Zassenhaus Lie Idempotents, q-Bracketing and a New Exponential/Logarithm Correspondence
G. Duchamp
, D. Krob2
and E.A. Vassilieva3
2LIAFA, Université Paris 7, 2, place Jussieu, 75251 Paris Cedex 05, France
DOI: 10.1023/A:1011263924121
Abstract
We introduce a new q-exponential/logarithm correspondance that allows us to solve a conjecture relating Zassenhauss Lie idempotents with other Lie idempotents related to the q-bracketing operator.
Pages: 251–277
Keywords: fer-zassenhauss formula; Lie idempotents; noncommutative symmetric functions; logarithm; exponential
Full Text: PDF
References
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2. I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, and J.-Y. Thibon, “Noncommutative symmetric functions,” Adv. in Math. 112 (1995), 218-348.
3. I. Gessel, “Multipartite P-partitions and inner products of skew Schur functions,” Contemporary Mathematics, 34 (1984), 289-301.
4. D. Krob, B. Leclerc, and J.-Y. Thibon, “Noncommutative symmetric functions II : Transformations of alphabets,” Int. J. of Alg. and Comput. 7 (1997), 181-264.
5. G, Duchamp, A. Klyachko, D. Krob, and J.-Y. Thibon, “Noncommutative symmetric functions III: Deformations of Cauchy products and convolution algebras,” Disc. Maths. and Theor. Comp. Sci. 1 (1997), 159-216.
6. D. Krob and J.-Y. Thibon, “Noncommutative symmetric functions IV : Quantum linear groups and Hecke algebras at q = 0,” J. of Alg. Comb. 6 (4) (1997), 339-376.
7. D. Krob and J.-Y. Thibon, “A crystalizable version of Uq (Gln),” in Proceedings of Formal Power Series and Algebraic Combinatorics, FPSAC'97, C. Krattenthaler (Ed.), Vol. 2, 326-338, Universit\ddot at Wien, 1997.
8. D. Krob and J.-Y. Thibon, “Noncommutative symmetric functions V: A degenerate version of Uq (Gln),” Int. J. of Alg. and Comput., to appear.
9. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Math. Monographs, 2nd edition, Oxford University Press, 1994.
10. A.I. Molev, Noncommutative Symmetric Functions and Laplace Operators for Classical Lie Algebras, Preprint, 1994.
11. C. Reutenauer, Free Lie Algebras, Oxford University Press, 1993.
12. L. Solomon, “A Mackey formula in the group ring of a Coxeter group,” J. Algebra 41 (1976), 255-268.
13. L. Solomon, “On the Poincaré-Birkhoff-Witt theorem,” J. Comb. Th. 4 (1968), 363-375.
14. M. Takeuchi, “A two-parameter quantization of GL(n),” Proc. Japan Acad., Ser. A (1990), 112-114, 1990.
15. R.M. Wilcox, “Exponential operators and parameter differentiation in Quantum Physics,” J. Math. Phys. 8 (1967), 962-982.