Kadeishvili, T. Measuring the noncommutativity of \(DG\)-algebras. (English) Zbl 1082.57029 J. Math. Sci., New York 119, No. 4, 494-512 (2004). 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. Reviewer: Maria Joita (Bucureşti) Cited in 10 Documents 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) Keywords:Hirsch algebra; Hopf algebra PDFBibTeX XMLCite \textit{T. Kadeishvili}, J. Math. Sci., New York 119, No. 4, 494--512 (2004; Zbl 1082.57029) Full Text: DOI