Lie identities for skew and symmetric elements of semiprime superalgebras with superinvolution.
For a semiprime associative superalgebra \(A\) the intersection of the prime ideals of \(A\) is zero and so \(A\) is a subdirect product of its prime images. When each prime image of \(A\) is a central order in a simple superalgebra at most \(n^2\)-dimensional over its center, we say that \(A\) is \(S(n)\). For an algebra \(A\) with involution \(*\) denote by \(H\) (respectively \(K\)) the set of all elements \(x\in A\) such that \(x^*=x\) (respectively \(x^*=-x\)). Now for an associative superalgebra \(A\) over a (commutative associative and unital) ring \(R\), denote by \(A^-\) the Lie superalgebra whose underlying \(R\)-module agrees with that of \(A\) and whose Lie superbracket satisfies \([a,b]=ab-(-1)^{\overline a\overline b}ba\) for any homogeneous elements \(a,b\) of degrees \(\overline a\) and \(\overline b\), respectively. For any \(m\geq 3\) and elements \(x_1,\dots,x_m\in A\) we can inductively define \([x_1,\dots,x_m]:=[[x_1,\dots,x_{m-1}],x_m]\) and so a subspace \(V\) of \(A\) is said to be Lie nilpotent if there exists an \(m\) such that \([x_1,\dots,x_m]=0\) for any \(x_1,\dots,x_m\in V\). In a similar way we can define Lie solvability of the subspace \(V\). The first main result of this paper is that if \(A\) is a nontrivial semiprime associative superalgebra over a commutative ring of scalars \(R\) with \({1\over 2}\in R\) and \(A\) has a superinvolution, then if \(H\) is Lie solvable \(A\) is \(S(2)\). The second main result is that if \(A\) is a nontrivial semiprime associative superalgebra over a commutative ring of scalars \(R\) with \({1\over 2}\in R\) and \(A\) has a superinvolution then: (a) \(H\) is Lie nilpotent if and only if the elements of \(H\) commute. (b) \(K\) is Lie nilpotent if and only if the elements of \(K\) commute. Some other interesting results can be found in section 3 of the paper. So for instance the authors characterize nontrivial prime associative superalgebras with superinvolution whose skew elements or symmetric elements are Lie nilpotent or Lie solvable.
scientific article; zbMATH DE number 6154886
scientific article