The B∞-Structure on the Derived Endomorphism Algebra of the Unit in a Monoidal Category

Consider a monoidal category that is at the same time abelian with enough projectives and such that projectives are flat on the right. We show that there is a $B_{\infty }$-algebra that is $A_{\infty }$-quasi-isomorphic to the derived endomorphism algebra of the tensor unit. This $B_{\infty }$-algebra is obtained as the co-Hochschild complex of a projective resolution of the tensor unit, endowed with a lifted $A_{\infty }$-coalgebra structure. We show that in the classical situation of the category of bimodules over an algebra, this newly defined $B_{\infty }$-algebra is isomorphic to the Hochschild complex of the algebra in the homotopy category of $B_{\infty }$-algebras.

Cyclic-Homology Chern–Weil Theory for Families of Principal Coactions

Viewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of Loday and Wodzicki. We work with families of principal coactions, and instantiate our noncommutative Chern–Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of Feng and Tsygan for the quantum-deformation family of the standard quantum Hopf fibrations.

Chain level proof for the isomorphism between Lie and Hochschild homologies

In this paper, we prove the commutativity of the square in the chain level of both Chevalley Eilenberg complex of Lie homology and Hochschild complex, where the antisymmetrization map is used between the complexes. Originally, Loday proved this isomorphism by constructing a certain map satisfying relations of a presimplicial homotopy to prove the commutativity of the square mentioned above. Here, we present a different approach for the proof of the commutativity without constructing that certain map satisfying the relations of the presimplicial homotopy.

Weyl n-algebras and the Kontsevich integral of the unknot

Given a Lie algebra with a scalar product, one may consider the latter as a symplectic structure on a [Formula: see text]-scheme, which is the spectrum of the Chevalley–Eilenberg algebra. In Sec. 1 we explicitly calculate the first-order deformation of the differential on the Hochschild complex of the Chevalley–Eilenberg algebra. The answer contains the Duflo character. This calculation is used in the last section. There we sketch the definition of the Wilson loop invariant of knots, which is, hopefully, equal to the Kontsevich integral, and show that for unknot they coincide. As a byproduct, we get a new proof of the Duflo isomorphism for a Lie algebra with a scalar product.

Universal operations in Hochschild homology

We provide a general method for finding all natural operations on the Hochschild complex of