Loop algebras of Zassenhaus algebras in characteristic three
The author continues the investigation of the central extensions of some loop algebras of Zassenhaus algebras over fields of characteristic \(p\). In this paper the case \(p=3\) is considered. When \(p=3\) the Zassenhaus algebra \(W=W(1;n)\) has an invariant symmetrical form, which allows to construct the infinite series of independent 2-cocycles (loop cocycles) on a special loop algebra \(L\) of \(W\) using cyclic 2-cohomology of the polynomial algebra \(\mathbb F_p[t]\). The description of 2-cocycles of Zassenhaus algebra obtained earlier by \textit{A. S. Dzhumadil'daev} [Mat. Sb., Nov. Ser. 126(168), 473-489 (1985); English transl. in Math. USSR, Sb. 54, 457-474 (1986; Zbl 0574.17009)] and \textit{R. Farnsteiner} [Can. J. Math. 39, 1078-1106 (1987; Zbl 0631.17005)] is given by means of which the infinite series of independent cocycles on loop algebra \(L\) (finite cohomology cocyles) can be constructed naturally. It is proved that the infinite dimensional central extension constructed by means of finite cohomology cocycles and loop cocycles is finitely presented. Generators and defining relations are indicated.
scientific article; zbMATH DE number 1310632
scientific article