Related papers: De Rham Cohomology and Hodge decomposition for Qua…

We introduce and survey a Betti form of the geometric Langlands conjecture, parallel to the de Rham form developed by Beilinson-Drinfeld and Arinkin-Gaitsgory, and the Dolbeault form of Donagi-Pantev, and inspired by the work of…

Let (\Gamma,d) be the 3D-calculus or the 4D_{\pm}-calculus on the quantum group SU_q(2). We describe all pairs (\pi, F) of a *-representation \pi of O(SU_q(2)) and of a symmetric operator F on the representation space satisfying a technical…

W-algebras are a class of non-commutative algebras related to the classical universal enveloping algebras. They can be defined as a subquotient of U(g) related to a choice of nilpotent element e and compatible nilpotent subalgebra m. The…

The authors study the Hodge theory of the exterior differential operator $d$ acting on $q$-forms on a smoothly bounded domain in $\RR^{N+1}$, and on the half space $\rnp$. The novelty is that the topology used is not an $L^2$ topology but a…

A review on spectral and differential-geometric properties of Delsarte transmutation operators in multidimension is given. Their differential geometrical and topological structure in multidimension is analyzed, the relationships with De…

The geometric description of open quantum systems via the Quantum Geometric Tensor (QGT) traditionally relies on the assumption that the physical states form a differentiable vector bundle over the parameter manifold. This framework becomes…

Algebraic quantum field theory is considered from the perspective of the Hochschild cohomology bicomplex. This is a framework for studying deformations and symmetries. Deformation is a possible approach to the fundamental challenge of…

Aim of this paper is to define a new type of cohomology for multiplicative Hom-Leibniz algebras which controls deformations of Hom-Leibniz algebra structure. The cohomology and the associated deformation theory for Hom-Leibniz algebras as…

For transcendental values of q the quantum tangent spaces of all left-covariant first order differential calculi of dimension less than four on the quantum group $\SLq 2$ are given. All such differential calculi $\Gamma $ are determined and…

In this article we propose a new and so-called holomorphic deformation scheme for locally convex algebras and Hopf algebras. Essentially we regard converging power series expansion of a deformed product on a locally convex algebra, thus…

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…

Real forms of a complex reductive group are classified in terms of Galois cohomology $H^1(\Gamma,G_{ad})$ where $G_{ad}$ is the adjoint group. Alternatively, the theory of the Cartan involution gives a description in terms of cohomology…

The ``local'' structure of a quantum group G_q is currently considered to be an infinite-dimensional object: the corresponding quantum universal enveloping algebra U_q(g), which is a Hopf algebra deformation of the universal enveloping…

We study the ring of differential operators D(X) on the basic affine space X=G/U of a complex semisimple group G with maximal unipotent subgroup U. One of the main results shows that the cohomology group H^*(X,O_X) decomposes as a finite…

We compute the cohomology with trivial coefficients of Lie algebras $\mathfrak{m}_0$ and $\mathfrak{m}_2$ of maximal class over the field $\mathbb{Z}_2$. In the infinite-dimensional case, we show that the cohomology rings…

Let $X\to Y^0$ be an abelian prime-to-$p$ Galois covering of smooth schemes over a perfect field $k$ of characteristic $p>0$. Let $Y$ be a smooth compactification of $Y^0$ such that $Y-Y^0$ is a normal crossings divisor on $Y$. We describe…

Let $G=\C^{n}\ltimes_{\phi} \C^{m}$ with a semi-simple action $\phi: \C^{n}\to GL_{m}(\C)$ (not necessarily holomorphic). Suppose $G$ has a lattice $\Gamma$. Then we show that in some conditions on $G$ and $\Gamma$, $G/\Gamma$ admits a…

We apply one of the formalisms of noncommutative geometry to $R^N_q$, the quantum space covariant under the quantum group $SO_q(N)$. Over $R^N_q$ there are two $SO_q(N)$-covariant differential calculi. For each we find a frame, a metric and…

A variety of three-dimensional left-covariant differential calculi on the quantum group $SU_q(2)$ is considered using an approach based on global $ U(1) $ -covariance. Explicit representations of possible $q $-Lie algebras are constructed…

We look more closely at the higher nonabelian de Rham cohomology of a smooth projective variety or family of varieties that had been defined in some previous papers. We formalize using $n$-stacks the notion of shape underlying this…