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\(T^*\)-extensions of Lie color algebras. (Chinese. English summary) Zbl 1324.17026

Summary: In this paper, the notion of \(T^*\)-extension of a Lie color algebra is introduced. Many properties of a Lie color algebra can be lifted to its \(T^*\)-extensions, such as nilpotency, solvability and decomposition. It is proved that every finite-dimensional nilpotent quadratic Lie color algebra \(A\) over an algebraically closed field of characteristic different from 2 is isometric to a \(T^*\)-extension of a nilpotent Lie color algebra \(B\), and the nilpotent length of \(B\) is at most half of that of \(A\). Moreover, the equivalence of \(T^*\)-extensions is investigated from the cohomological point of view.

MSC:

17B75 Color Lie (super)algebras
17B30 Solvable, nilpotent (super)algebras
17A45 Quadratic algebras (but not quadratic Jordan algebras)
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