Bulletin of the Belgian Mathematical Society - Simon Stevin

Some restrictions on the Betti numbers of a nilpotent Lie algebra

Peyman Niroomand and Francesco G. Russo

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We have recently shown that a nilpotent Lie algebra $L$ of dimension $n \ge1 $ satisfies the inequality $\mathrm{dim} \ H_2(L,\mathbb{Z}) \leq \frac{1}{2}(n+m-2)(n-m-1)+1$, where $\mathrm{dim} \ L^2=m \ge 1$ and $H_2(L,\mathbb{Z})$ is the 2-nd integral homology Lie algebra of $L$. Our first main result correlates this bound with the $i$-th Betti number $\mathrm{dim} \ H^i(L,\mathbb{C}^\times)$ of $L$, where $H^i(L,\mathbb{C}^\times)$ denotes the $i$-th complex cohomology Lie algebra of $L$. Our second main result describes a more general restriction, which follows an idea of Ellis in [G. Ellis, The Schur multiplier of a pair of groups, Appl. Categ. Structures 6 (1998), 355--371].

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 21, Number 3 (2014), 403-413.

First available in Project Euclid: 11 August 2014

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Zentralblatt MATH identifier

Primary: 17B30: Solvable, nilpotent (super)algebras 17B60: Lie (super)algebras associated with other structures (associative, Jordan, etc.) [See also 16W10, 17C40, 17C50] 17B99: None of the above, but in this section

Heisenberg algebras cohomology nilpotent Lie algebras Betti numbers Schur multiplier


Niroomand, Peyman; Russo, Francesco G. Some restrictions on the Betti numbers of a nilpotent Lie algebra. Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 3, 403--413. doi:10.36045/bbms/1407765880. https://projecteuclid.org/euclid.bbms/1407765880

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