Complete Reducibility Theorem (Semisimple Lie Algebra)

Introduction

A semisimple Lie algebra is a Lie algebra that is a direct sum of simple Lie algebras, a property that reflects its "maximal" structure within the category of Lie algebras. Such algebras are characterized by the existence of a faithful, finite-dimensional representation that is irreducible, meaning no non-trivial invariant subspaces exist. The Complete Reducibility Theorem asserts that any finite-dimensional representation of a semisimple Lie algebra over an algebraically closed field is reducible, i.e., decomposes into a direct sum of irreducible representations. This theorem is pivotal in representation theory, as it provides a foundational framework for understanding the structure of semisimple Lie algebras and their associated representations.

Definitions and Preliminaries

To establish the theorem, it is essential to define key concepts in Lie algebra theory. A Lie algebra over an algebraically closed field $ \mathbb{C} $ is a vector space $ \mathfrak{g} $ equipped with a bilinear, antisymmetric bracket $ [\cdot, \cdot] $ satisfying the Jacobi identity. A Lie algebra is semisimple if it is a direct sum of simple Lie algebras, and a simple Lie algebra is one that has no non-trivial subalgebras other than $ {0} $ and itself. The structure theorem for finite-dimensional semisimple Lie algebras states that any semisimple Lie algebra over $ \mathbb{C} $ is isomorphic to the direct sum of simple Lie algebras, each corresponding to a root space in the Cartan decomposition.

A representation of a Lie algebra $ \mathfrak{g} $ is a linear map $ \rho: \mathfrak{g} \to \mathfrak{gl}(V) $, where $ V $ is a finite-dimensional vector space, such that the bracket operation is preserved under the representation. A representation is irreducible if there are no non-trivial subspaces $ W \subset V $ invariant under the action of $ \mathfrak{g} $. The Complete Reducibility Theorem asserts that any finite-dimensional representation of a semisimple Lie algebra is reducible, implying that such representations can be decomposed into irreducible components.

Statement of the Theorem

The Complete Reducibility Theorem for semisimple Lie algebras over an algebraically closed field $ \mathbb{C} $ states that any finite-dimensional representation $ \rho: \mathfrak{g} \to \mathfrak{gl}(V) $ of a semisimple Lie algebra $ \mathfrak{g} $ is reducible. This means that there exists a non-trivial subspace $ W \subset V $ such that $ \rho(\mathfrak{g})W \subset W $, or equivalently, $ V = W \oplus W' $ for some $ W' \subset V $. The theorem is a cornerstone of representation theory, as it guarantees the existence of such decompositions for semisimple algebras, enabling the classification of representations through their irreducible components.

Proof Outline

The proof of the Complete Reducibility Theorem relies on the structure of semisimple Lie algebras and their representations. A key step involves the Jordan decomposition of elements in semisimple Lie algebras, which allows the decomposition of the Lie algebra into a direct sum of nilpotent subalgebras. For a semisimple Lie algebra $ \mathfrak{g} $, the Cartan decomposition further ensures that $ \mathfrak{g} $ can be written as $ \mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m} $, where $ \mathfrak{h} $ is a maximal abelian subalgebra and $ \mathfrak{m} $ is a nilpotent subalgebra.

To establish reducibility, consider the structure theorem for finite-dimensional representations of semisimple Lie algebras. If $ \rho: \mathfrak{g} \to \mathfrak{gl}(V) $ is a representation, then the action of $ \mathfrak{g} $ on $ V $ can be decomposed into invariant subspaces. For semisimple Lie algebras, this decomposition is guaranteed by the Weyl's theorem, which states that any finite-dimensional representation of a semisimple Lie algebra is completely reducible.

Another critical component of the proof involves the Jordan canonical form of elements in semisimple Lie algebras. For any element $ X \in \mathfrak{g} $, there exists a basis of $ V $ such that the matrix representation of $ X $ is diagonalizable. This diagonalizability ensures that the representation is reducible, as the Jordan form can be decomposed into blocks corresponding to eigenvalues.