Related papers: Projective representations of mapping class groups…

We study topological Hopf algebras that are holomorphically finitely generated (HFG) as Fr\'echet Arens--Micheal algebras in the sense of Pirkovskii. Some of them, but not all, can be obtained from affine Hopf algebras by applying the…

This paper is a sequel to the paper by A. Losev and Yu. Manin [LoMa1], in which new moduli stacks $\bar{L}_{g,S}$ of pointed curves were introduced. They classify curves endowed with a family of smooth points divided into two groups, such…

Combinatorial Hopf algebras give a linear algebraic structure to infinite families of combinatorial objects, a technique further enriched by the categorification of these structure via the representation theory of families of algebras. This…

Let $G$ be a discrete countable group, and let $\Gamma$ be an almost normal subgroup. In this paper we investigate the classification of (projective) unitary representations $\pi$ of $G$ into the unitary group of the Hilbert space…

Let G be a locally compact group, M(G) denote its measure algebra and L^1(G) denote its group algebra. Also, let pi:G->U(H) be a strongly continuous unitary representation, and let CB^{sigma}(B(H)) be the space of normal completely bounded…

Jensen, Su, and Yang described the projective indecomposable modules of the $0$-Schur algebra $\mathbf{S}_0(n,r)$ using its geometric realization. In this paper, the simple modules of $\mathbf{S}_0(n,r)$ are identified by computing the tops…

We introduce a family of periods of mixed Tate motives called dissection polylogarithms, that are indexed by combinatorial objects called dissection diagrams. The motivic coproduct on the former is encoded by a combinatorial Hopf algebra…

We provide new constraints for algebro-geometric subgroups of mapping class groups, namely images of fundamental groups of curves under complex algebraic maps to the moduli space of smooth curves. Specifically, we prove that the restriction…

We consider compact locally symmetric spaces $\Gamma\backslash G/H$ where $G/H$ is a non-compact semisimple symmetric space and $\Gamma$ is a discrete subgroup of $G$. We discuss some features of the joint spectrum of the (commutative)…

Let $\Sigma_{g,n}$ be the orientable genus $g$ surface with $n$ punctures, where $2-2g-n<0$. Let $$\rho: \pi_1(\Sigma_{g,n})\to GL_m(\mathbb{C})$$ be a representation. Suppose that for each finite covering map $f: \Sigma_{g', n'}\to…

We develop a calculus based on graph enumeration for $S_n$-equivariant motivic invariants of graphically stratified moduli spaces. We apply our theory to the Deligne--Mumford moduli space $\overline{\mathcal{M}}_{g, n}$ and to the space of…

We use quantum invariants to define an analytic family of representations for the mapping class group of a punctured surface. The representations depend on a complex number A with |A| <= 1 and act on an infinite-dimensional Hilbert space.…

We investigate the representation theory of the rational and trigonometric Cherednik algebra of type $GL_n$ by means of combinatorics on periodic (or cylindrical) skew diagrams. We introduce and study standard tableaux and plane partitions…

In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the…

In the aim to understand the generalization of Stirling numbers occurring in the bosonic normal ordering problem, several combinatorial models have been proposed. In particular, Blasiak \emph{et al.} defined combinatorial objects allowing…

Given a locally compact group $G$ and a unitary representation $\rho:G\to U({\mathcal H})$ on a Hilbert space ${\mathcal H}$, we construct a $C^*$-correspondence ${\mathcal E}(\rho)={\mathcal H}\otimes_{\mathbb C} C^*(G)$ over $C^*(G)$ and…

We define algebraic families of (all) morphisms which are purely algebraic analogs of quantum families of (all) maps introduced by P.M. Soltan. Also, algebraic families of (all) isomorphisms are introduced. By using these notions we…

We generalize the theory of the second invariant cohomology group $H^2_{\rm inv}(G)$ for finite groups $G$, developed in [Da2,Da3,GK], to the case of affine algebraic groups $G$, using the methods of [EG1,EG2,G]. In particular, we show that…

This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called…

We construct an invariant of closed oriented $3$-manifolds using a finite dimensional, involutory, unimodular and counimodular Hopf algebra $H$. We use the framework of normal o-graphs introduced by R. Benedetti and C. Petronio, in which…