On balanced systems of idempotents (Q2759309)

The notion of `balanced \(t\)-systems' in complex semisimple finite-dimensional associative algebras was introduced by the author in his previous papers [Mat. Sb. 189, No. 12, 83-102 (1998; Zbl 0937.16033) and Mat. Sb. 191, No. 4, 67-90 (2000; see the preceding review Zbl 1022.16017)]. In this paper, he considers `balanced bases' of the algebra \(M_n(\mathbb{C})\) of complex \(n\times n\)-matrices, that is, balanced \(2\)-systems \(\mathcal E\) that form a linear basis of \(M_n(\mathbb{C})\). He constructs balanced bases for \(M_{p^n}(\mathbb{C})\) with \(p\) any odd prime. His construction utilizes the same special basis of \(M_{p^n}(\mathbb{C})\) that was used by \textit{A. I. Kostrikin, I. A. Kostrikin}, and \textit{V. A. Ufnarovskij} [in Tr. Mat. Inst. Steklova 158, 105-120 (1981; Zbl 0526.17003)] to construct orthogonal decompositions of the corresponding Lie algebra (into a direct sum of Cartan subalgebras that are pairwise orthogonal with respect to the Killing form). In the case of \(M_{2^n}(\mathbb{C})\), the author establishes non-existence of balanced bases of a certain kind.