Irreducible representations of simple Lie algebras by differential operators

Abstract

We describe a systematic method to construct arbitrary highest-weight modules, including arbitrary finite-dimensional representations, for any finite dimensional simple Lie algebra $\mathfrak{g}$. The Lie algebra generators are represented as first order differential operators in $\frac{1}{2}\left(\dim \mathfrak{g}-\text{rank}\mathfrak{g}\right)$ variables. All rising generators $\mathbf{e}$ are universal in the sense that they do not depend on representation, the weights enter (in a very simple way) only in the expressions for the lowering operators $\mathbf{f}$. We present explicit formulas of this kind for the simple root generators of all classical Lie algebras.

References in corpus (3)

• Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations
• Split Casimir operator for simple Lie algebras, solutions of Yang-Baxter equations and Vogel parameters
• Mixed network calculus

Cited by in corpus (2)

• Hunt for 3-Schur polynomials
• Polynomial representations of classical Lie algebras and flag varieties