Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 064, 34 pages      arXiv:1101.2647      http://dx.doi.org/10.3842/SIGMA.2011.064

Structure Constants of Diagonal Reduction Algebras of gl Type

Sergei Khoroshkin a, b and Oleg Ogievetsky c, d, e
a) Institute of Theoretical and Experimental Physics, 117218 Moscow, Russia
b) Higher School of Economics, 20 Myasnitskaya Str., 101000 Moscow, Russia
c) J.-V. Poncelet French-Russian Laboratory, UMI 2615 du CNRS, Independent University of Moscow, 11 B. Vlasievski per., 119002 Moscow, Russia
d) Centre de Physique Théorique, Luminy, 13288 Marseille, France
e) On leave of absence from P.N. Lebedev Physical Institute, Theoretical Department, 53 Leninsky Prospekt, 119991 Moscow, Russia

Received January 14, 2011, in final form June 27, 2011; Published online July 09, 2011

Abstract
We describe, in terms of generators and relations, the reduction algebra, related to the diagonal embedding of the Lie algebra gln into gln⊕gln. Its representation theory is related to the theory of decompositions of tensor products of gln-modules.

Key words: reduction algebra; extremal projector; Zhelobenko operators.

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References

  1. Arnaudon D., Buffenoir E., Ragoucy E., Roche Ph., Universal solutions of quantum dynamical Yang-Baxter equations, Lett. Math. Phys. 44 (1998), 201-214, q-alg/9712037.
  2. Asherova R.M., Smirnov Yu.F., Tolstoy V.N., Projection operators for simple Lie groups. II. General scheme for construction of lowering operators. The groups SU(n), Teoret. Mat. Fiz. 15 (1973), 107-119 (English transl.: Theoret. and Math. Phys. 15 (1973), 392-401).
  3. Asherova R.M., Smirnov Yu.F., Tolstoy V.N., Description of a certain class of projection operators for complex semi-simple Lie algebras, Mat. Zametki 26 (1979), 15-25 (English transl.: Math. Notes 26 (1979), 499-504).
  4. Cherednik I., Quantum groups as hidden symmetries of classical representation theory, in Differential Geometric Methods in Theoretical Physics (Chester, 1988), Editor A.I. Solomon, World Sci. Publ., Teaneck, NJ, 1989, 47-54.
  5. Khoroshkin S., An extremal projector and dynamical twist, Teoret. Mat. Fiz. 139 (2004), 158-176 (English transl.: Theoret. and Math. Phys. 139 (2004), 582-597).
  6. Khoroshkin S., Ogievetsky O., Mickelsson algebras and Zhelobenko operators, J. Algebra 319 (2008), 2113-2165, math.QA/0606259.
  7. Khoroshkin S., Ogievetsky O., Diagonal reduction algebras of gl type, Funktsional. Anal. i Prilozhen. 44 (2010), no. 3, 27-49 (English transl.: Funct. Anal. Appl. 44 (2010), 182-198), arXiv:0912.4055.
  8. Lepowsky J., McCollum G.W., On the determination of irreducible modules by restriction to a subalgebra, Trans. Amer. Math. Soc. 176 (1973), 45-47.
  9. Olshanski G., Extension of the algebra U(g) for infinite-dimensional classical Lie algebras g, and the Yangians Y(gl(m)), Dokl. Akad. Nauk SSSR 297 (1987), 1050-1054 (English transl.: Soviet Math. Dokl. 36 (1988), 569-573).
  10. Zhelobenko D., Representations of reductive Lie algebras, Nauka, Moscow, 1994 (in Russian).

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