Loop cohomology. (English) Zbl 0728.20057

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.”


20N05 Loops, quasigroups
20J05 Homological methods in group theory
Full Text: EuDML


[1] A. A. Albert: Quasigroups II. Trans. Amer. Math. Soc., 55, (1944), 401 - 419. · Zbl 0063.00042
[2] M. André: Méthode simpliciale en algèbre homologique et algèbre commutative,/. Lecture Notes in Math. No. 32, Springer-Verlag, Berlin 1967. · Zbl 0154.01402
[3] M. Barr, J. Beck: Homology and standard constructions. Seminar on triples and categorical homology theory. Lecture Notes in Math. 80, 245-335, Springer, Berlin 1969. · Zbl 0176.29003
[4] R. H. Bruck: An extension theory for a certain class of loops. Bull. Amer. Math. Soc., 57, (1951), 11-26. · Zbl 0043.02002
[5] J. Duskin: Simplicial methods and the interpretation of ”triple” cohomology. Mem. Amer. Soc. No. 163) · Zbl 0376.18011
[6] Eilenberg, Maclane: Algebraic cohomology groups and loops. Duke Math. ]14, (1947), 435-463. · Zbl 0029.34102
[7] F. W. Lawvere: Functional semantics of algebraic systems. Proc. Nat. Acad. Sci., 50, (1963), 869-872. · Zbl 0119.25901
[8] C. R. Leedham-Green: Homology in varieties of groups I-III. Trans. Amer. Math. Soc.,762 (1971), 1-33. · Zbl 0232.18015
[9] C. R. Leedham-Green, T. C. Hurley: Homology in varieties of groups IV. Trans. Amen Math. Soc., 170 (1972), 293-303. · Zbl 0279.18012
[10] C. R. Leedham-Green, S. Mckay: Baer-invariants, isologism, varietal laws and homology. Acta Mathematica, 137 (1976), 99-150. · Zbl 0364.20036
[11] H. Niederreiter, K. Robinson: Bol loops of order \(pq\). Math. Proc. Camb. Phil. Soc. 89, (1981), 241-256. · Zbl 0463.20050
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