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Mat. Zametki, 2005, Volume 77, Issue 2, Pages 213–218 (Mi mz2485)  

On balanced bases

D. N. Ivanov

Tver State University

Abstract: It is proved that either a given balanced basis of the algebra $(n+1)M_1\oplus M_n$ or the corresponding complementary basis is of rank $n+1$. This result enables us to claim that the algebra $(n+1)M_1\oplus M_n$ is balanced if and only if the matrix algebra $M_n$ 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 $\operatorname{tr}XY$) and their intersection is the trivial subalgebra. Thus, the problem of whether or not the algebra $(n+1)M_1\oplus M_n$ 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 $A_{n-1}$ into the sum of Cartan subalgebras.

DOI: https://doi.org/10.4213/mzm2485

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English version:
Mathematical Notes, 2005, 77:2, 194–198

Bibliographic databases:

UDC: 512.64
Received: 13.05.2003

Citation: D. N. Ivanov, “On balanced bases”, Mat. Zametki, 77:2 (2005), 213–218; Math. Notes, 77:2 (2005), 194–198

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