How Euler would compute the Euler-Poincare characteristic of a Lie superalgebra

The Euler-Poincar'e characteristic of a finite-dimensional Lie algebra vanishes. If we want to extend this result to Lie superalgebras, we should deal with infinite sums. We observe that a suitable method of summation, which goes back to Euler, allows to do that, to a certain degree. The mathematics behind it is simple, we just glue the pieces of elementary homological algebra, first-year calculus and pedestrian combinatorics together, and present them in a (hopefully) coherent manner.

💡 Research Summary

The paper revisits the classical fact that the Euler‑Poincaré characteristic χ(L)=∑_{i}(-1)^{i}dim H^{i}(L) of a finite‑dimensional Lie algebra L always vanishes, and asks how one might extend this statement to Lie superalgebras, where the underlying cohomology groups are naturally graded by both degree and parity. Because the super‑graded cohomology typically yields an infinite alternating series, the ordinary sum is divergent and the characteristic is undefined in the usual sense. The author therefore turns to a historic summation technique invented by Leonhard Euler in the 18th century, now commonly called Euler summation. This method regularises an alternating series by averaging its successive partial sums and then taking a limit; it coincides with the Abel‑Euler or Cesàro means for alternating series and is known to assign finite values to many divergent expressions.

After a brief review of the homological algebra of ordinary Lie algebras—showing how exactness of the Chevalley‑Eilenberg complex forces the alternating sum of dimensions to cancel—the paper introduces the Chevalley‑Eilenberg complex for a Lie superalgebra 𝔤=𝔤_{\bar0}⊕𝔤_{\bar1}. The cochain spaces C^{i}(𝔤) inherit a parity, and the resulting cohomology groups H^{i}(𝔤) can be infinite‑dimensional even when 𝔤 itself is finite‑dimensional. Consequently the naïve alternating sum ∑_{i}(-1)^{i}dim H^{i}(𝔤) diverges.

The core of the work is the definition of an “Euler‑Poincaré characteristic for superalgebras”, \