Algebraic deformations arising from orbifolds with discrete torsion. (English) Zbl 1055.16010
Let \(R\) be a commutative ring with the group of units \(R^\times\), let \(G\) be a finite group defining an action on \(R\), and \(\alpha\colon G\times G\to R^\times\) be a 2-cocycle. The authors describe the Hochschild cohomology \(HH^*\), deformations and the behavior of the center of the crossed product ring \(A=R\#_\alpha G\) under deformations when \(R=\mathbb{C}[x,y,z]\) and \(G\) is Abelian. In the particular case \(G=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\) they compute all nontrivial groups \(HH^i(A)\), \(i=0,\dots,3\), a universal deformation formula \(F\), which provides a formal deformation \(A_F\) of \(A\) in the sense of A. Giaquinto and J. J. Zhang [J. Pure Appl. Algebra 128, No. 2, 133-151 (1998; Zbl 0938.17015)], and the center of the algebra \(A_F\). The authors underline that their results are closely related with earlier calculations of chiral numbers for orbifolds with discrete torsion by C. Vafa and E. Witten [J. Geom. Phys. 15, No. 3, 189-214 (1995; Zbl 0816.53053)].
Reviewer: Aleksandr G. Aleksandrov (Moskva)
MSC:
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 16S80 | Deformations of associative rings |
| 16S35 | Twisted and skew group rings, crossed products |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
Keywords:
crossed products; Hochschild cohomology; cyclic cohomology; deformations of associative rings and algebras; Koszul complexes; chiral numbers; Calabi-Yau threefolds; Hodge numbers; singularities; orbifolds; universal deformation formulasReferences:
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