Compatible left-symmetric algebraic structures on high rank Witt and Virasoro algebras

Abstract

We classify all graded compatible left-symmetric algebraic structures on high rank Witt algebras, and classify all non-graded ones satisfying a minor condition. Furthermore, graded compatible left-symmetric algebraic structures on high rank Virasoro algebras are also classified.

On structured spaces and their properties

“…There are many possible examples in this context. We just recall a few: 1) finitely generated abelian groups (which are all isomorphic, by the fundamental theorem of finitely generated abelian groups, to a group of the form Z n ⊕ Z/k 1 Z ⊕ ... ⊕ Z/k r Z (n is the rank, and the k i 's are powers of not necessarily distinct prime numbers); see also [19] for more details on the theory of abelian groups); 2) Finite simple groups (the list containing cyclic groups, sporadic groups, ...); 3) Finite dimensional C * -algebras (if A is a C * -algebra, then it is isomorphic to ⊕ e∈min A Ae, where min A indicates the set of minimal nonzero self-adjoint central projections of A); ... (see also the reference (in particular, [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]) for some papers on classification of various algebraic structures). So, it is natural to ask whether we can classify also structured spaces.…”