A basis for the right quantum algebra and the “1= q ” principle
Dominique Foata1
and Guo-Niu Han2
1Institut Lothaire 1 rue Murner 67000 Strasbourg France
2Université Louis Pasteur I.R.M.A. UMR 7501 7 rue René-Descartes 67084 Strasbourg France
2Université Louis Pasteur I.R.M.A. UMR 7501 7 rue René-Descartes 67084 Strasbourg France
DOI: 10.1007/s10801-007-0080-5
Abstract
We construct a basis for the right quantum algebra introduced by Garoufalidis, Lê and Zeilberger and give a method making it possible to go from an algebra subject to commutation relations (without the variable q) to the right quantum algebra by means of an appropriate weight-function. As a consequence, a strong quantum MacMahon Master Theorem is derived. Besides, the algebra of biwords is systematically in use.
Pages: 163–172
Keywords: keywords right quantum algebra; quantum macmahon master theorem
Full Text: PDF
References
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2. Etingof, P., & Pak, I. An algebraic extension of the MacMahon Master Theorem. Proceedings of the American Mathematical Society, to appear.
3. Foata, D., & Han, G.-N. (2007). A new proof of the Garoufalidis-Lê-Zeilberger quantum MacMahon Master Theorem. Journal of Algebra, 307, 424-431.
4. Garoufalidis, S., Lê, T.T.Q., & Zeilberger, D. (2006). The quantum MacMahon Master Theorem. Proceedings of the National Academy of Science of the United States of America, 103, 13928-13931.
5. Hai, P. H., & Lorenz, M. Koszul algebras and the quantum MacMahon Master Theorem. The Bulletin of the London Mathematical Society, to appear.
6. Kassel, C. (1995). Quantum groups. Graduate texts in mathematics (Vol. 155). New York: Springer.
7. Konvalinka, M., & Pak, I. Non-commutative extensions of the MacMahon Master Theorem. Advances in Mathematics, to appear.
8. Lauve, A., & Taft, E. J. (2007). A class of left quantum groups modeled after SLq (r). Journal of Pure and Applied Algebra, 208, 797-803.
9. Lothaire, M. (1983). Combinatorics on words. Encyclopedia of mathematics and its applications (Vol. 17). London: Addison-Wesley.
10. Lothaire, M. (2002). Algebraic combinatorics on words. Encyclopedia of mathematics and its applications (Vol. 90). Cambridge: Cambridge Univ. Press.
11. MacMahon, P. A. (1915). Combinatory analysis, Vol.
1. Cambridge Univ. Press. Reprinted by Chelsea Publ. Co., New York, 1960.
12. Parshall, B., & Wang, J.-P. (1991). Quantum linear groups. Memoirs of the American Mathematical Society (Vol. 89).
13. Rodríguez-Romo, S., & Taft, E. (2002). Some quantum-like Hopf algebras which remain noncommutative when q =
1. Letters in Mathematical Physics, 61, 41-50.
14. Rodríguez-Romo, S., & Taft, E. (2005). A left quantum group. Journal of Algebra, 286, 154-160.