scholarly journals BV-operators and the secondary Hochschild complex

2021 ◽  
Vol 358 (11-12) ◽  
pp. 1239-1258
Author(s):  
Mamta Balodi ◽  
Abhishek Banerjee ◽  
Anita Naolekar
Keyword(s):  
Hochschild Complex

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

2020 ◽  
Author(s):  
Piotr M. Hajac ◽  
Tomasz Maszczyk
Keyword(s):  
Cyclic Homology ◽  
Quantum Deformation ◽  
Homotopy Formula ◽  
Standard Quantum ◽  
Unital Algebra ◽  
Hopf Fibrations ◽  
Chain Homotopy ◽  
Cartan Model ◽  
Extension Algebra ◽  
Hochschild Complex

AbstractViewing 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.

scholarly journals DETECTING TORSION IN SKEIN MODULES USING HOCHSCHILD HOMOLOGY

2006 ◽  
Vol 15 (02) ◽  
pp. 259-277 ◽  
Author(s):  
MICHAEL McLENDON
Keyword(s):  
Tensor Product ◽  
Spectral Sequence ◽  
Boundary Surface ◽  
Hochschild Homology ◽  
Heegaard Splitting ◽  
Common Boundary ◽  
Skein Modules ◽  
Skein Module ◽  
Hochschild Complex

Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpreted as the skein module of the 3-manifold obtained by gluing the two handlebodies together along this surface. A spectral sequence associated to the Hochschild complex is constructed and conditions are given for the existence of algebraic torsion in the completion of the skein module of this 3-manifold.

Hochschild Complex

2004 ◽  
pp. 188-188
Author(s):  
Thomas Chen ◽  
Jürgen Fuchs ◽  
Steven Duplij ◽  
Evgeniy Ivanov ◽  
Steven Duplij ◽  
...  
Keyword(s):  
Hochschild Complex

scholarly journals The Hochschild complex of a twisting cochain

2016 ◽  
Vol 451 ◽  
pp. 302-356 ◽  
Author(s):  
Kathryn Hess
Keyword(s):  
Hochschild Complex

Chain level proof for the isomorphism between Lie and Hochschild homologies

2020 ◽  
pp. 2150113
Author(s):  
Zuhier Altawallbeh
Keyword(s):  
Chain Level ◽  
Hochschild Complex

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.

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

2021 ◽  
Author(s):  
Wendy Lowen ◽  
Michel Van den Bergh
Keyword(s):  
Monoidal Category ◽  
Projective Resolution ◽  
Homotopy Category ◽  
Endomorphism Algebra ◽  
Coalgebra Structure ◽  
The Right ◽  
Hochschild Complex

Abstract 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.

Higher operations on the Hochschild complex

1995 ◽  
Vol 29 (1) ◽  
pp. 1-5 ◽  
Author(s):  
A. A. Voronov ◽  
M. Gerstenhaber
Keyword(s):  
Hochschild Complex

scholarly journals Operations on the Hopf-Hochschild complex for module-algebras

2011 ◽  
Vol 13 (1) ◽  
pp. 259-272
Author(s):  
Donald Yau
Keyword(s):  
Module Algebras ◽  
Hochschild Complex

scholarly journals Weyl n-algebras and the Kontsevich integral of the unknot

2016 ◽  
Vol 25 (12) ◽  
pp. 1642008
Author(s):  
Nikita Markarian
Keyword(s):  
Lie Algebra ◽  
Wilson Loop ◽  
Scalar Product ◽  
Symplectic Structure ◽  
Loop Invariant ◽  
First Order ◽  
Order Deformation ◽  
Definition Of ◽  
Hochschild Complex

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.

scholarly journals ON THE OTHER SIDE OF THE BIALGEBRA OF CHORD DIAGRAMS

2007 ◽  
Vol 16 (05) ◽  
pp. 575-629 ◽  
Author(s):  
V. TOURTCHINE
Keyword(s):  
Spectral Sequence ◽  
Lie Algebras ◽  
Loop Space ◽  
The Other ◽  
Dimensional Sphere ◽  
Double Loop ◽  
Homology Groups ◽  
Chord Diagrams ◽  
Future Work ◽  
Hochschild Complex

In this paper we describe complexes whose homologies are naturally isomorphic to the first term of the Vassiliev spectral sequence computing (co)homology of the spaces of long knots in ℝd, d ≥ 3. The first term of the Vassiliev spectral sequence is concentrated in some angle of the second quadrant. In homological case the lower line of this term is the bialgebra of chord diagrams (or its superanalog if d is even). We prove in this paper that the groups of the upper line are all trivial. In the same bigradings we compute the homology groups of the complex spanned only by strata of immersions in the discriminant (maps having only self-intersections). We interpret the obtained groups as subgroups of the (co)homology groups of the double loop space of a (d - 1)-dimensional sphere. In homological case the last complex is the normalized Hochschild complex of the Poisson or Gerstenhaber (depending on parity of d) algebras operad. The upper line bigradings are spanned by the operad of Lie algebras. To describe the cycles in these bigradings, we introduce new homological operations on Hochschild complexes. We show in future work that these operations are in fact the Dyer–Lashof–Cohen operations induced by the action of the singular chains operad of little squares on Hochschild complexes.

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