Loop cohomology. (English) Zbl 0728.20057

Johnson, K. W.; Leedham-Green, Charles R.

Czech. Math. J. 40(115), No. 2, 182-194 (1990).

From the authors’ introduction: “Associated with every algebraic category there is a cohomology theory. In the category of groups cohomology theory is well-established, and one of the interpretations of the higher cohomology groups of a group leads to the considerations of extensions in the category of loops. In general very few algebraic tools are available in the category of loops and so there is a possibility that the cohomology theory will lead to interesting algebraic insights into categories of varieties of loops.
We remark that relative to L the multiplicative structure of a loop is brought into play only insofar as it determines the identity element, but for the other varieties the multiplication becomes more important. In §3 the group Ext(Q,A) of extensions of a Q-module A by a loop Q is calculated. In §4 the cohomology groups \(L^ n(Q,A)\) are defined and it is proved that they have the properties mentioned above. A brief description of the ‘standard’ cohomology theory of an algebraic category is given in §5. It is shown that the cohomology groups defined in §4 are indeed the standard cohomology groups. In §6 it is shown that L is balanced and that the category of commutative loops is not.”

Reviewer: P.Kannappan (Waterloo/Ontario)

MSC:

20N05	Loops, quasigroups
20J05	Homological methods in group theory

Keywords:

extensions; category of loops; varieties of loops; cohomology groups

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