On cohomology groups of current Lie algebras

Year 2026, Volume: 39 Issue: 39, 99 - 115, 10.01.2026

Rosendo Garcia Delgado

https://doi.org/10.24330/ieja.1747239

Abstract

In this work we state a result that relates the cohomology groups of a Lie algebra $\frak g$ and a current Lie algebra $\frak g \otimes \mathcal{S}$, by means of a short exact sequence similar to the universal coefficients theorem for modules, where $\mathcal{S}$ is a finite dimensional, commutative and associative algebra with unit over a field $\Bbb F$. Using this result we determine the cohomology group of $\frak g \otimes \mathcal{S}$ where $\frak g$ is a semisimple Lie algebra.

Keywords

Lie algebra cohomology , current Lie algebra , tensor product , associative and commutative algebra , semisimple Lie algebra

References

• F. A. Berezin and F. I. Karpelevic, Lie algebras with supplementary structure, Math. USSR Sbornik, 6(2) (1968), 185-203.
• H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, NJ, 1956.
• M. Chaktoura and F. Szechtman, A note on orthogonal Lie algebras in dimension 4 viewed as a current Lie algebras, J. Lie Theory, 23(4) (2013), 1101-1103.
• C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 63 (1948), 85-124.
• K. H. Neeb and F. Wagemann, The second cohomology of current algebras of general Lie algebras, Canad. J. Math., 60(4) (2008), 892-922.
• P. Zusmanovich, Low-dimensional cohomology of current Lie algebras and analogs of the Riemann tensor for loop manifolds, Linear Algebra Appl., 407 (2005), 71-104.
• P. Zusmanovich, Invariants of Lie algebras extended over commutative algebras without unit, J. Nonlinear Math. Phys., 17(1) (2010), 87-102.