The Betti numbers for Heisenberg Lie algebras
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The Betti numbers for Heisenberg Lie algebras María Alejandra Alvarez1 Received: 2 May 2018 / Accepted: 5 October 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this note we compute the Betti numbers for the Heisenberg Lie algebras. This is a well-known result due to Santharoubane. We re-derive this by considering Heisenberg Lie algebras as nilradicals of a certain subalgebra of a special linear Lie algebra and computing the dimensions of certain irreducible modules by representing them using Young diagrams. Keywords Heisenberg Lie algebras · Betti numbers · Young diagrams Mathematics Subject Classification 17B30 · 17B56
1 Introduction Computing explicitly Betti numbers of Lie algebras is a difficult task. This has been achieved for very few families of nilpotent Lie algebras. The computation of the Betti numbers for the Heisenberg Lie algebras is one of the nicest combinatorial results in the cohomology theory of Lie algebras and one of the first concerning this subject. This result, in characteristic zero, was obtained by Santharoubane in [13]. In characteristic two, Sköldberg obtained the Poincaré polynomial for Heisenberg Lie algebras in [14]. Later, Cairns and Jambor, in [5], computed the Betti numbers in arbitrary characteristic. Moreover, these authors re-obtain Santharoubane’s result by choosing a sufficiently large characteristic. Related to this problem is the computation of the adjoint Betti numbers for Heisenberg Lie algebras, which is obtained in [12] and then in [1] by using Young diagrams. In [3], Armstrong, Cairns and Jessup considered certain families of nilpotent Lie algebras and obtained their Betti numbers in characteristic zero. This result is re-derived in [2] by using Young diagrams. Moreover, the Betti numbers for these families in characteristic two are computed in [15]. Another result concerning
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María Alejandra Alvarez [email protected] Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta, Chile
123
Journal of Algebraic Combinatorics
Betti numbers is the one in [4], where Armstrong and Sigg considered filiform Lie algebras. In this work we show how to re-derive the Betti numbers for Heisenberg Lie algebras using a different method: by considering Heisenberg Lie algebras as nilradicals of a certain parabolic subalgebra of a special linear Lie algebra and by associating a Young diagram with every highest weight vector of the trivial homology. This method has already been used efficiently in [2]. Let g be a complex semisimple Lie algebra, h a Cartan subalgebra of g and = + ∪ − a system of roots of g. Let be the set of simple roots determining + . Consider a subset 0 ⊆ and define = + ∪ α ∈ | α ∈ span(c0 ) , where c0 is the complement of 0 in . Then, a parabolic subalgebra p of g can be parametrized by in the following manner: p=h⊕
gα .
α∈
If one defines g1 = h ⊕
gα and n =
α∈∩−
gα , then p = g1 ⊕ n. Moreover,
α∈ α ∈− /
g1 is a reductive subalgebr
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