tailieunhanh - Complemented invariant subspaces of structural matrix algebras

In this paper, we explore when the lattice of invariant subspaces of a structural matrix algebra can be complemented. We give several equivalent conditions for this lattice to be a Boolean algebra. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 993 – 1000 ¨ ITAK ˙ c TUB ⃝ doi: Complemented invariant subspaces of structural matrix algebras Mustafa AKKURT,1,∗ Emira AKKURT,1 George Phillip BARKER2 Department of Mathematics, Gebze Institute of Technology, Gebze, Kocaeli, Turkey 2 Department of Mathematics, University of Missouri Kansas City, Kansas City, Missouri, USA 1 Received: • Accepted: • Published Online: • Printed: Abstract: In this paper, we explore when the lattice of invariant subspaces of a structural matrix algebra can be complemented. We give several equivalent conditions for this lattice to be a Boolean algebra. Key words: Boolean algebra, structural matrix algebra, invariant subspace, lattice of invariant subspaces 1. Introduction Let V denote a vector space of finite dimension n over a field F . Let L(V ) denote the set of all subspace of V . Then L(V ) is a modular lattice under the operations ∩ and + . If W is a sublattice, it is also modular. Let Hom(V ) denote the algebra of all linear transformations of V onto itself. As usual, Hom(V ) can be identified with Mn (F ), the algebra of all n × n matrices over F . We assume that all algebras contain the identity map, I . Definition Let V be a sublattice of L(V ) and let R be a subalgebra of Hom(V ). We define AlgV = {θ ∈ Hom(V ) : W θ ⊂ W, for every W ∈ V} and LatR = {W ∈ L(V ) : W θ ⊂ W, for every θ ∈ R}. AlgV is a subalgebra of Hom (V ) and LatR is a sublattice of L(V ). In general, the containments LatAlgV ⊇ V and AlgLatR ⊇ R are proper. If equality holds, then V (respectively R ) is called reflexive (see [5]). Let F be a field and let ρ be a reflexive transitive relation on the set N = {1, ., n} for some n ≥ 2 (more information about ρ will be given in Section 2). The set Mn (F, ρ) = {A ∈ Mn (F ) : aij = 0 whenever (i, j) ∈ / ρ} is a subalgebra of Mn (F