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On algebras obtained by tensor product. (English) Zbl 1228.18007

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.)

MSC:

18D50 Operads (MSC2010)
15A69 Multilinear algebra, tensor calculus
17A30 Nonassociative algebras satisfying other identities
17D25 Lie-admissible algebras

Citations:

Zbl 0855.18006
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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
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