Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0606499

Differential Calculi on Some Quantum Prehomogeneous Vector Spaces

S. Sinel'shchikov·A. Stolin·L. Vaksman
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper is devoted to study of differential calculi over quadratic algebras, which arise in the theory of quantum bounded symmetric domains. We prove that in the quantum case dimensions of the homogeneous components of the graded vector spaces of k-forms are the same as in the classical case. This result is well-known for quantum matrices. The quadratic algebras, which we consider in the present paper, are q-analogues of the polynomial algebras on prehomogeneous vector spaces of commutative parabolic type. This enables us to prove that the de Rham complex is isomorphic to the dual of a quantum analogue of the generalized Bernstein-Gelfand-Gelfand resolution.

Explore related subjects

math.QAmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

S. Sinel'shchikov, A. Stolin, L. Vaksman. 2007-05-03. Differential Calculi on Some Quantum Prehomogeneous Vector Spaces. https://doi.org/10.1063/1.2746131

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

KEEP EXPLORING

Related papers

arXiv2026

An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence

We describe infinitesimal deformations of post-Lie algebras and post-Hopf algebras and prove that the universal enveloping algebra and primitive elements adjunction is compatible with the infinitesimal structure. In characteristic zero, this restricts to an equivalence for connected cocommutative infinitesimal post-Hopf algebras, extending the Cartier--Milnor--Moore theorem. We parametrize all infinitesimal post-Lie structures on $\mathfrak{sl}(2)$ and exhibit a second-order obstruction. From a geometric point of view, infinitesimal post-Lie algebras emerge from flat connections with covariantly constant torsion and an additional compatible symmetric $(1,2)$-tensor. This leads to explicit examples of infinitesimal post-Lie structures based on Heisenberg Lie algebras. On the Hopf side, we classify infinitesimal structures above the trivial action, prove rigidity for group algebras, and describe the structures on Sweedler's Hopf algebra. The subadjacent construction induces a Hopf algebra deformation with fixed coproduct. Finally, we prove that the quadratic operad of infinitesimal post-Lie algebras is Koszul, using a filtered distributive law between the operads of Lie algebras and bi-magmas.

math.QA↗
arXiv2026

Galois-modular obstructions to the categorification of near-group fusion rings

In this article we establish new obstructions to the categorification of near-group fusion rings by combining the values of irreducible characters of the Grothendieck ring with the Galois symmetry of the induced unit object $I(1)$ in the Drinfeld center. As a result, we show that a near-group ring of type $G+m$, where $G$ is an abelian group and both $|G|$ and $m$ are odd, can only categorify if $m=|G|$.

math.QA↗
arXiv2026

Reflective algebras over factorizable Hopf algebras

The reflective algebra $(R_H(A),K_{\mathrm{ref}}(A))$, introduced by Laugwitz, Walton and Yakimov, is a quasitriangular left $H$-comodule algebra constructed from a quasitriangular Hopf algebra $(H,R)$ and a left $H$-comodule algebra $A$. We study the structure and factorizability of finite-dimensional reflective algebras. When $(H,R)$ is factorizable, we show that $(R_H(A),K_{\mathrm{ref}}(A))$ is isomorphic, as a quasitriangular left $H$-comodule algebra, to $A\otimes H$ equipped with the trivial left $H$-coaction on $A$ and the universal $K$-matrix induced by $R_{21}R$. In this case, we also classify all universal $K$-matrices on $R_H(A)$. Finally, we prove that $(R_H(A),K_{\mathrm{ref}}(A))$ is factorizable if and only if $(H,R)$ is factorizable and $A$ is simple as an algebra.

math.QA↗