Tohoku Mathematical Journal
2022
June
SECOND SERIES VOL. 74, NO. 2

Tohoku Math. J.
74 (2022), 301-311

Title THE SECOND BETTI NUMBER OF DOUBLY WEIGHTED HOMOLOGY GROUPS OF A CERTAIN PRE LIE SUPERALGEBRA

Author Kentaro Mikami and Tadayoshi Mizutani

(Received October 31, 2019, revised August 17, 2020)
Abstract. The space of multi-vector fields on a manifold has a structure of ${\mathbb{Z}}$-graded Lie superalgebras under the Schouten bracket. (We call such an algebra a pre Lie superalgebra.) As in the case of Lie algebras, we have the associated homology groups similarly. We treat the case where the underlying manifold is a Euclidean $n$-space, and the coefficients are polynomials. Under such conditions, we show the vanishing of the second homology group of the chain complexes which are doubly indexed by the pairs of integers $(w,h)$, which we call weights. We specially treat the case $w=h$ in this article.

Mathematics Subject Classification. Primary 53Dxx; Secondary 17Bxx.
Key words and phrases. Betti number, homology groups, Lie superalgebras.

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