787 research outputs found
Functional identities of one variable
Let be a centrally closed prime algebra over a characteristic 0 field
, and let be the trace of a -linear map (i.e.,
where is a -linear map). If for
every , then is of the form
where each is the trace of a -linear map from into . For
infinite dimensional algebras and algebras of dimension this was proved
by Lee, Lin, Wang, and Wong in 1997. In this paper we cover the remaining case
where the dimension is . Using this result we are able to handle
general functional identities of one variable on ; more specifically, we
describe the traces of -linear maps that satisfy for every .Comment: 10 page
On Lie and associative algebras containing inner derivations
We describe subalgebras of the Lie algebra \mf{gl}(n^2) that contain all
inner derivations of (where and is an algebraically
closed field of characteristic 0). In a more general context where is a
prime algebra satisfying certain technical restrictions, we establish a density
theorem for the associative algebra generated by all inner derivations of .Comment: 11 pages, accepted for publication in Linear Algebra App
Functional identities on matrix algebras
Complete solutions of functional identities on the matrix algebra are
given. The nonstandard parts of these solutions turn out to follow from the
Cayley-Hamilton identity.Comment: 20 pages, comments welcome, version 2- applications have been adde
Lie Superautomorphisms on Associative Algebras, II
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.Comment: 19 pages, accepted for publication in Algebr. Represent. Theor
Quasi-identities on matrices and the Cayley-Hamilton polynomial
We consider certain functional identities on the matrix algebra that
are defined similarly as the trace identities, except that the "coefficients"
are arbitrary polynomials, not necessarily those expressible by the traces. The
main issue is the question of whether such an identity is a consequence of the
Cayley-Hamilton identity. We show that the answer is affirmative in several
special cases, and, moreover, for every such an identity and every central
polynomial with zero constant term there exists such that
the affirmative answer holds for . In general, however, the answer is
negative. We prove that there exist antisymmetric identities that do not follow
from the Cayley-Hamilton identity, and give a complete description of a certain
family of such identities.Comment: Version 2: 24 pages. This paper is a replacement of the paper
"Quasi-identities and the Cayley-Hamilton quasi-polynomial" by the first and
the third author. The changes are substantial. Version 3: 27 pages. The title
has been changed slightly, the exposition improved, references added, and
some typos remove
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