Nilpotent last-regular elements

Abstract: We say that an element $x$ in a ring $R$ is nilpotent last-regular if it is nilpotent of certain index $n+1$ and its last nonzero power $x^n$ is regular von Neumann, i.e., there exists another element $y\in R$ such that $x^nyx^n=x^n$. This type of elements naturally arise when studying certain inner derivations in the Lie algebra $\Skew(R,*)$ of a ring $R$ with involution $*$ whose indices of nilpotence differ when considering them acting as derivations on $\Skew(R,*)$ and on the whole $R$. When moving to the symmetric Martindale ring of quotients $Q^s_m(R)$ of $R$ we still obtain inner derivations with the same indices of nilpotence on $Q^s_m(R)$ and on the skew-symmetric elements $\Skew(Q^s_m(R),*)$ of $Q^s_m(R)$, but with the extra condition of being generated by a nilpotent last-regular element. This condition strongly determines the structure of $Q^s_m(R)$ and of $\Skew(Q^s_m(R),*)$. We will review the Jordan canonical form of nilpotent last-regular elements and show how to get gradings in associative algebras (with and without involution) when they have such elements.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar