an:05306408
Zbl 1160.17018
Szczesny, Matthew
On the structure and representations of the insertion-elimination Lie algebra
EN
Lett. Math. Phys. 84, No. 1, 65-74 (2008).
00219035
2008
j
17B65 17B10 17B66 17B81
Lie algebras; Hopf algebras; pre-Lie relation; lowest weight representations; Verma modules; Connes-Kreimer algebra; insertion-elimination Lie algebra
Summary: We examine the structure of the insertion-elimination Lie algebra on rooted trees introduced in \textit{A. Connes} and \textit{D. Kreimer} [Ann. Henri Poincar?? 3, No. 3, 411--433 (2002; Zbl 1033.81061)]. It possesses a triangular structure \({\mathfrak{g} = \mathfrak{n}_+ \oplus \mathbb{C}\cdot d \oplus \mathfrak{n}_-}\), like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a ``lowest weight'' \({\lambda \in \mathbb{C}}\). We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible.
Zbl 1033.81061