On the dimension of products of homogeneous subspaces in free lie algebras

Nil Mansuroglu, Nil Mansurolu, Ralph Stöhr

    Research output: Contribution to journalArticlepeer-review


    Let L be a free Lie algebra of finite rank over a field K and let L n denote the degree n homogeneous component of L. Formulae for the dimension of the subspaces [Lm, Ln] for all m and n were obtained by the second author and Michael Vaughan-Lee. In this note we consider subspaces of the form [Lm, Ln, Lk] = [[L m, Ln], Lk]. Surprisingly, in contrast to the case of a product of two homogeneous components, the dimension of such products may depend on the characteristic of the field K. For example, the dimension of [L2, L2, L1] over fields of characteristic 2 is different from the dimension over fields of characteristic other than 2. Our main results are formulae for the dimension of [Lm, Ln, Lk]. Under certain conditions on m, n and k they lead to explicit formulae that do not depend on the characteristic of K, and express the dimension of [Lm, Ln, Lk] in terms of Witt's dimension function. © 2013 World Scientific Publishing Company.
    Original languageEnglish
    Pages (from-to)205-213
    Number of pages8
    JournalInternational Journal of Algebra and Computation
    Issue number1
    Publication statusPublished - Feb 2013


    • Free Lie algebras
    • homogeneous subspaces


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