A spatial version of Wedderburn’s Principal Theorem

Abstract

In this article we verify that ‘Wedderburn’s Principal Theorem’ has a particularly pleasant spatial implementation in the case of cleft subalgebras of the algebra of all linear transformations on a finite-dimensional vector space. Once such a subalgebra A is represented by block upper triangular matrices with respect to a maximal chain of its invariant subspaces, after an application of a block upper triangular similarity, the resulting algebra is a linear direct sum of an algebra of block-diagonal matrices and an algebra of strictly block upper triangular matrices (i.e. the radical), while the block-diagonal matrices involved have a very nice structure. We apply this result to demonstrate that, when the underlying field is algebraically closed, and (Rad(A))μ(A)−1 ≠ {0} the algebra is unicellular, i.e. the lattice of all invariant subspaces of A is totally ordered by inclusion. The quantity μ(A) stands for the length of (every) maximal chain of non-zero invariant subspaces of A.