Abstract
It is proved that either a given balanced basis of the algebra (n + 1)M1 ⊕ Mn or the corresponding complementary basis is of rank n + 1. This result enables us to claim that the algebra (n + 1)M1 ⊕ Mn is balanced if and only if the matrix algebra Mn admits a WP-decomposition, i.e., a family of n + 1 subalgebras conjugate to the diagonal algebra and such that any two algebras in this family intersect orthogonally (with respect to the form tr XY) and their intersection is the trivial subalgebra. Thus, the problem of whether or not the algebra (n + 1)M1 ⊕ Mn is balanced is equivalent to the well-known “Winnie-the-Pooh problem” on the existence of an orthogonal decomposition of a simple Lie algebra of type An−1 into the sum of Cartan subalgebras.
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