The Euler operator on the space of Lie elements in free Novikov algebra
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Date
2026
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SDU University
Abstract
This thesis is devoted to the study of special subspaces in the free Novikov algebra, namely the spaces of multilinear Lie and symmetric elements. The main aim of the work is to investigate the intersection of these subspaces and to study conditions for recognizing Lie elements in the free Novikov algebra. The first main result concerns the spaces Ln and Tn. Here Ln denotes the space of multilinear Lie elements constructed by means of the commutator [a, b] = a · b − b · a, while Tn denotes the space of multilinear symmetric, or Tortken, elements constructed by means of the symmetrized product a ◦ b = a · b + b · a. Using the differential realization of the free Novikov algebra and the null Lagrangian criterion in terms of the Euler operator, we prove that for degrees n ≤ 7 this intersection is trivial: Ln ∩ Tn = {0}. The case n = 3 is treated explicitly, while the cases n = 4, 5, 6, 7 are verified by computer computations. The second part of the thesis is devoted to the problem of constructing a criterion for Lie elements. We consider an operator associated with replacing the Novikov product by the Lie commutator and then expressing the result in a basis of the corresponding multilinear component. In small degrees, polynomial conditions satisfied by the elements of Ln are obtained. These computations provide the first steps toward a possible criterion for Lie elements in the free Novikov algebra.
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the Euler operator applied to the element f, algerbra, Novikov, Lie elements
Citation
Yersaliyeva A / The Euler operator on the space of Lie elements in free Novikov algebra / SDU University / Department of Mathematics and Natural Sciences