ABSTRACT
Throughout F will be a field of characteristic zero and A a PI-algebra over F. Here we present the results recently obtained on the asymptotics of the codimensions of A in case A is a finite dimensional (associative or Lie) algebra. Let F⟨X⟩ be the free associative algebra freely generated by the set X = {x 1, X 2,...} over F and let Id(A) be the T-ideal of identities of the associative algebra A. The algebra https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332630/bdf2b6ad-8ca8-4d57-9bb0-ce1a8aabf4d0/content/inline13.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is the free algebra of countable rank in the variety generated by A. Since in characteristic zero every identity is equivalent to a system of multilinear identities, an important role is played by the multilinear part of https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332630/bdf2b6ad-8ca8-4d57-9bb0-ce1a8aabf4d0/content/inline14.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>: for n ≥ 1, if Vn is the space of multilinear polynomials in x 1 ...,xn , then one defines https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332630/bdf2b6ad-8ca8-4d57-9bb0-ce1a8aabf4d0/content/inline15.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and cn (A) is called the n-th codimension of A. Similarly, starting with the free Lie algebra L⟨X⟩ ⊆ F⟨X⟩ one defines the n-th Lie codimension cL n(B) of the Lie algebra B.
