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This article is cited in 4 scientific papers (total in 4 papers)
On balanced systems of idempotents
D. N. Ivanov M. V. Lomonosov Moscow State University
Abstract:
By definition, a balanced basis of an associative semisimple finite-dimensional algebra over the field of complex numbers $\mathbb C$ is a system of idempotents $\{e_i\}$ such that it forms a linear basis and the $\operatorname{Tr}e_i$ and $\operatorname{Tr}e_ie_j$ are independent of $i$, $j$, $i\ne j$, where $\operatorname{Tr}$ is the trace of the regular
representation of the algebra. In the present paper balanced bases are constructed in the matrix algebra $\mathrm M_{p^n}(\mathbb C)$, where $p$ is an odd prime. For matrix
algebras such bases have so far been known only in the cases $\mathrm M_2(\mathbb C)$ and $\mathrm M_3(\mathbb C)$. It is proved that there are no balanced bases of certain ranks having a regular elementary Abelian 2-group of automorphisms in the algebras $\mathrm M_{2^n}(\mathbb C)$, $n>1$. In addition, the balanced 1-systems of $n+1$ idempotents of rank $r$ in the algebra $\mathrm M_{rn}(\mathbb C)$ are classified.
Received: 15.03.2000
Citation:
D. N. Ivanov, “On balanced systems of idempotents”, Mat. Sb., 192:4 (2001), 73–86; Sb. Math., 192:4 (2001), 551–564
Linking options:
https://www.mathnet.ru/eng/sm558https://doi.org/10.1070/sm2001v192n04ABEH000558 https://www.mathnet.ru/eng/sm/v192/i4/p73
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Abstract page: | 395 | Russian version PDF: | 200 | English version PDF: | 24 | References: | 68 | First page: | 1 |
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