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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

On certain properties of branching coefficients for affine Lie algebras


Authors: M. Ilyin, P. Kulish and V. Lyakhovsky
Translated by: the authors
Original publication: Algebra i Analiz, tom 21 (2009), nomer 2.
Journal: St. Petersburg Math. J. 21 (2010), 203-216
MSC (2000): Primary 17B10, 17B20
Published electronically: January 21, 2010
MathSciNet review: 2549451
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Abstract: It is demonstrated that the decompositions of integrable highest weight modules of a simple Lie algebra (classical or affine) with respect to its reductive subalgebra obey a set of algebraic relations leading to recursive properties for the corresponding branching coefficients. These properties are encoded in a special element $ \Gamma _{\mathfrak{g} \supset \mathfrak{a}}$ of the formal algebra $ \mathcal{E}_{\mathfrak{a}}$ that describes the injections $ \mathfrak{a}\to \mathfrak{g}$ and is called a fan. In the simplest case where $ \mathfrak{a} = \mathfrak{h} (\mathfrak{g})$, the recursion procedure generates the weight diagram of a module $ L_{\mathfrak{g}}$. When the recursion described by a fan is applied to highest weight modules, it provides a highly efficient tool for explicit calculations of branching coefficients.


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Additional Information

M. Ilyin
Affiliation: Department of Theoretical Physics, St. Petersburg State University, St. Petersburg 198904, Russia
Email: milyin-5@mail.ru

P. Kulish
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: kulish@pdmi.ras.ru

V. Lyakhovsky
Affiliation: Department of Theoretical Physics, St. Petersburg State University, St. Petersburg 198904, Russia
Email: lyakh1507@nm.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-10-01090-3
PII: S 1061-0022(10)01090-3
Keywords: Integrable highest weight modules, simple Lie algebra, reductive subalgebra, branching coefficients, fan, weight diagram.
Received by editor(s): September 14, 2008
Published electronically: January 21, 2010
Additional Notes: The second author was supported by RFFI grant 09-01-00504
The third author was supported by RFFI grant 09-01-00504 and the National Project RNP.2.1.1./1575
Article copyright: © Copyright 2010 American Mathematical Society