Search results for "Representation theory"
showing 10 items of 197 documents
A Classification of Modular Functors via Factorization Homology
2022
Modular functors are traditionally defined as systems of projective representations of mapping class groups of surfaces that are compatible with gluing. They can formally be described as modular algebras over central extensions of the modular surface operad, with the values of the algebra lying in a suitable symmetric monoidal $(2,1)$-category $\mathcal{S}$ of linear categories. In this paper, we prove that modular functors in $\mathcal{S}$ are equivalent to self-dual balanced braided algebras $\mathcal{A}$ in $\mathcal{S}$ (a categorification of the notion of a commutative Frobenius algebra) for which a condition formulated in terms of factorization homology with coefficients in $\mathcal{…
The distinguished invertible object as ribbon dualizing object in the Drinfeld center
2022
We prove that the Drinfeld center $Z(\mathcal{C})$ of a pivotal finite tensor category $\mathcal{C}$ comes with the structure of a ribbon Grothendieck-Verdier category in the sense of Boyarchenko-Drinfeld. Phrased operadically, this makes $Z(\mathcal{C})$ into a cyclic algebra over the framed $E_2$-operad. The underlying object of the dualizing object is the distinguished invertible object of $\mathcal{C}$ appearing in the well-known Radford isomorphism of Etingof-Nikshych-Ostrik. Up to equivalence, this is the unique ribbon Grothendieck-Verdier structure on $Z(\mathcal{C})$ extending the canonical balanced braided structure that $Z(\mathcal{C})$ already comes equipped with. The duality fun…
Le cône diamant symplectique
2009
Resume Si n + est le facteur nilpotent d'une algebre semi-simple g , le cone diamant de g est la description combinatoire d'une base d'un n + module indecomposable naturel. Cette notion a ete introduite par N.J. Wildberger pour sl ( 3 ) , le cone diamant de sl ( n ) est decrit dans Arnal (2006) [2] , celui des algebres semi-simples de rang 2 dans Agrebaoui (2008) [1] . Dans cet article, nous generalisons ces constructions au cas des algebres de Lie sp ( 2 n ) . Les tableaux de Young semi-standards symplectiques ont ete definis par C. De Concini (1979) [4] , ils forment une base de l'algebre de forme de sp ( 2 n ) . Nous introduisons ici la notion de tableaux de Young quasi standards symplec…
Complex group algebras of finite groups: Brauer's Problem 1
2007
Abstract Brauer's Problem 1 asks the following: What are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to present a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number m of isomorphic summands, then its dimension is bounded in terms of m . We prove that this is true for every finite group if it is true for the symmetric groups. The problem for symmetric groups reduces to an explicitly stated question in number theory or combinatorics.
Codimensions of algebras and growth functions
2008
Abstract Let A be an algebra over a field F of characteristic zero and let c n ( A ) , n = 1 , 2 , … , be its sequence of codimensions. We prove that if c n ( A ) is exponentially bounded, its exponential growth can be any real number >1. This is achieved by constructing, for any real number α > 1 , an F-algebra A α such that lim n → ∞ c n ( A α ) n exists and equals α. The methods are based on the representation theory of the symmetric group and on properties of infinite Sturmian and periodic words.
Coprime actions and correspondences of Brauer characters
2017
We prove several results giving substantial evidence in support of the conjectural existence of a Glauberman–Isaacs bijection for Brauer characters under a coprime action. We also discuss related bijections for the McKay conjecture.
Unbounded C*-seminorms, biweights and *-representations of partial *-algebras: a review
2006
The notion of (unbounded) C*-seminorms plays a relevant role in the representation theory of *-algebras and partial *-algebras. A rather complete analysis of the case of *-algebras has given rise to a series of interesting concepts like that of semifinite C*-seminorm and spectral C*-seminorm that give information on the properties of *-representations of the given *-algebra A and also on the structure of the *-algebra itself, in particular when A is endowed with a locally convex topology. Some of these results extend to partial *-algebras too. The state of the art on this topic is reviewed in this paper, where the possibility of constructing unbounded C*-seminorms from certain families of p…
On Almost Nilpotent Varieties of Linear Algebras
2020
A variety \(\mathcal {V}\) is almost nilpotent if it is not nilpotent but all proper subvarieties are nilpotent. Here we present the results obtained in recent years about almost nilpotent varieties and their classification.
CHEVALLEY COHOMOLOGY FOR KONTSEVICH'S GRAPHS
2005
We introduce the Chevalley cohomology for the graded Lie algebra of polyvector fields on $R^d$. This cohomology occurs naturally in the problem of construction and classification of fomalities on the sapce $ R^d$. Considering only graphs formalities, we define the Chevalley cohomology directly on spaces of graphs. We obtain some simple expressions for the Chevalley coboundary operator and we give examples and applications.
On algebraic supergroups, coadjoint orbits and their deformations
2004
In this paper we study algebraic supergroups and their coadjoint orbits as affine algebraic supervarieties. We find an algebraic deformation quantization of them that can be related to the fuzzy spaces of non-commutative geometry.