Representation Properties of Compact Lie Algebras (Integralness of Weights)

Introduction

Compact Lie algebras are a fundamental object in Lie algebra theory, characterized by their finite dimensionality and the existence of a non-degenerate invariant inner product. These algebras are closely tied to compact Lie groups, whose Lie algebras correspond via the exponential map. The study of representations of compact Lie algebras is pivotal in understanding their structural properties, as representations encode information about the algebra’s symmetry and its associated Lie groups. Among the key features of representations are the weights, which are the eigenvalues of the action of the Lie algebra on a given representation space. The integralness of these weights is a critical property that distinguishes certain representations from others and has profound implications for the classification and structure of compact Lie algebras.

Definitions and Preliminaries

A compact Lie algebra is a Lie algebra that is isomorphic to the Lie algebra of a compact Lie group. Such algebras are finite-dimensional, and their structure is governed by the Cartan decomposition and the root system. The Cartan decomposition decomposes the algebra into a sum of subalgebras associated with a Cartan subalgebra, and the root space decomposition further breaks down the algebra into eigenspaces of the Cartan subalgebra. For compact Lie algebras, the Killing form is non-degenerate, which ensures the existence of an invariant inner product and allows for the construction of orthogonal bases.

A representation of a compact Lie algebra is a linear map from the algebra to the space of bounded linear operators on a complex vector space, satisfying the Lie bracket relation. Finite-dimensional representations are particularly significant, as they admit a complete set of eigenvectors and are classified by their highest weights. The integralness of weights in a representation refers to the property that the eigenvalues of the adjoint action on a given vector are integers. This property is crucial in the study of modular representations and the classification of irreducible representations.

Properties of Compact Lie Algebras

Compact Lie algebras possess several key properties that distinguish them from other Lie algebras. First, they are nilpotent if and only if their Killing form is trivial, which is a condition satisfied by all compact Lie algebras. Second, the root system of a compact Lie algebra is finite, and the roots are indexed by a finite set of integers. This finiteness ensures that the representation theory is well-behaved, with the existence of a finite number of irreducible representations. The structure theorem for finite-dimensional compact Lie algebras states that any compact Lie algebra is isomorphic to the Lie algebra of a compact Lie group, and this isomorphism is determined by the Cartan subalgebra and the root system.

The Killing form plays a central role in the classification of representations. For compact Lie algebras, the Killing form is non-degenerate, which allows for the construction of orthogonal bases and the computation of invariant inner products. This non-degeneracy ensures that the representation spaces are finite-dimensional and that the weights of a representation are well-defined. Moreover, the Killing form's non-degeneracy implies that the representation is irreducible if and only if the algebra is semisimple, a property that holds for all compact Lie algebras.

Integralness of Weights in Representations

The integralness of weights