Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-23T07:32:29.306Z Has data issue: false hasContentIssue false
  • English
  • Français

On the conjectural Leibniz cohomology for groups

Published online by Cambridge University Press:  30 November 2012

Simon Covez
Simon Covez*
Affiliation:
University of Luxembourg, Campus Kirchberg Mathematics Research Unit, BLG, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Grand Duchy of Luxembourgsimon.covez@uni.lu
Get access

Abstract

This article presents results which are consistent with conjectures about Leibniz (co)homology for discrete groups, due to J. L. Loday in 2003. We prove that rack cohomology has properties very close to the properties expected for the conjectural Leibniz cohomology. In particular, we prove the existence of a graded dendriform algebra structure on rack cohomology, and we construct a graded associative algebra morphism H•(−) → HR•(−) from group cohomology to rack cohomology which is injective for ● = 1.

Keywords

Leibniz algebradendriform algebraZinbiel algebrarack cohomologygroup cohomologyconjectural Leibniz K-theory
Type
Research Article
Information
Journal of K-Theory , Volume 10 , Issue 3 , December 2012 , pp. 519 - 563
Copyright
Copyright © ISOPP 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Carter, J. S., Saito, M., Quandle homology theory and cocycle knot invariants, Topology and geometry of manifolds (Athens, GA, 2001), Proc. Sympos. Pure Math. 71, Amer. Math. Soc., Providence, RI, 2003, 249–268.CrossRefGoogle Scholar
2.Covez, S., The local integration of Leibniz algebras, ArXiv e-prints http://arxiv.org/abs/1011.4112.Google Scholar
3.Cuvier, C., Algèbres de Leibnitz: définitions, propriétés, Ann. Sci. École Norm. Sup. (4) 27(1), (1994), 1–45.CrossRefGoogle Scholar
4.Etingof, P., Graña, M., On rack cohomology, J. Pure Appl. Algebra 177(1), (2003), 49–59.CrossRefGoogle Scholar
5.Feĭgin, B. L., Tsygan, B. L., Additive K-theory, K-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math. 1289, Springer, Berlin, 1987, 67–209.Google Scholar
6.Fenn, R., Rourke, C., Racks and links in codimension two, J. Knot Theory Ramifications 1(4), (1992), 343–406.CrossRefGoogle Scholar
7.Fenn, R., Rourke, C., Sanderson, B., Trunks and classifying spaces, Appl. Categ. Structures 3(4), (1995), 321–356.Google Scholar
8.Kinyon, M. K., Leibniz algebras, Lie racks, and digroups, J. Lie Theory 17(1), (2007), 99–114.Google Scholar
9.Loday, J.-L., Une version non commutative des algèbres de Lie: les algèbres de Leibniz, R.C.P. 25, Vol. 44 (French) (Strasbourg, 1992), Prépubl. Inst. Rech. Math. Av., 1993/41, Univ. Louis Pasteur, Strasbourg, 1993, 127–151.Google Scholar
10.Loday, J.-L., Cup-product for Leibniz cohomology and dual Leibniz algebras, Math. Scand. 77(2), (1995), 189–196.Google Scholar
11.Loday, J.-L., Overview on Leibniz algebras, dialgebras and their homology, Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995), Fields Inst. Commun. 17, Amer. Math. Soc., Providence, RI, 1997, 91–102.Google Scholar
12.Loday, J.-L., Cyclic homology, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 301, Springer-Verlag, Berlin, 1998, appendix E by Ronco, María O., Chapter 13 by the author in collaboration with Teimuraz Pirashvili.Google Scholar
13.Loday, J.-L., Algebraic K-theory and the conjectural Leibniz K-theory, K-Theory 30(2), (2003), 105–127, special issue in honor of Hyman Bass on his seventieth birthday. Part II.Google Scholar
14.Loday, J.-L., Some problems in operad theory, Operads and universal algebras (Tianjin, China, July 2010), Proc. Int. Conf. in Nankai Series in Pure, Applied Mathematics and Theoretical Physics 9, World Scientific, 2012, 139–146.Google Scholar
15.Loday, J.-L., Frabetti, A., Chapoton, F., Goichot, F., Dialgebras and related operads, Lecture Notes in Mathematics 1763, Springer-Verlag, Berlin, 2001.Google Scholar
16.Loday, J.-L., Quillen, D., Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv. 59(4), (1984), 569–591.Google Scholar
17.Lane, S. Mac, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, New York, 1998.Google Scholar
18.Przytycki, J. H., Sikora, A. S., Distributive Products and Their Homology, ArXiv e-prints http://arxiv.org/abs/1105.3700.Google Scholar