Space of Tortken elements in free Novikov algebra

Abstract

In this paper, we study the one-generated case of the space of Tortken elements in a free Novikov algebra. Using the differential-monomial realization, we prove that the dimensions of the homogeneous components are given by dimT₁ = dimT₂ =1 and dim T_n = p(n-2) for all n ≥ 3, where p(n-2) denotes the partition number. We also describe the first homogeneous components in the circle language and analyze the relations among their elements in low degrees. In particular, we show that the elements are linearly independent up to degree 5, while the first nontrivial relation appears in degree 6. We derive explicit identities in degrees 6 and 7 and prove a recursive inclusion for homogeneous components in the one-generated case for n ≤ 7. These results give a more precise description of the low-degree structure of the space of Tortken elements and suggest a regular pattern that may persist in higher degrees. We also formulate a conjecture extending the recursive inclusion to all degrees.