From the introduction: Let \(G = \langle X,E\rangle\) be an undirected graph without loops with the finite set of vertices \(X = \{x_1, \ldots, x_n\}\) and the set of edges \(E\) \((E\subseteq X\times X)\). We denote the elements of \(E\) by \(\{x, y\}\). Consider a variety \(M\) of Lie algebras over a ring \(R\). A partially commutative Lie algebra in \(M\) with a defining graph \(G\) is a Lie algebra \(\mathcal L_R(X;G)\) defined as \[ \mathcal L_R(X;G) = \langle X\mid [x_i, x_j ] = 0 \Leftrightarrow \{x_i, x_j\} \in E; M\rangle \] in \(M\). Thus, in this algebra, the variety identities and the defining relations hold together. If there is no ambiguity denote this algebra just by \(\mathcal L(X;G)\). For simplicity (to avoid using the notation for the set of edges), we write \(\{x_i, x_j\} \in G\) instead of \(\{x_i, x_j\} \in E\).
Usually, the varieties whose identities do not imply additional relations of vertices’ commutation are considered. It means that two vertices commute if and only if they are adjacent in \(G\). In this paper, we study partially commutative metabelian Lie algebras. These algebras clearly possess the property indicated above. It follows, for example, from the structure of the bases for partially commutative metabelian Lie algebras (see Theorem 3.3).
Along with the variety of Lie algebras, one can consider other varieties of algebras and groups. The most actively studied objects of this kind are partially commutative groups which are defined by commutativity relations in the variety of all groups. Some papers are devoted to universal theories of partially commutative metabelian groups. Also partially commutative associative algebras were studied.
This paper is organized as follows. In Sec. 2, preliminary definitions and results are given. In Sec. 3, we find bases for partially commutative metabelian Lie (see Theo- rem 3.3). This theorem is used a great deal in the paper but it is also interesting in itself. In Sec. 4, we prove Theorem 4.7 and Theorem 4.9 which give information on the centralizers of some elements and on the annihilators of some elements in the derived subalgebra of partially commutative metabelian Lie algebras. We need these results for the study of the universal theories of partial commutative metabelian Lie algebras.
The main result of the paper is Theorem 5.9 which is proved in Sec. 5. This is a criterion for coincidence of the universal theories of partially commutative metabelian Lie algebras whose defining graphs are trees. It is easy to verify if this condition holds. So, the problem of universal equivalence for the algebras defined above is algorithmically solvable.