A characterization of *-minimal algebras with involution.
Let \((A,*)\) be a *-PI algebra with involution over a field of characteristic zero and let \(c_m(A,*)\) denote its \(m\)-th *-codimension. \textit{A. Giambruno} and \textit{M. Zaicev}, [J. Algebra 222, No. 2, 471-484 (1999; Zbl 0944.16023)], proved that, if \(A\) is finite dimensional, then there exists \(\lim_{m\to+\infty}\root m\of {c_m(A,*)}\), and it is an integer, which is called the *-exponent of \(A\). As a consequence of the presence of this invariant, in a natural manner [\textit{O. M. Di Vincenzo} and \textit{R. La Scala}, J. Algebra 317, No. 2, 642-657 (2007; Zbl 1138.16008)] the definition of *-minimal algebra was introduced. The authors characterize, up to *-PI equivalence, *-minimal algebras. Theorem. A finite-dimensional algebra with involution is *-minimal if and only if it is *-PI equivalent to a block triangular matrix algebra with involution, denoted by \(UT_*(A_1,\dots,A_n)\), where \((A_1,\dots,A_n)\) is a suitable \(n\)-tuple of *-simple algebras.
scientific article; zbMATH DE number 6108405
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