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Measuring the noncommutativity of \(DG\)-algebras. (English) Zbl 1082.57029

The author introduces the notion of Hirsch algebra (a Hirsch algebra is a \(DG\)-algebra \((A,d,\cdot )\) equipped with a collection of higher multiplications \(E_{mn}:A^{\otimes m}\times A^{\otimes n}\rightarrow A\) such that \(E_{01}=E_{10}=id,\) \(E_{0k}=E_{k0}=0\) for \(k>1\) and which satisfy the generalized Hirsch formula) and he compares this structure with the \(B\left( \infty \right) \) -algebra structure, \(DG\)-Lie algebra structure, homotopy \(G\)-algebra structure and strong homotopy commutative structure. Finally the author presents some applications: multiplicative twisted tensor products, deformation of algebras, and degeneracy of \(A\left( \infty \right) \)-algebras.

MSC:

57T30 Bar and cobar constructions
16E45 Differential graded algebras and applications (associative algebraic aspects)
18D50 Operads (MSC2010)
57T05 Hopf algebras (aspects of homology and homotopy of topological groups)
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