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

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical…

We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S_3, the dihedral group D_4 and the quaternion group Q. Poincare' duality holds in every case, and under some…

Let A be a Hopf algebra and $Gamma$ be a bicovariant first order differential calculus over A. It is known that there are three possibilities to construct a differential Hopf algebra $Gamma^wedge$ that contains $Gamma$ as its first order…

We compute the noncommutative de Rham cohomology for the finite-dimensional q-deformed coordinate ring $C_q[SL_2]$ at odd roots of unity and with its standard 4-dimensional differential structure. We find that $H^1$ and $H^3$ have three…

We obtain an interesting realization of the de Rham cohomological operators of differential geometry in terms of the noncommutative q-superoscillators for the supersymmetric quantum group GL_{qp} (1|1). In particular, we show that a unique…

We study *-differential calculi over compact quantum groups in the sense of S.L. Woronowicz. Our principal results are the construction of a Hodge operator commuting with the Laplacian, the derivation of a corresponding Hodge decomposition…

We study the cohomology $H^*_{\lambda \omega}(G/\Gamma, {\mathbb C})$ of the deRham complex $\Lambda^*(G/\Gamma)\otimes{\mathbb C}$ of a compact solvmanifold $G/\Gamma$ with a deformed differential $d_{\lambda \omega}=d + \lambda\omega$,…

Let $\Gamma$ be an $N^2$-dimensional bicovariant first order differential calculus on a Hopf algebra $SL_q(N)$. There are three possibilities to construct a differential Z-graded Hopf algebra $\Gamma^\wedge$ which contains $\Gamma$ as its…

Lecture notes. Introduction to the cohomology of algebras, Lie algebras, Lie bialgebras and quantum groups. Contains a new derivation of the classification of classical r-matrices in terms of deformation cohomology, and a calculation of the…

For a simply connected (non-nilpotent) solvable Lie group $G$ with a lattice $\Gamma$ the de Rham and Dolbeault cohomologies of the solvmanifold $G/\Gamma$ are not in general isomorphic to the cohomologies of the Lie algebra $\mathfrak g$…

We consider the possible covariant external algebra structures for Cartan's 1-forms on GL_q(N) and SL_q(N). We base upon the following natural postulates: 1. the invariant 1-forms realize an adjoint representation of quantum group; 2. all…

For the split group $G_{2}$ defined over $\mathbb{Z},$ we show that the de Rham cohomology ring of $B(G_{2})_{\mathbb{F}_{2}}$ is isomorphic to the singular cohomology ring with $\mathbb{F}_{2}$-coefficients of $B(G_{2})_{\mathbb{C}}.$ For…

Let G be a general (not necessarily finite dimensional compact) Lie group, let g be its Lie algebra, let Cg be the cone on g in the category of differential graded Lie algebras, and consider the functor which assigns to a chain complex V…

We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds, of which symplectic manifolds are an important class of examples. Quantum de Rham cohomology is defined as the cohomology of d_h. We also…

We explore the indecomposable submodule structure of quantum Grassmann super-algebra $\Omega_q(m|n)$ and its truncated objects $\Omega_q(m|n,\textbf{r})$ in the case when $q=\varepsilon$ is an $\ell$-th root of unity. A net-like…

We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds (of which symplectic manifolds are an important class of examples). Quantum de Rham cohomology, which is a deformation quantization of de…

We describe Laplacian operators on the quantum group SUq (2) equipped with the four dimensional bicovariant differential calculus of Woronowicz as well as on the quantum homogeneous space S2q with the restricted left covariant three…

This paper applies the decomposition theorem in intersection cohomology to geometric invariant theory quotients, relating the intersection cohomology of the quotient to that of the semistable points for the action. Suppose a connected…

Let $G$ be a simply connected solvable Lie group with a lattice $\Gamma$ and the Lie algebra $\g$ and a representation $\rho:G\to GL(V_{\rho})$ whose restriction on the nilradical is unipotent. Consider the flat bundle $E_{\rho}$ given by…

In this article, we establish results concerning the cohomology of Zariski dense subgroups of solvable linear algebraic groups. We show that for an irreducible solvable $\mathbb{Q}$-defined linear algebraic group $\mathbf{G}$, there exists…