On rack cohomology

By Pavel Etingof, M. Graña. 2002

99 citations recorded by the source corpus.

Abstract We prove that the lower bounds for Betti numbers of the rack, quandle and degeneracy cohomology given in Carter et al. (J. Pure Appl. Algebra, 157 (2001) 135) are in fact equalities. We compute as well the Betti numbers of the twisted cohomology introduced in Carter et al. (Twisted quandle cohomology theory and cocycle knot invariants, math. GT/0108051). We also give a group-theoretical interpretation of the second cohomology group for racks.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f3ab6564f1d440dbf40f8eef615de88d7a083496

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

Extensions of Quandles and Cocycle Knot Invariants (2001, unknown): “Computations are found in [9, 10, 14, 31, 34], for example.”
Finite-dimensional pointed Hopf algebras with alternating groups are trivial (2008, background): “study of Nichols algebras of decomposable Yetter-Drinfeld modules in the analogous study of indecomposable ones. We stress that the computation of a second rack-cohomology group is a diﬃcult task. By [EG], it coincides with a ﬁrst group-cohomology group, but this does not make the problem easier. Tw”
From racks to pointed Hopf algebras (2002, background): “roup with trivial action then H2(X,A) = H1(G X,Fun(X,A)). Here the space Fun(X,A) of all functions from X to Ais a trivial left GX; the right action is given by (f.x)(z) = f(x⊲z). Proof. (Sketch, see [EG].) Consider the map Φ : X→ GX, x→ x. Let f : GX → Fun(X,A) and take r(f) : X× X → A, where r(f)(”
Yang–Baxter deformations of quandles and racks (2004, methodology): “eformations were taken into account. •Diagonal deformations have been more fully developed in [6], where quandle cohomology was used to construct state-sum invariants of knots. P.Etingof and M.Gran˜a [11] have calculated rack cohomology H∗(Q,A) assuming |Inn(Q)|invertible in A. Our calculation of H2”
Cocycle knot invariants from quandle modules and generalized quandle homology (2003, unknown): “d surfaces and their diagrams, Surveys and monographs, Amer. Math. Soc. 55 (1998). [13] R. Dijkgraaf; E. Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990) 393–429. [14] P. Etingof; M. Gran˜a, On rack cohomology, J. of Pure and Applied Algebra 177 (2003) 49–59. [15”
Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation (2015, background): “This generalizes the analogous result for racks established by Etingof and Graña [16].”