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Associator dependent algebras and Koszul duality

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Abstract

We resolve a 10-year-old open question of Loday of describing Koszul operads that act on the algebra of octonions. In fact, we obtain the answer by solving a more general classification problem: we find all Koszul operads among those encoding associator dependent algebras.

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... The abovementioned result of Anan'in and Kemer was refined by Drenski and Vladimirova [36] who studied in great detail varieties of associative algebras defined by identities of degree and their lattices of subvarieties. These latter results were recently used by Bremner and the first author of this paper to classify Koszul quotients of the associative operad in [9]. Similarly, we were able to use the main result of the present paper to classify Koszul quotients of the Novikov operad. ...
... Recall that Dzhumadildaev [17] proved that the Novikov operad is not Koszul, so this result describes all ways in which a Novikov algebra can be regarded as an algebra over an Koszul operad. (This may be compared with a similar problem of Loday [28] asking to determine Koszul operads that act on the algebra of octonions, a question that motivated the paper [9].) Specifically, we prove the following theorem. ...
... It follows that our operad vanishes from the arity 4 onwards, and its Poincaré series is t + t 2 + 1 3 t 3 . It has the same Poincaré series as that of operads considered in [9,Prop. 3.13], and our argument will be very similar to the one of that statement. ...
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We prove that a variety of Novikov algebras has a distributive lattice of subvarieties if and only if the lattice of its subvarieties defined by identities of degree three is distributive, thus answering, in the case of Novikov algebras, a question of Bokut from about fifty years ago. As a byproduct, we classify all Koszul operads with one binary generator of which the Novikov operad is a quotient.
... This idea has been discussed in similar terms in [26] in the context of Lie-admissible algebras and has roots in foundational articles on left-symmetric algebras [60,62]. In [8] there are considered algebras satisfying identities like those treated in this section from the more sophisticated point of view of Koszul duality of operads. The unrelated duality given by the adjoint multiplication considered here is motivated by considerations from affine differential geometry [19]. ...
... 7.8 Although this does not seem to be realized widely, for anti-Hermitian matrices the inequality (7.4) is closely related to Vinberg's results on invariant norms on compact simple Lie algebras in[63] applied in the special case of su(n). Precisely, Vinberg shows that, for an invariant norm, || · || on a compact simple Lie algebra g the quantity θ(x) = sup 0 =y∈g by definition, θ([x, y]) ≤ θ(x)θ (y), taking g to be a compact simple Lie algebra of matrices, e.g. ...
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