zbMATH — the first resource for mathematics

On the structure and representations of the insertion-elimination Lie algebra. (English) Zbl 1160.17018
Summary: We examine the structure of the insertion-elimination Lie algebra on rooted trees introduced in A. Connes and D. Kreimer [Ann. Henri PoincarĂ© 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.

17B65 Infinite-dimensional Lie (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B66 Lie algebras of vector fields and related (super) algebras
17B81 Applications of Lie (super)algebras to physics, etc.
Full Text: DOI arXiv
[1] Connes A. and Kreimer D. (2002). Insertion and elimination: the doubly infinite Lie algebra of Feynman graphs. Ann. Henri Poincar 3(3): 411–433 · Zbl 1033.81061 · doi:10.1007/s00023-002-8622-9
[2] Foissy L. (2002). Finite-dimensional comodules over the Hopf algebra of rooted trees. J. Algebra 255: 89–120 · Zbl 1017.16031 · doi:10.1016/S0021-8693(02)00110-2
[3] Kac V.G. (1988). Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras. World Scientific Publishing, Singapore
[4] Kac V.G. (1994). Infinite Dimensional Lie Algebras. Cambridge University Press, Cambridge · Zbl 0929.17023
[5] Kreimer D. (1998). On the Hopf algebra structure of perturbative quantum field theory. Adv. Theor. Math. Phys. 2: 303–334 · Zbl 1041.81087
[6] Kreimer D. and Mencattini I. (2004). Insertion and elimination Lie algebra: the Ladder case. Lett. Math. Phys. 67(1): 61–74 · Zbl 1081.17013 · doi:10.1023/B:MATH.0000027749.09118.a5
[7] Kreimer D. and Mencattini I. (2005). The structure of the Ladder insertion–elimination Lie algebra. Commun. Math. Phys. 259: 413–432 · Zbl 1129.81060 · doi:10.1007/s00220-005-1340-7
[8] Mencattini, I.: Thesis, Boston University (2005)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.