Nilpotency criteria for multiplicative Lie algebras
The structure of a multiplicative Lie algebra has been introduced by \textit{G. J. Ellis} [J. Aust. Math. Soc., Ser. A 54, No. 1, 1-19 (1993; Zbl 0777.20001)]. A multiplicative Lie algebra is a (possibly nonabelian) group with an extra binary function \(\{.,.\}\) satisfying the identities: \[ \{x,x\}=1,\quad\{x,yy'\}=\{x,y\}^y\{x,y'\},\quad\{xx',y\}={}^x\{x',y\}\{x,y\}, \] \[ \{y,x,{}^xz\}\{x,z,{}^zy\}\{z,y,{}^yx\}=1,\;{}^z\{x,y\}=\{^zx,^zy\},\text{ where }\{x,y,z\}=\{\{x,y\},z\},\;{}^yx=yxy^{-1}. \] The two main examples of multiplicative Lie algebras are a group with the binary function \(\{.,.\}\) defined as the group commutator \([x,y]\equiv xyx^{-1} y^{-1}\) and a Lie ring with \(\{.,.\}\), defined as the Lie bracket. The notions of ideal, nilpotency, solubility can be defined in terms of a new binary function. The authors prove several nilpotency criteria for multiplicative Lie algebras which extend known results about groups and Lie algebras. The main results of the paper are some variations of Gruenberg's theorem, stating nilpotency of a soluble finitely generated Engel group (Lie ring) [\textit{K. W. Gruenberg}, Proc. Camb. Philos. Soc. 49, 377-380 (1953; Zbl 0050.01901), Theorem 1]. The authors also prove an analog of the Hirsch-Plotkin theorem for multiplicative Lie algebras.
scientific article; zbMATH DE number 920032
scientific article