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.
Highlights
Groups and algebras play a distinguished role in modern physics, since they describe symmetries
The progress at the end of the previous century reveals that the true symmetries of the stringy models involve affine [4] and double affine algebras [5–12], which are clever generalization of the simple Lie algebras
In affine case the variables and differential operators are promoted to chiral free fields, and are used in the free-field realization of Kac–Moody algebras in [16,17]
Summary
Groups and algebras play a distinguished role in modern physics, since they describe symmetries. Lie algebra generators in the monomial representation have simple form: the first order differential operators with polynomial coefficients variable. The raising operators (e in our notation) do not depend on the highest weight λ of the representation In other words, these operators have the same form in all irreducible representations of the algebra g and the corresponding formulas do not contain the Dynkin labels of the highest weight. The generators of sln+2 (n ∈ N) are realized in terms of 2n + 1 variables, that are coordinates on the generalized flag manifold These formulas appears to be useful in classifying singular vectors and analysis of singular vectors in Cauchy–Riemann case of generalised flag manifolds. In this paper we do not use flag structures and leave corresponding questions for future studies
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