Quantum $\frak {gl}_\infty$, infinite $q$-Schur algebras and their representations

Jie Du; Qiang Fu

Home/Archive/arXiv

arXiv · 0708.2525

Quantum $\frak {gl}_\infty$, infinite $q$-Schur algebras and their representations

Jie Du·Qiang Fu

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we investigate the structure and representations of the quantum group ${\mathbf{U}(\infty)}=\mathbf U_\upsilon(\frak{gl}_\infty)$. We will present a realization for $\mathbf{U}(\infty)$, following Beilinson--Lusztig--MacPherson (BLM) \cite{BLM}, and show that the natural algebra homomorphism $ζ_r$ from $\mathbf{U}(\infty)$ to the infinite $q$-Schur algebra ${\boldsymbol{\mathcal S}}(\infty,r)$ is not surjective for any $r\geq 1$. We will give a BLM type realization for the image $\mathbf{U}(\infty,r):=ζ_r(\mathbf{U}(\infty))$ and discuss its presentation in terms of generators and relations. We further construct a certain completion algebra $\hat{\boldsymbol{\mathcal K}}^\dagger(\infty)$ so that $ζ_r$ can be extended to an algebra epimorphism $\tildeζ_r:\hat{\boldsymbol{\mathcal K}}^\dagger(\infty)\to{\boldsymbol{\mathcal S}}(\infty,r)$. Finally we will investigate the representation theory of ${\bf U}(\infty)$, especially the polynomial representations of ${\bf U}(\infty)$.

Explore related subjects

math.QA math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Jie Du, Qiang Fu. 2011-06-23. Quantum $\frak {gl}_\infty$, infinite $q$-Schur algebras and their representations. https://arxiv.org/abs/0708.2525

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Generalised Joyal disks, $Θ_d$-colored $(d+1)$-operads, and higher Deligne conjecture

The classical Deligne conjecture (which by now has found several different proofs) says that the Hochschild cohomological complex of a small dg category over a field $k$ has a structure of a $C(E_2,k)$-algebra. A similar statement for any small dg weak $d$-category is called the generalised Deligne conjecture, it says the cohomological Hochschild complex of a dg weak $d$-category (appropriately defined) admits an action of the operad $C(E_{d+1},k)$. This paper is the first in a series of papers, in which our goal is to construct a $(d+1)$-algebra providing a solution to the generalised Deligne conjecture, on the level of complexes. This paper contains the combinatorial core of this $(d+1)$-algebra structure, for any $d\ge 1$. Namely, we construct a colored $(d+1)$-operad in sets (in the sense of M.Batanin), denoted by $\mathbf{seq}_d$. Its category of colors (= the category of unary operations) is the category $Θ_d$ of A.Joyal [J], dual to the category of Joyal $d$-disks [J], [Be2,3]. For $d=1$, our $Δ$-colored operad $\mathbf{seq}_1$ coincides with the operad $\mathbf{seq}$ of D.Tamarkin [T3]. We prove that the construction indeed gives rise to a $(d+1)$-operad, and that this $Θ_d$-colored $(d+1)$-operad $\mathbf{seq}_d$ is contractible in the dg and in the topological condensations, for any $d\ge 1$. The contractibility of a $(d+1)$-operad $\mathcal{O}$ is a key property, which, due to the Batanin symmetrisation theorem [Ba1,2], endows any its ($d$-terminal) algebra with an $E_{d+1}$-algebra structure.

math.QA↗

arXiv2026

Graded Satake diagrams and super-symmetric pairs

We list classical spherical subalgebras in basic matrix Lie superalgebras which are quantizable to coideal subalgebras in the standard quantum supergroups, for any choice of Borel subalgebra. We classify the corresponding Satake-type diagrams and prove that each of them defines a family of proper spherical subalgebras.

math.QA↗

arXiv2026

Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras

Let $G=FΓ$ be a factorization of a finite group, with neither factor assumed normal and with $F\capΓ$ allowed to be nontrivial. We prove that $\exp(G)$ divides $\operatorname{lcm}(|F|,|Γ|)$, or equivalently that $\gcd([G:F],[G:Γ])\exp(G)$ divides $|G|$. This answers a cohomological divisibility question posed by Natale. Combining the group-theoretic divisibility with Natale's exponent bound and a lifting argument, we prove Kashina's exponent conjecture, in the arbitrary-field formulation of Etingof and Gelaki, for every finite-dimensional semisimple and cosemisimple Hopf algebra $H$ over a field $k$ for which $\operatorname{Rep}(H\otimes_k\overline{k})$ is group-theoretical. The same argument proves the corresponding degree-three cohomological divisibility for coefficients in an arbitrary $G$-module. For complex group-theoretical categories, we also establish Frobenius-Schur exponent divisibility under a cohomological factorization hypothesis, without assuming a fiber functor. We derive applications to low-dimensional Hopf algebras and abelian extensions.

math.QA↗