Twisted homology of quantum SL(2)

Twisted quandle homology theory and cocycle knot invariants

The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group cocycles. Explicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The corresponding generalization of the quandle cocycle knot invariants is given, by using the…

Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras

We show that many noetherian Hopf algebras A have a rigid dualising complex R with R isomorphic to ^νA^1 [d]. Here, d is the injective dimension of the algebra and νis a certain k-algebra automorphism of A, unique up to an inner automorphism. In honour of the finite dimensional theory which is hereby generalised we call νthe Nakayama automorphism of A. We prove that ν= S^2\XXi, where S is the antipode of A and \XXi is the left winding automorphism of A determined by the left integral of A. The Hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are s…

扭转杨代数与型C的steinberg变种

Twisted Yangians and Steinberg varieties of type C

研究型C广义steinberg变种的equivariant同调，并证明扭转杨代数AIII_{2n}^{(τ)}到它的代数同态映射

Twisted Poincare duality for some quadratic Poisson algebras

We exhibit a Poisson module restoring a twisted Poincare duality between Poisson homology and cohomology for the polynomial algebra R=C[X_1,...,X_n] endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a corollary we compute the Poisson cohomology of R, and so retrieve a result obtained by direct methods (so completely different from ours) by Monnier.

Khovanov-Rozansky 同调、Bott-Samelson 空间与扭曲上同调

Khovanov-Rozansky homologies, Bott-Samelson spaces and twisted cohomology

本文比较了 Rasmussen 的 Khovanov-Rozansky 同调表述与 Kitchloo 基于 Bott-Samelson 簇和 Borel 等变上同调扭曲的链环不变量，证明这些几何构造可产生具有特殊化势或万有势的等变整系数 sl(n) 链环同调。

The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podles sphere

We continue the investigation of twisted homology theories in the context of dimension drop phenomena. This work unifies previous equivariant index calculations in twisted cyclic cohomology. We do this by proving the existence of the resolvent cocycle, a finitely summable analogue of the JLO cocycle, under weaker smoothness hypotheses and in the more general setting of `modular' spectral triples. As an application we show that using our twisted resolvent cocycle, we can obtain a local index formula for the Podles sphere. The resulting twisted cyclic cocycle has non-vanishing Hochschild class w…

Homologically Twisted Invariants Related to (2+1)- and (3+1)-Dimensional State-Sum Topological Quantum Field Theories

Motivated by suggestions of Paolo Cotta-Ramusino's work at the physical level of rigor relating BF theory to the Donaldson polynomials, we provide a construction applicable to the Turaev/Viro and Crane/Yetter invariants of *a priori* finer invariants dependent on a choice of (co)homology class on the manifold

Cyclic structures in algebraic (co)homology theories

This note discusses the cyclic cohomology of a left Hopf algebroid ($\times_A$-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special case of this theory. A generalisation of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti Yetter-…

On the noncommutative spin geometry of the standard Podles sphere and index computations

The purpose of the paper is twofold: First, known results of the noncommutative spin geometry of the standard Podles sphere are extended by discussing Poincare duality and orientability. In the discussion of orientability, Hochschild homology is replaced by a twisted version which avoids the dimension drop. The twisted Hochschild cycle representing an orientation is related to the volume form of the distinguished covariant differential calculus. Integration over the volume form defines a twisted cyclic 2-cocycle which computes the q-winding numbers of quantum line bundles. Second, a "twisted" …

On the Hochschild homology of quantum SL(N)

We show that the standard quantized coordinate ring A of quantum SL(N) satisfies van den Bergh's analogue of Poincare duality for Hochschild (co)homology with dualizing bimodule being A_sigma, the A-bimodule which is A as k-vector space with right multiplication twisted by the modular automorphism sigma of the Haar functional. This implies that H_{N^2-1} (A, A_sigma)=k, generalizing our previous results for quantum SL(2).