On the Codimension Growth of Finite-Dimensional Lie Algebras

Abstract We study the exponential growth of the codimensions cn(L) of a finite-dimensional Lie algebra L over a field of characteristic zero. We show that if the solvable radical of L is nilpotent then lim n → ∞ c n ( L ) exists and is an integer.

Simple and semisimple Lie algebras and codimension growth

Standard polynomials are characterized by their degree and exponent

Abstract By the Giambruno–Zaicev theorem (Giambruno and Zaicev, 1999) [5] , the exponent exp ( A ) of a p.i. algebra A exists, and is always an integer. In Berele and Regev (2001) [2] it was shown that the exponent exp ( St n ) of the standard polynomial St n of degree n is not smaller than the exponent of any polynomial of degree n. Here it is proved that exp ( St n ) is strictly larger than the exponent of any other polynomial of degree n which is not a multiple of St n .

Involution codimensions and trace codimensions of matrices are asymptotically equal

We calculate the asymptotic growth oft n (M p (F),*) andc n (M p (F),*), the trace and ordinary *-codimensions ofp×p matrices with involution. To do this we first calculate the asymptotic growth oft n and then show thatc n ⋍t n .