Structures of Not-finitely Graded Lie Algebras Related to Generalized Virasoro Algebras

Abstract

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.

Structures of not-finitely graded Lie algebras related to generalized Heisenberg–Virasoro algebras

“…Similar to the Lie algebras studied in [3], the Lie algebra HV has the following significant features:…”

“…Nevertheless, as stated in [3], due to the fact that Γ may not be finitely generated (as a group), HV may not be finitely generated as a Lie algebra. The classical techniques (such as those in [5]) cannot be directly applied to our situation here.…”

“…Not-finitely graded Lie algebras are important objects in Lie theory, whose structure and representation theories are subjects of studies with more challenges than those of finitely graded Lie algebras. In [3], the authors have studied the structure theory of a class of not-finitely graded Lie algebras related to generalized Virasoro algebras.…”

“…In recent years, some researches on Lie algebras concerning their derivation algebras, automorphisms, second cohomology groups have been undertaken by many authors (see, e.g., [3][4][5][11][12][13][14][15][16][17]). It is well known that central extensions, which are determined by second cohomology groups, are closely related to the structures of Lie algebras (see, e.g., [6,7]).…”

Lie bialgebras of not-finitely graded Lie algebras related to generalized Virasoro algebras

“…for α, β ∈ Γ and i, j ∈ Z + , where Γ is any nontrivial additive subgroup of C. Its structures, such as derivation algebras, automorphism groups and second cohomology groups, have been studied in [1], where it is shown that the Lie algebra W (Γ) has the unique universal central extensionŴ (Γ) = W (Γ) ⊕ CC with one-dimensional center CC and relation [L α,i , L β,j ] = (β − α)L α+β,i+j + (j − i)L α+β,i+j+1 + δ α+β,0 δ i+j,0 α 3 − α 12 C for α, β ∈ Γ, i, j ∈ Z + . One of the interesting aspects, this Lie algebra turns out to be not finitely graded in the sense that there exists an abelian, necessarily infinite, group G such thatŴ (Γ) = g∈GŴ (Γ) g is G-graded for which [Ŵ (Γ) g ,Ŵ (Γ) h ] ⊆Ŵ (Γ) g+h and dimŴ (Γ) g < ∞ for g, h ∈ G.…”