A conjecture on cluster automorphisms of cluster algebras

Automorphism groups of pseudo $H$-type algebras

In the present paper, we determine the group of automorphisms of pseudo $H$-type Lie algebras, which are two-step nilpotent Lie algebras closely related to the Clifford algebras $\Cl(\mathbb R^{r,s})$.

The Nakayama automorphism of a self-injective preprojective algebra

We give a simple proof, using Auslander-Reiten theory, that the preprojective algebra of a basic hereditary algebra of finite representation type is Frobenius. We then describe its Nakayama automorphism, which is induced by the Nakayama functor on the module category of our hereditary algebra.

Galois Automorphisms on Harish-Chandra Series and Navarro's Self-Normalizing Sylow $2$-Subgroup Conjecture

G. Navarro has conjectured a necessary and sufficient condition for a finite group $G$ to have a self-normalizing Sylow 2-subgroup, which is given in terms of the ordinary irreducible characters of $G$. In a previous article, the author has reduced the proof of this conjecture to showing that certain related statements hold for simple groups. In this article, we describe the action of Galois automorphisms on the Howlett-Lehrer parametrization of Harish-Chandra induced characters. We use this to complete the proof of the conjecture by showing that the remaining simple groups satisfy the require…

Automorphisms of Coxeter groups and Lusztig's conjectures for Hecke algebras with unequal parameters

Let $(W,S)$ be a Coxeter system, let $G$ be a finite solvable group of automorphisms of $(W,S)$ and let $φ$ be a weight function which is invariant under $G$. Let $φ_G$ denote the weight function on $W^G$ obtained by restriction from $φ$. The aim of this paper is to compare the ${\mathbf{a}}$-function, the set of Duflo involutions and the Kazhdan-Lusztig cells associated to $(W,φ)$ and to $(W^G,φ_G)$, provided that Lusztig's Conjectures hold.

Cluster automorphism groups of cluster algebras with coefficients

We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. For this, we introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra (i.e. the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients, cluster algebras with universal geom…

Twisted automorphisms of group algebras

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".

Convolution algebra of diagram automorphism fixed quiver variety

We study the convolution algebra $H_{*}(Z^θ_{W})$ of homology on diagram automorphism fixed point quiver variety and prove that there exists an algebra homomorphism from the universal enveloping algebra of the diagram automorphism fixed algebra of the split quiver to $H_{*}(Z^θ_{W})$.

Automorphisms of cluster algebras of rank 2

We compute the automorphism group of the affine surfaces with the coordinate ring isomorphic to a cluster algebra of rank 2.

带图自同构作用的扭曲多圈代数的整基

Integral bases for twisted multiloop algebras with diagram automorphism actions

该研究针对带图自同构作用的扭曲多圈代数，构造了其泛包络代数的整形式及显式整基，解决了相关代数结构的整基存在性与构造问题。

Broué's isotypy conjecture for the sporadic groups and their covers and automorphism groups

Let B be a p-block of a finite group G with abelian defect group D such that S\unlhd G, S'=S, G/Z(S)\le\Aut(S) and S/Z(S) is a sporadic simple group. We show that B is isotypic to its Brauer correspondent in N_G(D) in the sense of Broué. This has been done by [Rouquier, 1994] for principal blocks and it remains to deal with the non-principal blocks.

Periodic automorphisms of Takiff algebras, contractions, and $θ$-groups

Let $q$ be an algebraic Lie algebra and $q<m>$ a (generalised) Takiff algebra. Any finite order automorphism $θ$ of $q$ induces an automorphisms of $q<m>$ of the same order, denoted $Θ$. We study invariant-theoretic properties of representations of the fixed point subalgebra of $Θ$ on other eigenspaces of $Θ$ in $q<m>$.

On automorphisms of enveloping algebras

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 $\mathfrak{g}$ …

Stabilizers of Subspaces under Similitudes of the Klein Quadric, and Automorphisms of Heisenberg Algebras

We determine the groups of automorphisms and their orbits for nilpotent Lie algebras of class 2 and small dimension, over arbitrary fields (including the characteristic 2 case).

Weight-one elements of vertex operator algebras and automorphisms of categories of generalized twisted modules

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.