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Published in:
–Lie Algebras Canonically Associated to Probability Measures on with All Moments Read chapter Read first chapter

2016 | OriginalPaper | Chapter

Quantum Plactic and Pseudo-Plactic Algebras

Author : Todor Popov

Published in: Lie Theory and Its Applications in Physics

Publisher: Springer Singapore

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Abstract

We review the Robinson–Schensted–Knuth correspondence in the light of the quantum Schur–Weyl duality. The quantum plactic algebra is defined to be a Schur functor mapping a tower of left modules of Hecke algebras into a tower of \({U_q{\mathfrak {gl}}}\)-modules. The functions on the quantum group carry a \({U_q{\mathfrak {gl}}}\)-bimodule structure whose combinatorial spirit emerges in the RSK algorithm. The bimodule structure on the algebra of biletter words is used for a functorial formulation of the quantum pseudo-plactic algebra. The latter algebra has been proposed by Daniel Krob and Jean-Yves Thibon as a higher noncommutative analogue of the quantum torus.

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Footnotes
1
It is worth noting that a noncommutative word \(w=w_1 w_2 \ldots w_r\) can be written as a monomial in commutative biletters \(\left( \begin{array}{c}w_i\\ i \end{array}\right) \), e.g., \(\Big [\begin{array}{cccc}w_1&{} w_2 &{}\ldots &{}w_r\\ 1 &{}2&{} \ldots &{} r \end{array}\Big ]\).
 
2
Although some of the Schur modules \(V_{\lambda }=S_{\lambda }(V)\) are vanishing, those with height \(ht(\lambda )>\dim V\).
 
3
Another way to get \({\mathfrak {Lie}}^3_q(V)\) is through the parabolic subalgebra \( \mathfrak {gl}(V) \ltimes {\mathfrak n}(V) \subset \mathfrak {so}_{1+2\dim V}\). Its radical \({\mathfrak n}(V)\) defines \(U_q {\mathfrak n}(V)\) uniquely from the quantum Serre relations of \(U_q \mathfrak {so}_{2n+1}\).
 
Literature
1.
E. Date, M. Jimbo and T. Miwa, Representations of U\(_q\) (gl(n,\(\mathbb{C}\))) at \(q = 0\) and the Robinson- Schensted correspondence, World Sci. Publ. (1990), 185-211.
2.
M. Dubois-Violette and T. Popov, Lett. Math. Phys. 61 (2002), 159-170.
3.
D. Krob and J.-Y. Thibon, Noncommutative symmetric functions IV: Quantum linear groups and Hecke algebras at \(q= 0\), Journal of Algebraic Combinatorics 6(1997), 339-376.
4.
A. Lascoux, B. Leclerc, and J.-Y. Thibon, The plactic monoid, Algebraic Combinatoric on Words (2002), 10pp.
5.
J.-L. Loday, Série de Hausdorff, idempotents eulériens et algèbres de Hopf, Exposition. Math. 12 (1994), 165-178.
6.
J.-L. Loday and T. Popov, Parastatistics Algebra, Young Tableaux and Super Plactic Monoid, International Journal of Geometric Methods in Modern Physics 5 (2008), 1295-1314.
7.
J.-L. Loday, T. Popov, Hopf Structures on Standard Young Tableaux, in: Proceedings “Lie Theory and Its Applicatioins in Physics”, V. Dobrev ed., AIP conference series Vol. 1243(2010), 265–275.
8.
C. Malvenuto and C. Reutenauer, Duality between Quasi-Symmetric Functions and the Solomon Descent Algebra, Journal of Algebra 177 (1995), 967–982.
9.
S. Poirier and C. Reutenauer, Algèbres de Hopf de tableaux, Ann. Sci. Math. Québec 19 (1995), 79-90.
Metadata
Title
Quantum Plactic and Pseudo-Plactic Algebras
Author
Todor Popov
Copyright Year
2016
Publisher
Springer Singapore
Book
Lie Theory and Its Applications in Physics

Print ISBN: 978-981-10-2635-5

Electronic ISBN: 978-981-10-2636-2

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-981-10-2636-2_32

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