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Twisting in group algebras of finite groups. (English. Russian original) Zbl 0813.20005

Funct. Anal. Appl. 27, No. 4, 240-244 (1993); translation from Funkts. Anal. Prilozh. 27, No. 4, 17-23 (1993).
From the introduction: Let \(G\) be a finite group, and let \(k[G]\) be its group algebra over an algebraically closed field \(k\) of characteristic zero. By quantization of a discrete group we shall mean the preparation of a new diagonal by conjugating the old one by an element \(\varphi\) of a tensor square of the group algebra. In the present paper we study an equation for \(\varphi\). Under certain additional conditions on the element \(\varphi\) we prove that the setting of \(\varphi\) is equivalent to the choice of a subgroup with a special 2-cocycle on it satisfying some condition. This condition on the 2-cocycle, namely the nondegeneracy condition, can also be stated in the case of a Lie group. A connected Lie group with a nondegenerate cocycle turns out to be always solvable.
Reviewer: T.Mollov (Plovdiv)

MSC:

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
16S20 Centralizing and normalizing extensions
20J05 Homological methods in group theory
57T10 Homology and cohomology of Lie groups

References:

[1] V. I. Arnold, ”Remarks on perturbation theory for problems of Mathieu type,” Usp. Mat. Nauk,38, No. 4, 189–203 (1983).
[2] V. I. Arnold, ”Small denominators I. Mappings of the circumference onto itself,” Trans. Amer. Math. Soc.,46, 213–284 (1965). · Zbl 0152.41905
[3] O. G. Galkin, ”Phase-locking for Mathieu-type vector fields on the torus,” Funkts. Anal. Prilozhen.,26, No. 1, 1–8 (1992). · Zbl 0828.20025 · doi:10.1007/BF01077066
[4] A. Khinchin, Continued fractions, Groningen, Nordhorff (1963). · Zbl 0117.28601
[5] C. Baesens, J. Guckenheimer, S. Kim, and R. S. MacKay, ”Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos,” Phys. D,49, 387–475 (1991). · Zbl 0734.58036 · doi:10.1016/0167-2789(91)90155-3
[6] R. E. Ecke, J. D. Farmer, and D. K. Umberger, ”Scaling of the Arnold tongues,” Nonlinearity,2, 175–196 (1989). · Zbl 0689.58017 · doi:10.1088/0951-7715/2/2/001
[7] J. Franks and M. Misiurewicz, ”Rotation sets of toral flows,” Proc. Amer. Math. Soc.,109, 243–249 (1990). · Zbl 0701.57016 · doi:10.1090/S0002-9939-1990-1021217-5
[8] O. G. Galkin, ”Resonance regions for Mathieu type dynamical systems on a torus,” Phys. D,39, 287–298 (1989). · Zbl 0695.58025 · doi:10.1016/0167-2789(89)90011-0
[9] C. Grebogi, E. Ott, and J. A. Yorke, ”Attractors on ann-torus: quasiperiodicity versus chaos,” Phys. D,15, 354–373 (1985). · Zbl 0577.58023 · doi:10.1016/S0167-2789(85)80004-X
[10] G. R. Hall, ”Resonance zones in two-parameter families of circle homeomorphisms,” SIAM J. Math. Anal.,15, 1075–1081 (1984). · Zbl 0554.58040 · doi:10.1137/0515083
[11] S. Kim, R. S. MacKay, and J. Guckenheimer, ”Resonance regions for families of torus maps,” Nonlinearity,2, 391–404 (1989). · Zbl 0678.58034 · doi:10.1088/0951-7715/2/3/001
[12] M. Misiurewicz and K. Ziemian, ”Rotation sets for maps of tori,” J. London Math. Soc.,40, 490–506 (1989). · Zbl 0663.58022 · doi:10.1112/jlms/s2-40.3.490
[13] S. Newhouse, J. Palis, and F. Takens, ”Bifurcations and stability of families of diffeomorphisms,” Publ. Math. IHES,57, 5–72 (1983). · Zbl 0518.58031
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