of $\mathfrak{g}$-modules determines algebra $\mathfrak{g}$ for a class of Lie algebras. Second, given a semi-simple Lie algebra $\mathfrak{g}$ …
In this article, we determine the automorphism groups of $14$ holomorphic vertex operator algebras of central charge $24$ obtained by applying the $\mathbb{Z}_2$-orbifold construction to the Niemeier lattice vertex operator algebras and lifts of the $-1$-isometries.
In this article, we give a method of calculating the automorphism groups of the vertex operator algebras $V_L^+$ associated with even lattices $L$. For example, by using this method we determine the automorphism groups of $V_L^+$ for even lattices of rank one, two and three, and even unimodular lattices.
We study the quantum automorphism group of the lexicographic product of two finite regular graphs, providing a quantum generalization of Sabidussi's structure theorem on the automorphism group of such a graph.
Every finite dimensional Hopf algebra is a Frobenius algebra, with Frobenius homomorphism given by an integral. The Nakayama automorphism determined by it yields a decomposition with degrees in a cyclic group. For a family of pointed Hopf algebras, we determine necessary and sufficient conditions for this decomposition to be strongly graded.
Many important quantum algebras such as quantum symplectic space, quantum Euclidean space, quantum matrices, $q$-analogs of the Heisenberg algebra and the quantum Weyl algebra are semi-commutative. In addition, enveloping algebras $U(L_+)$ of even Lie color algebras are also semi-commutative. In this paper, we generalize work of Montgomery and examine the $X$-inner automorphisms of such algebras. The theorems and examples in our paper show that for algebras $R$ of this type, the non-identity $X$-inner automorphisms of $R$ tend to have infinite order. Thus if $G$ is a finite group of automorphi…
Automorphisms of algebras $R$ from a very large axiomatic class of quantum nilpotent algebras are studied using techniques from noncommutative unique factorization domains and quantum cluster algebras. First, the Nakayama automorphism of $R$ (associated to its structure as a twisted Calabi-Yau algebra) is determined and shown to be given by conjugation by a normal element, namely, the product of the homogeneous prime elements of $R$ (there are finitely many up to associates). Second, in the case when $R$ is connected graded, the unipotent automorphisms of $R$ are classified up to minor excepti…
In this paper, we study the algebra automorphisms and isomorphisms for a family of "symmetric" multiparameter quantized Weyl algebras $\A$ and some related algebras in the generic case. First, we compute the Nakayama automorphism for $\A$ and give a necessary and sufficient condition for $\A$ to be Calabi-Yau. We also prove that $\A$ is cancellative. Then we determine the automorphism group for $\A$ and its polynomial extension $\E$. As an application, we solve the isomorphism problem for $\{\A\}$ and $\{\E\}$. Similar results will be established for the Maltisiniotis multiparameter quantized …
Given a weight-one element $u$ of a vertex operator algebra $V$, we construct an automorphism of the category of generalized $g$-twisted modules for automorphisms $g$ of $V$ fixing $u$. We apply this construction to the case that $V$ is an affine vertex operator algebra to obtain explicit results on these automorphisms of categories. In particular, we give explicit constructions of certain generalized twisted modules from generalized twisted modules associated to diagram automorphisms of finite-dimensional simple Lie algebras and generalized (untwisted) modules.
