Remm, Elisabeth; Goze, Michel On algebras obtained by tensor product. (English) Zbl 1228.18007 J. Algebra 327, No. 1, 13-30 (2011). Given a Lie algebra \({\mathfrak g}\) and a commutative associative algebra \(A\) both over a field \(k\), the tensor product \(A\otimes_k{\mathfrak g}\) is a Lie algebra for the current bracket\[ [a\otimes X,b\otimes Y]=ab\otimes [X,Y]. \]This fact has a generalization to operads in the following sense. Given a quadratic operad \({\mathcal P}\), denote by \({\mathcal P}^!\) its quadratic dual in the sense of V. Ginzburg and M. Kapranov, [“Koszul duality for operads”, Duke Math. J. 76, No. 1, 203–272 (1994; Zbl 0855.18006)]. Ginzburg and Kapranov show in loc. cit. that for a \({\mathcal P}\)-algebra \(A\) and a \({\mathcal P}^!\)-algebra \(B\) over \(k\), the tensor product \(A\otimes_k B\) has a natural Lie algebra structure. The above special case arises for \({\mathcal P}={\mathcal L}ie\), because \({\mathcal P}^!={\mathcal C}om\).In the article under review, the authors extend this scheme to other operads. Namely, given a quadratic operad \({\mathcal P}\) with one (generating) binary operation, the main result is the construction of a quadratic operad \(\widetilde{\mathcal P}\) such that tensor products \(A\bigotimes_k B\) of a \({\mathcal P}\)-algebra \(A\) and a \(\widetilde{\mathcal P}\)-algebra \(B\) are naturally \({\mathcal P}\)-algebras and such that \(\widetilde{\mathcal P}\) is maximal with respect to this property. (The maximality condition serves to exclude trivial anwers to the problem.) Reviewer: Friedrich Wagemann (Nantes) Cited in 4 Documents MSC: 18D50 Operads (MSC2010) 15A69 Multilinear algebra, tensor calculus 17A30 Nonassociative algebras satisfying other identities 17D25 Lie-admissible algebras Keywords:quadratic operad; dual quadratic operad; algebra structure on the tensor product Citations:Zbl 0855.18006 PDFBibTeX XMLCite \textit{E. Remm} and \textit{M. Goze}, J. Algebra 327, No. 1, 13--30 (2011; Zbl 1228.18007) Full Text: DOI arXiv References: [1] Ginzburg, V.; Kapranov, M., Koszul duality for operads, Duke Math. J., 76, 1, 203-272 (1994) · Zbl 0855.18006 [2] Goze, M.; Remm, E., Lie-admissible algebras and operads, J. Algebra, 273, 1, 129-152 (2004) · Zbl 1045.17007 [3] Goze, M.; Remm, E., A class of nonassociative algebras, Algebra Colloq., 14, 2, 313-326 (2007) · Zbl 1230.17001 [4] Markl, M., Models for operads, Comm. Algebra, 24, 4, 1471-1500 (1996) · Zbl 0848.18003 [5] Markl, M.; Shnider, S.; Stasheff, J., Operads in Algebra, Topology and Physics, Math. Surveys Monogr., vol. 96 (2002), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI, x+349 pp · Zbl 1017.18001 [6] Markl, M.; Remm, E., Algebras with one operation including Poisson and other Lie-admissible algebras, J. Algebra, 299, 1, 171-189 (2006) · Zbl 1101.18004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.