Lie maps on triangular algebras without assuming unity
In this paper all algebras will be considered over a commutative unital ring. The authors characterize Lie centralizer, Lie derivations and generalized Lie \(2\)-derivations at zero products on triangular algebras without assuming unity. The Lie centralizers, Lie derivations and generalized Lie \(2\)-derivations are described for triangular algebras. Several examples are presented which illustrate limitations on extending some of the theory developed. The obtained results are used for upper triangular matrix algebras (not necessarily unital) and nest algebras. Let \(A\) be an algebra. A linear mapping \(\Psi : A \to A\) is called a \textit{centralizer} if \(\Psi(ab) = \Psi(a)b = a\Psi(b)\) for any \(a, b \in A\). Lie product on \(A\) is defined as \([a, b] = ab-ba\) for any \(a, b \in A\). We say that the linear mapping \(\Phi : A \to A\) is a \textit{Lie centralizer} if \(\Phi([a, b]) = [\Phi(a), b]\) for any \(a,b\in A\). Every centralizer is a Lie centralizer but the converse is not necessarily true. One can define a \textit{Lie derivation} in a natural way, considering associative algebra \(A\) as a Lie algebra with bracket operation. Let \(\Gamma: A\to A\) be a linear mapping and \(\Delta\) be a Lie derivation on \(A\). Then \(\Gamma\) is a \textit{generalized Lie \(2\)-derivation associated with the Lie derivation \(\Delta\)} if \(\Gamma([a, b]) = [\Gamma(a), b] + [a, \Delta(b)]\) for any \(a, b \in A\). Every Lie derivation \(\Delta\) is a generalized Lie \(2\)-derivation associated with \(\Delta\). This notion is inspired by the notion of module derivation comparing with a ring derivation. In fact, the paper aimed to study algebras, related with modulo constructions (so called \textit{triangular algebras} according to J. Lewin). Suppose \(A, B\) are algebras, \(M\)~ an \(A,B\)-bimodule, then \textit{triangular algebra} associated is \[ \mathrm{Tri}(A,M,B)= \left\{ \left( \begin{array}{cc} a & m \\ 0 & b \\ \end{array} \right) \| a\in A, b\in B, m\in M \right\}. \] Such algebras associated with bimodule structures which are building blocks for more general situations (this construction can be extended for the nonassociative case; consult papers of I. Shestakov and E. Zelmanov). More general structures are discussed in [\textit{A. Belov-Kanel} et al., Trans. Am. Math. Soc. 364, No. 10, 5525--5569 (2012; Zbl 1286.16019); \textit{A. V. Il'tyakov}, Sib. Math. J. 32, No. 6, (1991; Zbl 0777.17028); translation from Sib. Mat. Zh. 32, No. 6(190), 61--76 (1991); \textit{A. V. Il'tyakov}, Nova J. Algebra Geom. 1, No. 3, 207--259 (1992; Zbl 0892.17007); \textit{A. Ya. Vajs} and \textit{E. I. Zel'manov}, Sov. Math. 33, No. 6, 38--47 (1990; Zbl 0695.17014); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1989, No. 6(325), 42--51 (1989)] The constructions and theorems in this paper are mostly aimed to study triangular algebras.
scientific article; zbMATH DE number 7397613
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