New Automorphisms of Generic Matrix Algebras and Polynomial Algebras

Automorphisms of Polynomial Algebras and Dirichlet Series

Let Fq[x, y] be the polynomial algebra in two variables over the finite field Fq with q elements. We give an exact formula and the asymptotics for the number pn of automorphisms (f, g) of Fq[x, y] such that max{deg(f), deg(g)} = n. We describe also the Dirichlet series generating function p(s) = ∑

The inversion formula for automorphisms of the Weyl algebras and polynomial algebras

Let An be the n’th Weyl algebra and Pm be a polynomial algebra in m variables over a field K of characteristic zero. The following characterization of the algebras {An⊗Pm} is proved: an algebra A admits a finite set δ1, . . . , δs of commuting locally nilpotent derivations with generic kernels and ∩i=1ker(δi) = K iff A ≃ An ⊗ Pm for some n and m with 2n + m = s, and vice versa. The inversion fo...

NILPOTENT GRAPHS OF MATRIX ALGEBRAS

Let $R$ be a ring with unity. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set ~$Z_N(R)^* = {0neq x in R | xy in N(R) for some y in R^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in N(R)$, or equivalently, $yx in N(R)$, where $N(R)$ denoted the nilpotent elements of $R$. Recently, it has been proved that if $R$ is a left A...

Automorphisms of Operator Algebras

Dimension of Automorphisms with Fixed Degree for Polynomial Algebras

Let K[x, y] be the polynomial algebra in two variables over an algebraically closed field K. We generalize to the case of any characteristic the result of Furter that over a field of characteristic zero the set of automorphisms (f, g) of K[x, y] such that max{deg(f), deg(g)} = n ≥ 2 is constructible with dimension n + 6. The same result holds for the automorphisms of the free associative algebr...