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

SIGMA 15 (2019), 026, 10 pages      arXiv:1709.08394      https://doi.org/10.3842/SIGMA.2019.026

Contravariant Form on Tensor Product of Highest Weight Modules

Andrey I. Mudrov
Department of Mathematics, University of Leicester, University Road, LE1 7RH Leicester, UK

Received August 23, 2018, in final form March 25, 2019; Published online April 07, 2019

Abstract
We give a criterion for complete reducibility of tensor product $V\otimes Z$ of two irreducible highest weight modules $V$ and $Z$ over a classical or quantum semi-simple group in terms of a contravariant symmetric bilinear form on $V\otimes Z$. This form is the product of the canonical contravariant forms on $V$ and $Z$. Then $V\otimes Z$ is completely reducible if and only if the form is non-degenerate when restricted to the sum of all highest weight submodules in $V\otimes Z$ or equivalently to the span of singular vectors.

Key words: highest weight modules; contravariant form; tensor product; complete reducibility.

pdf (350 kb)   tex (17 kb)

References

1. Asherova R.M., Smirnov Yu.F., Tolstoy V.N., Projection operators for simple Lie groups, Teoret. and Math. Phys. 8 (1971), 813-825.
2. Bernstein J.N., Gelfand S.I., Tensor products of finite and infinite dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), 245-285.
3. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
4. Drinfeld V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
5. Drinfeld V.G., Quasi-Hopf algebras, Leningrad Math. J. 1 (1989), 1419-1457.
6. Erdmann K., Wildon M.J., Introduction to Lie algebras, Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd., London, 2006.
7. Etingof P., Varchenko A., Dynamical Weyl groups and applications, Adv. Math. 167 (2002), 74-127, arXiv:math.QA/0011001.
8. Jantzen J.C., Lectures on quantum groups, Graduate Studies in Mathematics, Vol. 6, Amer. Math. Soc., Providence, RI, 1996.
9. Khoroshkin S.M., Tolstoy V.N., Extremal projector and universal $R$-matrix for quantized contragredient Lie (super)algebras, in Quantum Groups and Related Topics (Wrocław, 1991), Math. Phys. Stud., Vol. 13, Kluwer Acad. Publ., Dordrecht, 1992, 23-32.
10. Kostant B., On the tensor product of a finite and an infinite dimensional representation, J. Funct. Anal. 20 (1975), 257-285.
11. Kulish P.P., Mudrov A.I., Dynamical reflection equation, in Quantum Groups, Contemp. Math., Vol. 433, Amer. Math. Soc., Providence, RI, 2007, 281-309, arXiv:math.QA/0405556.
12. Mudrov A., Equivariant vector bundles over quantum spheres, arXiv:1710.05690.
13. Mudrov A., Equivariant vector bundles over quantum projective spaces, Teoret. and Math. Phys. 198 (2019), 284-295, arXiv:1805.02557.
14. Shapovalov N.N., On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra, Funct. Anal. Appl. 6 (1972), 307-312.
15. Zhelobenko D.P.., Representations of reductive Lie algebras, Nauka, Moscow, 1994.