Differential Novikov algebras

In this talk, we consider Novikov algebras with derivation and algebras obtained from its dual operad. It turns out that the obtained dual operad has a connection with bicommutative algebras. The motivation for this work comes from the white and black Manin product of the Novikov operad with itself.

Main Results. In recent years, Novikov algebras have yielded surprising results in combinatorial algebra and combinatorics. For example, the Specht property for Novikov algebras was established in [V. Dotsenko, N. Ismailov, U. Umirbaev, Polynomial identities in Novikov algebras, Mathematische Zeitschrift, 303(3), 60, (2023).]. One of the main combinatorial results on the free Novikov algebras is the monomial basis of Nov⟨X⟩, see [A. S. Dzhumadil’daev, C. Löfwall, Trees, free right-symmetric algebras, free Novikov algebras and identities, Homology, Homotopy Appl., 4(2), 165–190 (2002).]. Recently, it was shown that every Novikov algebra can be embedded into appropriate commutative algebra with a derivation [L. A. Bokut, Y. Chen, Z. Zhang, Gröbner–Shirshov bases method for Gelfand–Dorfman–Novikov algebras, Journal of Algebra its Applications, 16(1), 1750001, 22 pp. (2017)]. The analog of this result for noncommutative Novikov algebras is given in [B. Sartayev, P. Kolesnikov, Noncommutative Novikov algebras, European Journal of Mathematics, 9(2), 35, (2023)], i.e., every noncommutative Novikov algebra can be embedded into an appropriate associative algebra with derivation.The defining identities of the variety of Novikov algebras first appeared in [I. M. Gelfand, I. Ya. Dorfman, Hamilton operators and associated algebraic structures, Functional analysis and its application, 13, no. 4, 13–30 (1979)]. This work also contains the first mention of a differential associative-commutative algebra under the operation

which satisfies the following identities:

The motivation for this work comes from the white and black Manin product of Novikov operad with itself. Since Nov = Nov^!, we have

where Nov is an operad derived from the variety of Novikov algebras. Since there is a one-to-one correspondence between a variety of algebras Var and the quadratic operad derived from it, we will use the same terminology for both in the future. On one hand, the white Manin product of the Novikov operad with itself gives an operad that can be embedded into the operad Nov with a derivation. On the other hand, the black Manin product of the Novikov operad with itself defines a class of algebras with a rich algebraic structure, and both operads are dual to each other.

Let Der Nov be the operad obtained from Nov ◦ Nov, and let Der Nov⟨X⟩ be the free algebra of the variety Der Nov generated by a set X. One of the recent combinatorial results on Der Nov⟨X⟩ is the following chain of inclusions:

Here X^(ω,ω) = (X^(ω))^(ω) = {x^(n,m) | x ∈ X, n, m ∈ Z₊}, a variable x^(n,m) represents dⁿ∂^m(x). The elements of Der Nov⟨X⟩ are exactly those polynomials in Com⟨X^(ω,ω)⟩ that can be presented as linear combinations of monomials