Related papers: Mixed tensor products and Capelli-type determinant…

A result of Farahat and Higman shows that there is a ``universal'' algebra, $\mathrm{FH}$, interpolating the centres of symmetric group algebras, $Z(\mathbb{Z}S_n)$. We explain that this algebra is isomorphic to $\mathcal{R} \otimes…

Let A be a Hopf algebra and H a coalgebra. We shall describe and classify up to an isomorphism all Hopf algebras E that factorize through A and H: that is E is a Hopf algebra such that A is a Hopf subalgebra of E, H is a subcoalgebra in E…

We study the ranges of the maps of convolution $u\otimes v\mapsto u\ast v$ and a `twisted' convolution $u\otimes v\mapsto u\ast \check{v}$ ($\check{u}(s)=u(s^{-1})$) and on the Haagerup tensor product of a Fourier algebra of a compact group…

Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings and a unitary local system V on it. We consider a differential graded Lie algebra (DGLA) of forms with holomorphic logarithmic singularities…

We propose a universal matrix Capelli identity and explain how to derive Capelli identities for all quantum immanants in the Reflection Equation algebra and in the universal enveloping algebra U(gl_(M|N)).

A global representation is a compatible collection of representations of the outer automorphism groups of the groups belonging to some collection of finite groups $\mathscr{U}$. Global representations assemble into an abelian category…

We prove the equivalence of two tensor products over a category of W*-algebras with normal (not necessarily unital) *-homomorphisms, defined by Guichardet and Dauns, respectively. This structure differs from the standard tensor product…

Given a finite generating set $T=\{g_0,\dots, g_n\}$ of a group $G$, and a representation $\rho$ of $G$ on a Hilbert space $V$, we investigate how the geometry of the set $D(T,\rho)=\{ [x_0 : \dots : x_n] \in\mathbb C\mathbb P^n \mid \sum…

The problem of whether the cohomological support map of a finite dimensional Hopf algebra has the tensor product property has attracted a lot of attention following the earlier developments on representations of finite group schemes. Many…

Let $\mathfrak{g}$ be a Lie algebra over an algebraically closed field $\Bbbk$ of characteristic zero. Define the universal grading group $\mathcal{C}(\mathfrak{g})$ as having one generator $g_{\rho}$ for each irreducible…

The principle of tannakian duality states that any neutral tannakian category is tensorially equivalent to the category Rep_k G of finite dimensional representations of some affine group scheme G and field k, and conversely. Originally…

We define the Hopf superalgebra U_T sl(1,1), which is a variant of the quantum supergroup U_q sl(1,1), and its tensor product representations V_1^{\otimes n} for n>0. We construct families of DG algebras A, B and R_n, and consider the DG…

The universal enveloping algebra $U(\mathfrak{sl}_2)$ of $\mathfrak{sl}_2$ is a unital associative algebra over $\mathbb C$ generated by $E,F,H$ subject to the relations \begin{align*} [H,E]=2E, \qquad [H,F]=-2F, \qquad [E,F]=H.…

This paper focuses on the $GL_n$ tensor product algebra, which encapsulates the decomposition of tensor products of arbitrary finite dimensional irreducible representations of $GL_n$. We will describe an explicit basis for this algebra.…

In this paper, we compute the homology group and cohomology algebra of various polyhedral product objects uniformly from the point of view of diagonal tensor product. As applications, we introduce the polyhedral product method into…

By conformal welding, there is a pair of univalent functions $(f,g)$ associated to every point of the complex K\"ahler manifold $\Mob(S^1)\bk\Diff_+(S^1)$. For every integer $n\geq 1$, we generalize the definition of Faber polynomials to…

Extending work of Saneblidze-Umble and others, we use diagonals for the associahedron and multiplihedron to define tensor products of A-infinity algebras, modules, algebra homomorphisms, and module morphisms, as well as to define a bimodule…

We review the construction of the multiparametric quantum group $ISO_{q,r}(N)$ as a projection from $SO_{q,r}(N+2) $ and show that it is a bicovariant bimodule over $SO_{q,r}(N)$. The universal enveloping algebra $U_{q,r}(iso(N))$,…

Every irreducible finite-dimensional representation of the quantized enveloping algebra U_q(gl_n) can be extended to the corresponding quantum affine algebra via the evaluation homomorphism. We give in explicit form the necessary and…

This paper develops a unified framework for observables in n-plectic geometry, extending the L_infty-algebra of Hamiltonian (n-1)-forms to Hamiltonian forms of all degrees via a degree-shifting Grassmann variable u that encodes submanifold…