Representations of Truncated Current Lie Algebras

Let g denote a Lie algebra, and let ĝ denote the tensor product of g with a ring of truncated polynomials. The Lie algebra ĝ is called a truncated current Lie algebra. The highest-weight theory of ĝ is investigated, and a reducibility criterion for the Verma modules is described. Let g be a Lie algebra over a field k of characteristic zero, and fix a positive integer N . The Lie algebra (1) ĝ = g ⊗k k[t]/t N+1 k[t], over k, with the Lie bracket given by [x⊗ t, y ⊗ t ] = [x, y ]⊗ t for all x, y ∈ g and i, j > 0, is called a truncated current Lie algebra, or sometimes a generalised Takiff algebra or a polynomial Lie algebra. We describe a highest-weight theory for ĝ, and the reducibility criterion for the universal objects of this theory, the Verma modules. Representations of truncated current Lie algebras have been studied in [2, 3, 5, 6], and have applications in the theory of soliton equations [1] and in the representation theory of affine Kac–Moody Lie algebras [8]. A highest-weight theory is defined by a choice of triangular decomposition. Choose an abelian subalgebra h ⊂ g that acts diagonally upon g via the adjoint action, and write g = h ⊕ ( ⊕ α∈∆ g ) for the eigenspace decomposition, where ∆ ⊂ h, and for all α ∈ ∆, [h, x ] = 〈α, h〉x, for all h ∈ h and x ∈ g. A triangular decomposition of g is, in essence, a division of the eigenvalue set ∆ into two opposing halves (2) ∆ = ∆+ ⊔∆−, −∆+ = ∆−, There are additional hypotheses (see [4]) — in particular, there must exist some finite subset of ∆+ that generates ∆+ under addition. This excludes, for example, the imaginary highest-weight theory of an affine Lie algebra (see [7], Appendix B).