Free Subgroups and Free Subsemigroups of Division Rings Original Research Article

LetWcolon, equalsW(m, n) be the general Lie algebra of Cartan type and imagecolon, equalsimage(m, n) the corresponding divided power algebra on whichWacts. Translative isomorphisms, i.e., isomorphisms ofWintoWplus sign, dot abovegl(m, image) of the formAmaps toA+phi(A),Aset membership, variantW, phi(A)set membership, variantgl(m, image), are investigated. The mixed product theorems of the author (1986,Sci. Sinica Ser. A29, No. 6, 570–581) are generalized. Under certain conditions, translative isomorphisms induce left-symmetric structures onW. Classes of left-symmetric structures onWare constructed, among which are restricted left-symmetric structures on the restricted Lie algebraW(m, 1).