LIE POLYNOMIALS IN FREE SPECIAL TORTKEN ALGEBRAS

This paper studies Lie elements and symmetric (Tortken) elements in a free Novikov algebra and examines whether a nonzero multilinear element can belong to both classes simultaneously. We use the Euler operator and the null Lagrangian criterion to test membership in the symmetric subspace for elements represented in the standard multilinear Lie basis. For the Lie component, we employ left-normed commutators with a fixed first variable, which form a convenient basis of the multilinear part. The case is worked out explicitly by expanding the commutators in the Novikov product and applying the Euler operator. For degrees , the corresponding linear systems are obtained and solved computationally in Wolfram Mathematica and Albert. The computations show that the intersection of the multilinear Lie subspace with the subspace of symmetric elements is trivial for all . Thus, up to degree 7 there is no nonzero multilinear element in a free Novikov algebra that is simultaneously Lie and symmetric. These results provide a starting point for studying the problem in higher degrees.

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