Full Projective Oscillator Representations of Special Linear Lie Algebras and Combinatorial Identities

Zhenyu Zhou; Xiaoping Xu

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arXiv · 2210.13878

Full Projective Oscillator Representations of Special Linear Lie Algebras and Combinatorial Identities

Zhenyu Zhou·Xiaoping Xu

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Abstract

Using the projective oscillator representation of sl(n+1) and Shen's mixed product for Witt algebras, Zhao and the second author (2011) constructed a new functor from sl(n)-Mod to sl(n+1)-Mod. In this paper, we start from n = 2 and use the functor successively to obtain a full projective oscillator realization of any finite-dimensional irreducible representation of sl(n+1). The representation formulas of all the root vectors of sl(n+1) are given in terms of first-order differential operators in n(n+1)/2 variables. One can use the result to study tensor decompositions of finite-dimensional irreducible modules by solving certain first-order linear partial differential equations, and thereby obtain the corresponding physically interested Clebsch-Gordan coefficients and exact solutions of Knizhnik-Zamolodchikov equation in WZW model of conformal field theory.

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BibTeX RIS

Zhenyu Zhou, Xiaoping Xu. 2022-10-25. Full Projective Oscillator Representations of Special Linear Lie Algebras and Combinatorial Identities. https://arxiv.org/abs/2210.13878

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