G-structure on the cohomology of Hopf algebras

Farinati, Marco; Solotar, Andrea

doi:10.1090/S0002-9939-04-07274-0

G-structure on the cohomology of Hopf algebras
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by Marco A. Farinati and Andrea L. Solotar

Proc. Amer. Math. Soc. 132 (2004), 2859-2865

DOI: https://doi.org/10.1090/S0002-9939-04-07274-0

Published electronically: June 2, 2004

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Abstract:

We prove that $\mathrm {Ext} ^{\bullet }_A(k,k)$ is a Gerstenhaber algebra, where $A$ is a Hopf algebra. In case $A=D(H)$ is the Drinfeld double of a finite-dimensional Hopf algebra $H$, our results imply the existence of a Gerstenhaber bracket on $H^{\bullet }_{GS}(H,H)$. This fact was conjectured by R. Taillefer. The method consists of identifying $H^{\bullet }_{GS}(H,H)\cong {\mathrm {Ext}}^{\bullet }_A(k,k)$ as a Gerstenhaber subalgebra of $H^{\bullet }(A,A)$ (the Hochschild cohomology of $A$).

References

D. J. Benson, Representations and cohomology. I, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1998. Basic representation theory of finite groups and associative algebras. MR 1644252
Claude Cibils and Marc Rosso, Hopf bimodules are modules, J. Pure Appl. Algebra 128 (1998), no. 3, 225–231. MR 1626345, DOI 10.1016/S0022-4049(97)00060-1
Murray Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MR 161898, DOI 10.2307/1970343
Murray Gerstenhaber and Samuel D. Schack, Bialgebra cohomology, deformations, and quantum groups, Proc. Nat. Acad. Sci. U.S.A. 87 (1990), no. 1, 478–481. MR 1031952, DOI 10.1073/pnas.87.1.478
Murray Gerstenhaber and Samuel D. Schack, Algebras, bialgebras, quantum groups, and algebraic deformations, Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990) Contemp. Math., vol. 134, Amer. Math. Soc., Providence, RI, 1992, pp. 51–92. MR 1187279, DOI 10.1090/conm/134/1187279
Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR 1243637, DOI 10.1090/cbms/082
Stefan Schwede, An exact sequence interpretation of the Lie bracket in Hochschild cohomology, J. Reine Angew. Math. 498 (1998), 153–172. MR 1629858, DOI 10.1515/crll.1998.048
Dragoş Ştefan, Hochschild cohomology on Hopf Galois extensions, J. Pure Appl. Algebra 103 (1995), no. 2, 221–233. MR 1358765, DOI 10.1016/0022-4049(95)00101-2
Rachel Taillefer, Cohomology theories of Hopf bimodules and cup-product, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 3, 189–194 (English, with English and French summaries). MR 1817359, DOI 10.1016/S0764-4442(00)01811-5
Rachel Taillefer, thesis, Université Montpellier 2 (2001).
Rachel Taillefer, Injective Hopf bimodules, cohomologies of infinite dimensional Hopf algebras and graded-commutativity of the Yoneda product. ArXivMath math.KT/0207154.