Related papers: Fourier transform on graded Lie algebras

Humphreys' conjecture on blocks parametrises the blocks of reduced enveloping algebras $U_\chi({\mathfrak g})$, where ${\mathfrak g}$ is the Lie algebra of a reductive algebraic group over an algebraically closed field of characteristic…

Let $G$ be a finite group and $p$ be a prime. We study the kernel of the map, between the Burnside ring of $G$ and the Grothendieck ring of $\mathbb{F}_p[G]$-modules, taking a $G$-set to its associated permutation module. We are able, for…

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group $G$, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a…

In this paper we study weighted versions of Fourier algebras of compact quantum groups. We focus on the spectral aspects of these Banach algebras in two different ways. We first investigate their Gelfand spectrum, which shows a connection…

Let $\mathfrak{g}$ be a Lie algebra in characteristic zero equipped with a vector space decomposition $\mathfrak{g}=\mathfrak{g}^-\oplus \mathfrak{g}^+$, and let $s$ and $t$ be commuting formal variables. We prove that the…

In this paper, we consider the relation between two nonabelian Fourier transforms. The first one is defined in terms of the Langlands-Kazhdan-Lusztig parameters for unipotent elliptic representations of a split p-adic group and the second…

In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We…

Let G be a finite group. The Plesken Lie algebra L[G] is a subalgebra of the complex group algebra C[G] and admits a direct-sum decomposition into simple Lie algebras based on the ordinary character theory of G. In this paper we review the…

This paper has two parts. The main goal, carried out in Part I, is to survey some recent work by the authors in which "forced" grading constructions have played a significant role in the representation theory of semisimple algebraic groups…

We reformulate Fourier-space crystallography in the language of cohomology of groups. Once the problem is understood as a classification of linear functions on the lattice, restricted by a particular group relation, and identified by gauge…

In this paper, we focus on how we can interpret the actions of the elements in the Gelfand spectrum of a weighted Fourier algebra on connected Lie groups. They can be viewed as evaluations on specific points of the complexification of the…

In this note, we consider a non-commutative analogy of the classical Fefferman multiplier problem for the ball. More precisely, if $\chi$ is the characteristic function of the unit interval $I=[0,1],$ we investigate a family of differential…

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra containing pointed generating invariant closed convex cones. We determine those derivations $D$ of $\mathfrak{g}$ which induce a 3-grading of the form $\mathfrak{g} =…

In a recent paper by Zhao and the author, the Lie algebras $A[D]=A\otimes F[D]$ of Weyl type were defined and studied, where $A$ is a commutative associative algebra with an identity element over a field $F$ of any characteristic, and…

We explore the graded and filtered formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie…

Inspired by the commutator and anticommutator algebras derived from algebras graded by groups, we introduce noncommutatively graded algebras. We generalize various classical graded results to the noncommutatively graded situation concerning…

Let $G$ be a classical group of Lie rank $l$ and let $C$ be an algebraically closed field of characteristic zero. For $l$ differential indeterminates $\boldsymbol{v}=(v_1,\dots,v_l)$ over $C$ we constructed in a previous paper a general…

Let G be a reductive connected linear algebraic group over an algebraically closed field of positive characteristic and let g be its Lie algebra. First we extend a well-known result about the Picard group of a semisimple group to reductive…

This paper presents a survey of the results and ideas behind the classification of the fine gradings, up to equivalence, on the simple finite dimensional Lie algebras over an algebraically closed field of characteristic zero. It provides an…

For a finite-dimensional representation V of a group G we introduce and study the notion of a Lie element in the group algebra k[G]. The set L(V) \subset k[G] of Lie elements is a Lie algebra and a G-module acting on the original…