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 $${\mathfrak {g}}$$ g . The Lie algebra generators are represented as first order differential operators in $$\frac{1}{2} \left( \dim {\mathfrak {g}} - \text {rank} \, {\mathfrak {g}}\right) $$ 1 2 dim g - rank g variables. All rising generators $$\mathbf{e}$$ 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 $$\mathbf{f}$$ f . We present explicit formulas of this kind for the simple root generators of all classical Lie algebras.

Polynomial representations of classical Lie algebras and flag varieties

“…The resulting formulas (32),(34),(36),(38) for arbitrary weight λ coincide with those, derived from the Dynkin diagrams in [16]. This paper organized as follows.…”

“…the generators ( 13) coincide with the generators, found in [16]. The generators describe irreducible representations in a universal way, in a sense that the vector fields nearly independent on a representation.…”

“…The first and second column represents pairs of g and g algebras. The last column provides explicit form of generators (16) in representation ρ forming commuting algebra g and satisfying (3). Note that the algebra generators A a,b , B a,b , C a,b are different for so and sp algebras.…”

“…Our purpose is to build a uniform and universal construction in the most elementary way, using no detailed knowledge about particular algebras understandable to a broad audience. In current paper we explain how the results, obtained in this way for classical algebras in [16], are related to the language of coset theories, gauge transformations and flag varieties which is perhaps more familiar to some readers.…”

3-Schurs from explicit representation of Yangian $$ \textrm{Y}\left({\hat{\mathfrak{gl}}}_1\right) $$. Levels 1–5

“…where the normalization of Schur polynomials and time-variables are different from the classical choice. As soon as Schur functions S R [p] have two kinds of variables -Young diagram and time-varibles, this allows them to be the transformation matrices in Schur-Weyl duality, since R are natural for the action of symmetric group, while linear algebras act as differential operators in times (a la free-field representations of [44][45][46][47]).…”