On the dimension of H-strata in quantum matrices

Primitive ideals in quantum Schubert cells: dimension of the strata

The aim of this paper is to study the representation theory of quantum Schubert cells. Let $\g$ be a simple complex Lie algebra. To each element $w$ of the Weyl group $W$ of $\g$, De Concini, Kac and Procesi have attached a subalgebra $U_q[w]$ of the quantised enveloping algebra $U_q(\g)$. Recently, Yakimov showed that these algebras can be interpreted as the quantum Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of $U_q[w]$. More precisely, it follows from the Stratification Theorem of Goodearl and Letzter that the primitive spectrum of $U_q[w]$ admits …

Spectra and catenarity of multiparameter quantum Schubert cells

We study the ring theory of the multiparameter deformations of the quantum Schubert cell algebras obtained from 2-cocycle twists. This is a large family, which extends the Artin-Schelter-Tate algebras of twisted quantum matrices. We classify set theoretically the spectra of all such multiparameter quantum Schubert cell algebras, construct each of their prime ideals by contracting from explicit normal localizations, and prove formulas for the dimensions of their Goodearl-Letzter strata for base fields of arbitrary characteristic and all deformation parameters that are not roots of unity. Furthe…

Representation theory of Jordanian algebra

We describe the complete set of pairwise non-isomorphic irreducible modules S(a) over the algebra R given by the defining relation xy-yx=yy, and the rule how they could be glued to indecomposables. Namely, we show that Ext_k^1(S(a),S(b))=0, if a not equal to b. Also the set of all representations is described subject to the Jordan normal form of Y. We study then properties of the image algebras in the endomorphism ring. Among facts we prove is that they are all basic algebras. Along this line we establish an analogue of the Gerstenhaber-Taussky-Motzkin theorem on the dimension of algebras gene…

Quantized coordinate rings and related noetherian algebras

This paper contains a survey of some ring-theoretic aspects of quantized coordinate rings, with primary focus on the prime and primitive spectra. For these algebras, the overall structure of the prime spectrum is governed by a partition into strata determined by the action of a suitable group of automorphisms of the algebra. We discuss this stratification in detail, as well as its use in determining the primitive spectrum -- under suitable conditions, the primitive ideals are precisely those prime ideals which are maximal within their strata. The discussion then turns to the global structure o…

通过 incidence 变换研究变形的几何

Geometry of Deformations via Incidence Varieties

本文通过构造 GL-等变双线性 incidence 变换，统一研究了有限维向量空间上四种代数结构的几何变形问题，并证明了其与上同调理论的同构关系。