Lie superautomorphisms of prime associative superalgebras are considered. A definitive result is obtained for central simple superalgebras: their Lie superautomorphisms are of standard forms, except when the dimension of the superalgebra in question is 2 or 4.
COBISS.SI-ID: 16299353
This book provides an easy-to-read introduction to noncommutative rings and algebras. the emphasis is on clarity of exposition and simplicity of the proofs, with several being different from those in other texts on the subject.
COBISS.SI-ID: 17143897
Let $A$ be a centrally closed prime algebra over a characteristic 0 field $k$, and let $q: A \to A$ be the trace of a $d$-linear map (i.e., $q(x)=M(x, \dots ,x)$ where $M: A^d \to A$ is a $d$-linear map). If $[q(x),x] = 0$ for every $x \in A$, then $q$ is of the form $q(x) = \sum_{i=0}^{d} \mu_i(x)x^i$ where each $\mu_i$ is the trace of a $(d-i)$-linear map from $A$ into $k$. For infinite dimensional algebras and algebras of dimension $) d^2$ this was proved by Lee, Lin, Wang, and Wong in 1997. In this paper we cover the remaining case where the dimension is $ \le d^2$. Using this result we are able to handle general functional identities of one variable on $A$; more specifically, we describe the traces of $d$-linear maps $q_i: A \to A$ that satisfy $\sum_{i=0}^m x^i q_i(x)x^{m-i} \in k$ for every $x \in A$.
COBISS.SI-ID: 16842329
We describe the general form of bijective comparability preserving transformations of the Hilbert space effect algebra, thus improving several known characterizations of ortho-order automorphisms.
COBISS.SI-ID: 16568409
Hua's fundamental theorem of geometry of matrices describes the general form of bijective maps on the space of all $m\times n$ matrices over a division ring $\mathbb{D}$ which preserve adjacency in both directions. Motivated by several applications we study a long standing open problem of possible improvements. There are three natural questions. Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only and still get the same conclusion? Can we relax the bijectivity assumption? Can we obtain an analogous result for maps acting between the spaces of rectangular matrices of different sizes? A division ring is said to be EAS if it is not isomorphic to any proper subring. For matrices over EAS division rings we solve all three problems simultaneously, thus obtaining the optimal version of Hua's theorem. In the case of general division rings we get such an optimal result only for square matrices and give examples showing that it cannot be extended to the non-square case.
COBISS.SI-ID: 16947545