Abstract
This paper addresses several structural aspects of the insertion–elimination algebra \({\mathfrak{g}}\), a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras of \({\mathfrak{g}}\), the automorphism group of \({\mathfrak{g}}\), the derivation Lie algebra of \({\mathfrak{g}}\), and a generating set. Several results are stated in terms of Lie algebras admitting a triangular decomposition and can be used to reproduce results for the generalized Virasoro algebras.
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References
Barnes D.: On the cohomology of soluble Lie algebras. Math. Zeitschr. 101(5), 343–349 (1967)
Benkart G.M., Isaacs I.M., Osborn J.M.: Lie algebras with self-centralizing ad-nilpotent elements. J. Alg. 57(2), 279–309 (1979)
Chapoton F.: Free pre-Lie algebras are free as Lie algebras. Bull. Can. Math. 53(3), 425–437 (2010)
Chibrikov E.S.: A right normed basis for free Lie algebras and Lyndon-Shirshov words. J. Alg. 302(2), 593–612 (2006)
Connes A., Kreimer D.: Insertion and elimination: the doubly infinite Lie algebra of Feynman graphs. Ann. Henri Poincare 3(3), 411–433 (2002)
Dokovic D., Zhao K.: Derivations, isomorphisms, and second cohomology of generalized Witt algebras. TAMS 350(2), 643–664 (1998)
Farnsteiner R.: Derivations and central extensions of finitely generated graded Lie algebras. J. Alg. 118(1), 33–45 (1988)
Foissy L.: Finite-dimensional comodules over the Hopf algebra of rooted trees. J. Alg. 255(1), 89–120 (2002)
Hoffman M.: Combinatorics of rooted trees and hope algebras. TAMS 355(9), 3795–3811 (2003)
Kreimer D., Mencattini I.: Insertion and elimination Lie algebra: the Ladder case. Lett. Math. Phys. 67(1), 61–74 (2004)
Kreimer D., Mencattini I.: The structure of the Ladder insertion–elimination Lie algebra. Commun. Math. Phys. 259(2), 413–432 (2005)
Moody, R., Pianzola, A.: Lie algebras with triangular decompositions. In: Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1995)
Patera J., Zassenhaus H.: The higher rank Virasoro algebras. Commun. Math. Phys. 136(1), 1–14 (1991)
Su Y., Zhao K.: Generalized Virasoro and super-Virasoro algebras and modules of the intermediate series. J. Alg. 252(1), 1–19 (2002)
Szczesny M.: On the structure and representations of the insertion–elimination Lie algebra. Lett. Math. Phys. 84(1), 65–74 (2008)
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Ondrus, M., Wiesner, E. Automorphisms and Derivations of the Insertion–Elimination Algebra and Related Graded Lie Algebras. Lett Math Phys 106, 925–949 (2016). https://doi.org/10.1007/s11005-016-0848-4
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DOI: https://doi.org/10.1007/s11005-016-0848-4