Homotopy Commutative Algebra and 2-Nilpotent Lie Algebra
The homotopy transfer theorem due to Tornike Kadeishvili induces the structure of a homotopy commutative algebra, or \(C_{\infty }\) -algebra, on the cohomology of the free 2-nilpotent Lie algebra. The latter \(C_{\infty }\) -algebra is shown to be genera
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Abstract The homotopy transfer theorem due to Tornike Kadeishvili induces the structure of a homotopy commutative algebra, or C∞ -algebra, on the cohomology of the free 2-nilpotent Lie algebra. The latter C∞ -algebra is shown to be generated in degree one by the binary and the ternary operations.
1 Introduction Every Universal Enveloping Algebra (UEA) Ug of a finite dimensional positively graded Lie algebra g belongs to the class of Artin-Schelter regular algebras(see e.g. [4]). As every finitely generated graded connected algebra, Ug has a free minimal resolution which is canonically built from the data of its Yoneda algebra E := Ext U g(K, K). By construction the Yoneda algebra E is isomorphic (as algebra) to the cohomology of the Lie algebra (with coefficients in the trivial representation provided by the ground field K) E = Ext •U g(K, K) ∼ = H • (g, K)
(1)
equipped with wedge product between cohomological classes in H • (g, K). The homotopy transfer theorem of Kadeishvili [7] implies that the Yoneda algebra E = Ext•U g(K, K) has the structure of homotopy associative algebra, or A∞ -algebra. M. Dubois-Violette (B) Laboratoire de Physique Théorique, UMR 8627, Université Paris XI, Bâtiment 210, 91405 Orsay Cedex, France e-mail: [email protected] T. Popov Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussée, 1784 Sofia, Bulgaria e-mail: [email protected] A. Makhlouf et al. (eds.), Algebra, Geometry and Mathematical Physics, Springer Proceedings in Mathematics & Statistics 85, DOI: 10.1007/978-3-642-55361-5_5, © Springer-Verlag Berlin Heidelberg 2014
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Since E ∼ = H • (g, K) is the cohomology of the exteriour algebra g∗ which is graded-commutative, it has the structure of homotopy commutative and associative algebra, or C∞ -algebra. Throughout the text g will be the free 2-nilpotent graded Lie algebra, with degree one generators in the finite dimensional vector space V over a field K of characteristic 0, 2 V. g=V⊕ The UEA U(V ⊕ ∧2 V ) arises naturally in physics in the universal Fock-like space of the parastatistics algebra introduced by Green [5] (see also [3]). Here we will concentrate on the case when V is an ordinary (even) vector space V , when the algebra Ug is the parafermionic algebra. The aim of this note is to describe the Yoneda algebra E of the UEA Ug, i.e., the cohomology H • (g, K) with its C∞ -structure induced by the isomorphism (1) through the homotopy transfer. The cohomology space H • (g, K) has a natural GL(V )-action. The decomposition of the GL(V )-module H • (g, K) into irreducible Schur modules Vλ is known since the work of Józefiak and Weyman [6]; it contains all GL(V )-modules with selfconjugated Young diagrams λ = λ once and exactly once. The decomposition of E = H • (g, K) into Schur modules provides a powerful tool to handle its C∞ -algebra structure.
2 Artin-Schelter Regularity Let g be the 2-nilpotent graded Lie algebra g = V ⊕ dimensional vector space V having Li
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