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The \(k\)-ary algebras and their operads. (Les algèbres \(k\)-aires et leurs opérades.) (French. Abridged English version) Zbl 0837.17003

The author studies algebras with \((k + 1)\)-ary operations giving two generalizations of the notion of associativity. In analogy to the article by V. Ginzburg and M. Kapranov [Duke Math. J. 76, 203-272 (1994)], he defines and investigates \((k + 1)\)-ary quadratic operads. Also, he constructs a Hochschild homology theory for partially associative \((k + 1)\)-ary algebras \((k\) is odd) and joins the triviality of the homology of a free partially associative \((k + 1)\)-ary algebra with some properties of the corresponding operad.

MSC:

17A42 Other \(n\)-ary compositions \((n \ge 3)\)
18G99 Homological algebra in category theory, derived categories and functors
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