Verbal width in the Nottingham group and related Lie algebras
\textit{M. A. Virasoro} introduced his celebrated algebraic structure (known as Virasoro algebra) in order to describe some physical phenomena, which are related to the harmonic oscillator in [``Subsidiary conditions and ghosts in dual-resonance models'', Phys. Rev. D. 1, No. 10, 2933--2936 (1970; \url{doi:10.1103/PhysRevD.1.2933})]. Virasoro algebras appeared in different ways in many dynamical systems, because their structure is very interesting and deserves a separate interest, not only in mathematical physics, but also in group theory and Lie theory. It should be mentioned (from the mathematical perspective) that a well-known result of \textit{M. Lazard} [Ann. Sci. Éc. Norm. Supér. (3) 71, 101--190 (1954; Zbl 0055.25103)] explores in fact a correspondence between nilpotent groups and nilpotent Lie rings. The present contribution shows some peculiarities of Virasoro algebras in connection with a profinite group, known as Nottingham group, and introduced by \textit{D. L. Johnson} [J. Aust. Math. Soc., Ser. A 45, No. 3, 296--302 (1988; Zbl 0666.20016)]. Virasoro algebras and some of their subalgebras are in fact related to the Nottingham group. Theorem 3.7 of the paper under review shows that the Nottingham group in zero characteristic is verbally elliptic, that is, all the words of such a group have finite width. Theorems 2.2 and 2.3 show that arbitrary multilinear polynomials have corresponding properties of width in the context of Virasoro algebras. Theorem 2.4 describes additional properties of structural nature for Virasoro algebras.
scientific article; zbMATH DE number 7318820
scientific article