Differential Identities and Polynomial Growth of the Codimensions

Differential identities and varieties of almost polynomial growth

“…Then we say that V has polynomial growth if there exist C, t such that c L n (V) ≤ Cn t and that V has almost polynomial growth if c L n (V) is not polynomially bounded but every proper subvariety has polynomial growth. In [31] the authors proved that there exists only two L-varieties generated by finite dimensional L-algebras of almost polynomial growth. Next we are going to describe such varieties.…”

“…Notice that although J is an L-invariant ideal of A (see [15]), it may does not exist an L-invariant Wedderburn-Malcev decomposition, i.e., it may happen that all semisimple subalgebras B of A that satisfy (2) are not L-subalgebras of A. For example, the algebra U T δ 2 of 2 × 2 upper triangular matrices where L acts as the 1-dimensional Lie algebra spanned by the inner derivation δ induced by e 12 has no L-invariant Wedderburn-Malcev decomposition (see [31,Example 2]). Things are different inside var L (U T ε 2 ), in fact at the end of the section, we will prove that, up to T L -equivalence, we can always assume that a subvariety of var L (U T ε 2 ) is generated by an L-algebra with an L-invariant Wedderburn-Malcev decomposition.…”

“…2 there is nothing to prove, so let suppose that A generates a proper L-subvariety of var L (U T ε 2 ). Thus, by Theorem 3, c L n (A) is polynomially bounded and by [31,Theorem 8] we may assume…”

“…In that paper, they show that such algebra generates an L-variety of almost polynomial growth. Finally, in [31] it was proved that the L-codimension sequence of an L-variety V generated by a finite dimensional L-algebra is polynomially bounded if and only if U T 2 , U T ε 2 / ∈ V, where U T 2 stands for the algebra of 2 × 2 upper triangular matrices and L acts trivially on it.…”

“…We shall prove that A ∼ T L N ε k and this will complete the proof. Since c L n (A) is polynomially bounded, by [31,Theorem 8] we may assume that…”