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We show that every Kac-Moody algebra is birationally equivalent to a smash biproduct of two copies of a Weyl algebra together with a polynomial algebra. We also show that the same is true for quantized Kac-Moody algebras where one replaces Weyl algebras with their quantum analogues. [ABSTRACT FROM AUTHOR]

We show that if G is a compact Lie group and $$\mathfrak {g}$$ is its Lie algebra, then there is a map from the Hopf-cyclic cohomology of the quantum enveloping algebra $$U_q(\mathfrak {g})$$ to the twisted cyclic cohomology of quantum group algebra $${\mathcal O}(G_q)$$ . We also show that the Schmüdgen-Wagner index cocycle associated with the volume form of the differential calculus on the standard Podleś sphere $${\mathcal O}(S^2_q)$$ is in the image of this map. [ABSTRACT FROM AUTHOR]

We prove that for an inclusion of unital associative but not necessarily commutative k-algebras B ⊆ A we have long exact sequences in Hochschild homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in André-Quillen homology, provided that the quotient B-module A=B is flat. We also prove that for an arbitrary r-flat morphism φ : B → A with an H-unital kernel, one can express the Wodzicki excision sequence and our Jacobi-Zariski sequence in Hochschild homology and cyclic (co)homology as a single long exact sequence. [ABSTRACT FROM AUTHOR]

For module algebras and module coalgebras over an arbitrary bialgebra, we define two types of bivariant cyclic cohomology groups called bivariant Hopf cyclic cohomology and bivariant equivariant cyclic cohomology. These groups are defined through an extension of Connes' cyclic category Λ. We show that, in the case of module coalgebras, bivariant Hopf cyclic cohomology specializes to Hopf cyclic cohomology of Connes and Moscovici and its dual version by fixing either one of the variables as the ground field. We also prove an appropriate version of Morita invariance for both of these theories. [ABSTRACT FROM AUTHOR]

We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show that this (co)homology, called Hopf-Hochschild (co)homology, can also be defined as a derived functor on the category of representations of an equivariant analogue of the enveloping algebra of a crossed product algebra. We investigate the relationship of our theory with Hopf cyclic cohomology and also prove Morita invariance of the Hopf-Hochschild (co)homology. [ABSTRACT FROM AUTHOR]