Document Zbl 1054.16028 - zbMATH Open
On rack cohomology. (English) Zbl 1054.16028
A rack is a set provided with a binary operation subject to axioms that abstract the main properties of the conjugation in a group. Racks were used in topology because they give rise to solutions of the quantum Yang-Baxter equation. There exist rack cohomology theories as well; with different levels of generality, they can be found in J. S. Carter, D. Jelsovsky, S. Kamada, and M. Saito [J. Pure Appl. Algebra 157, No. 2-3, 135-155 (2001; Zbl 0977.55013)], J. S. Carter, M. Elhamdadi, and M. Saito [Algebr. Geom. Topol. 2, 95-135 (2002; Zbl 0991.57005)], N. Andruskiewitsch and M. Graña [Adv. Math. 178, No. 2, 177-243 (2003; Zbl 1032.16028)], N. Jackson [Extensions of racks and quandles, math.CT/0408040].
In the present paper, the authors show that the lower bounds for Betti numbers given by J. S. Carter et al. [Zbl 0977.55013] are in fact equalities. They also compute the Betti numbers for the cohomology introduced by J. S. Carter et al. [Zbl 0991.57005]. Finally, they also give a group-theoretical interpretation of the second cohomology group for racks. This second cohomology group plays a rôle in the classification of pointed Hopf algebras, see M. Graña [J. Algebra 231, No. 1, 235-257 (2000; Zbl 0970.16017)].
MSC:
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 20J06 | Cohomology of groups |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 18G60 | Other (co)homology theories (MSC2010) |
| 55N35 | Other homology theories in algebraic topology |
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
References:
| [1] | N. Andruskiewitsch, M. Graña, From racks to pointed Hopf algebras, to appear.; N. Andruskiewitsch, M. Graña, From racks to pointed Hopf algebras, to appear. · Zbl 1032.16028 |
| [2] | J.S. Carter, M. Elhamdadi, M. Saito, Twisted quandle cohomology theory and cocycle knot invariants,; J.S. Carter, M. Elhamdadi, M. Saito, Twisted quandle cohomology theory and cocycle knot invariants, · Zbl 0991.57005 |
| [3] | Carter, J. S.; Jelsovsky, D.; Kamada, S.; Saito, M., Quandle homology groups, their Betti numbers, and virtual knots, J. Pure Appl. Algebra, 157, 135-155 (2001) · Zbl 0977.55013 |
| [4] | J.S. Carter, M. Saito, Quandle homology theory and cocycle knot invariants,; J.S. Carter, M. Saito, Quandle homology theory and cocycle knot invariants, · Zbl 1053.57002 |
| [5] | C. Chevalley, Théorie des groupes de Lie. Tome III. Théorèmes généraux sur les algèbres de Lie, Hermann & Cie, Paris, 1955.; C. Chevalley, Théorie des groupes de Lie. Tome III. Théorèmes généraux sur les algèbres de Lie, Hermann & Cie, Paris, 1955. · Zbl 1469.17032 |
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| [7] | M. Graña, On Nichols algebras of low dimension, in: N. Andruskiewitsch, W. Ferrer Santos, H.-J. Schneider (Eds.), New Trends in Hopf Algebra Theory, Contemp. Math. 267 (2000) 111-136.; M. Graña, On Nichols algebras of low dimension, in: N. Andruskiewitsch, W. Ferrer Santos, H.-J. Schneider (Eds.), New Trends in Hopf Algebra Theory, Contemp. Math. 267 (2000) 111-136. · Zbl 0974.16031 |
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| [9] | R.A. Litherland, S. Nelson, The Betti numbers of some finite racks,; R.A. Litherland, S. Nelson, The Betti numbers of some finite racks, · Zbl 1017.55014 |
| [10] | Lu, J.-H.; Yan, M.; Zhu, Y.-C., On set-theoretical Yang-Baxter equation, Duke Math. J., 104, 1-18 (2000) · Zbl 0960.16043 |
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