Quasifinite representations of a family of Lie algebras of Block type
The author investigates representations of Lie algebras \(B(s,G)\) over an algebraically closed field \(\mathbb F\) of characteristic zero, where \(s\) is 0 or 1 and \(G\) is an additive subgroup of \(\mathbb F\). \(B(s,G)\) has basis \(\{ x^{a,i} \mid a \in G, i \in \mathbb Z_+\}\), and \([x^{a,i},x^{b,j}] = s(b-a)x^{a+b,i+j} + ((a-1+s)j -(b-1+s)i)x^{a+b,i+j-1}\). It is shown that \(B(s,G)\) has a nontrivial quasifinite module (i.e. \(G\)-graded with all components finite dimensional) if and only if \(s = 1\) and \(G\) is isomorphic to \(\mathbb Z\). These modules are highest or lowest weight modules. A necessary and sufficient condition for an irreducible highest weight \(B(1,\mathbb Z)\)-module to be quasifinite is determined, and it is shown that the unitary ones are trivial.
scientific article; zbMATH DE number 2093435
scientific article