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Esposito, Chiara; Schnitzer, Jonas; Waldmann, Stefan

A universal construction of universal deformation formulas, Drinfeld twists and their positivity. (English) Zbl 1425.17019

Pac. J. Math. 291, No. 2, 319-358 (2017).
Summary: We provide an explicit construction of star products on \(\mathcal{U}(\mathfrak{g})\)-module algebras by using the Fedosov approach. This allows us to give a constructive proof to Drinfeld’s theorem and to obtain a concrete formula for Drinfeld twists. We prove that the equivalence classes of twists are in one-to-one correspondence with the second Chevalley-Eilenberg cohomology of the Lie algebra \(\mathfrak{g}\). Finally, we show that for Lie algebras with Kähler structure we obtain a strongly positive universal deformation of \(^*\)-algebras by using a Wick-type deformation. This results in a positive Drinfeld twist.

Cited in 7 Documents

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
53D55 Deformation quantization, star products
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory

Keywords:

star product; Drinfeld twist; universal deformation formula
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\textit{C. Esposito} et al., Pac. J. Math. 291, No. 2, 319--358 (2017; Zbl 1425.17019)
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References:

[1] 10.1016/0003-4916(78)90224-5 · Zbl 0377.53024
[2] 10.1016/0003-4916(78)90225-7 · Zbl 0377.53025
[3] 10.1088/0264-9381/14/1A/008 · Zbl 0881.58021
[4] 10.1023/A:1007598606137 · Zbl 0943.53051
[5] 10.1016/S0393-0440(03)00055-X · Zbl 1071.53054
[6] ; Drinfeld, Dokl. Akad. Nauk SSSR, 273, 531 (1983)
[7] ; Drinfeld, Zap. Nauchn. Sem. LOMI, 155, 18 (1986)
[8] 10.1007/978-3-319-09290-4 · Zbl 1301.81003
[9] ; Fedosov, Dokl. Akad. Nauk SSSR, 291, 82 (1986)
[10] 10.4310/jdg/1214455536 · Zbl 0812.53034
[11] ; Fedosov, Deformation quantization and index theory. Mathematical Topics, 9 (1996) · Zbl 0867.58061
[12] 10.2307/1970484 · Zbl 0123.03101
[13] 10.2307/1970553 · Zbl 0182.05902
[14] 10.1016/S0022-4049(97)00041-8 · Zbl 0938.17015
[15] 10.1007/BF00400441 · Zbl 0522.58019
[16] 10.1016/j.aim.2005.12.006 · Zbl 1163.17303
[17] 10.1007/BF02099631 · Zbl 0866.58037
[18] 10.1007/s00220-012-1657-y · Zbl 1271.53081
[19] 10.1007/978-1-4612-0783-2
[20] 10.1023/B:MATH.0000027508.00421.bf · Zbl 1058.53065
[21] 10.1007/BF02099427 · Zbl 0887.58050
[22] 10.1142/S0129055X05002297 · Zbl 1138.53316
[23] 10.1007/978-3-540-72518-3 · Zbl 1139.53001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.