Related papers: Differential operators on quantized flag manifolds…
We formulate a Beilinson-Bernstein type derived equivalence for a quantized enveloping algebra at a root of 1 as a conjecture. It says that there exists a derived equivalence between the category of modules over a quantized enveloping…
We give a proof of Lusztig's conjectural multiplicity formula for non-restricted modules over the De Concini-Kac type quantized enveloping algebra at the $\ell$-th root of unity, where $\ell$ is an odd prime power satisfying certain…
The quantized flag manifold, which is a $q$-analogue of the ordinary flag manifold, is realized as a non-commutative scheme, and we can define the category of $D$-modules on it using the framework of non-commutative algebraic geometry;…
We establish some properties of the ring of differential operators on the quantized flag manifold. Especially, we give an explicit description of its localization on an affine open subset in terms of the quantum Weyl algebra ($q$-analogue…
In this paper we construct explicitly natural (from the geometrical point of view) Fock space representations (contragradient Verma modules) of the quantized enveloping algebras. In order to do so, we start from the Gauss decomposition of…
We study a family of modules over Kac-Moody algebras realized in multi-valued functions on a flag manifold and find integral representations for intertwining operators acting on these modules. These intertwiners are related to some…
We show that the quantized flag manifold at a root of unity has natural affine open covering parametrized by the elements of the Weyl group. In particular, the quantized flag manifold turns out to be a quasi-scheme in the sense of…
Let $\mathfrak{g}$ be a semi-simple Lie algebra with fixed root system, and $U_q(\mathfrak{g})$ the quantization of its universal enveloping algebra. Let $\mathcal{S}$ be a subset of the simple roots of $\mathfrak{g}$. We show that the…
The discussions in the present paper arise from exploring intrinsically the structure nature of the quantum $n$-space. A kind of braided category $\Cal {GB}$ of $\La$-graded $\th$-commutative associative algebras over a field $k$ is…
We study certain Poisson structures related to quantized enveloping algebras. In particular, we give a description of the Poisson structure of a certain manifold associated to the ring of differential operators.
Quantum algebra of differential operators are studied
We study irreducible *-representations of a certain quantization of the algebra of polynomial functions on a generalized flag manifold regarded as a real manifold. All irreducible *-representations are classified for a subclass of flag…
It is shown that quantized irreducible flag manifolds possess a canonical $q$-analogue of the de Rham complex. Generalizing the well known situation for the standard Podle\'s' quantum sphere this analogue is obtained as the universal…
We introduce the notion of a twisted differential operator of given radius relative to an endomorphism $$\sigma$$ of an affinoid algebra A. We show that this notion is essentially independent of the choice of the endomorphism $$\sigma$$. As…
The algebra of quantum differential operators on graded algebras was introduced by V. Lunts and A. Rosenberg. D. Jordan, T. McCune and the second author have identified this algebra of quantum differential operators on the polynomial…
The universal enveloping algebra U(g) of a Lie algebra g acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or "quantum group") is a deformation of a universal…
Generalized flag manifolds endowed with the Bruhat-Poisson bracket are compact Poisson homogeneous spaces, whose decompositions in symplectic leaves coincide with their stratifications in Schubert cells. In this note it is proved that the…
The paper is concerned with cohomology of the small quantum group at a root of unity, and of its upper triangular subalgebra, with coefficients in a tilting module. We relate it to a certain t-structure on the derived category of…
An example of a finite dimensional factorizable ribbon Hopf C-algebra is given by a quotient H=u_q(g) of the quantized universal enveloping algebra U_q(g) at a root of unity q of odd degree. The mapping class group M_{g,1} of a surface of…
We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q=-1. This allows us to define graded modules over the Hecke algebra at q=-1…