Commuting traces and commutativity preserving maps on triangular algebras

Let A be a triangular algebra. The problem of describing the form of a bilinear map satisfying B(x,x)x=xB(x,x) for all xA is considered. As an application, commutativity preserving maps and Lie isomorphisms of certain triangular algebras (e.g., upper triangular matrix algebras and nest algebras) are determined.     

Jordan maps on triangular algebras

Let T be a triangular algebra and R′ be an arbitrary ring. Suppose that M:T→R^′ and M^∗:R^′→T are surjective maps such that

     

k-Commuting maps on triangular algebras

In this paper, k-commuting maps on certain triangular algebras are determined. As an application we show that every k-commuting map on an upper triangular matrix algebra over a unital commutative ring of 2-torsion free or a nest algebra is proper.     

Fixed-point-free automorphisms of Lie algebras

In this paper we have proved that the Lie algebraL over a fieldF of characteristic 0 which admits a fixed-point-free automorphism of finite order is solvable, andL is nilpotent ifL admits a fixed-point-free automorphism of prime order. The problem seems to be open if the fieldF has prime characteristic.     

Markov traces on cyclotomic Temperley-Lieb algebras

In this note, we use generalized Tchebychev polynomials to define a trace function which satisfies certain conditions. Such a trace will be called the Markov trace. In particular, we obtain formulae for the weights of the Markov trace. As a corollary, we get a combinatorial identity. This generalizes Jones's 1983 result on Temperley-Lieb algebras.

     

Local automorphisms of semisimple algebras and group algebras

Let F be a field of characteristic not 2, and let A be a finite-dimensional semisimple F -algebra. All local automorphisms of A are characterized when all the degrees of A are larger than 1. If F is further assumed to be an algebraically closed field of characteristic zero, K a finite group, F K the group algebra of K over F , then all local automorphisms of F K are also characterized.     

On triangular algebras with noncommutative diagonals

We construct a triangular algebra whose diagonals form a noncommutative algebra and its lattice of invariant projections contains only two nontrivial projections. Moreover we prove that our triangular algebra is maximal.     

Spectral automorphisms in CB-effect algebras

Spectral automorphisms have been introduced in [IVANOV, A.—CARAGHEORGHEOPOL, D.: Spectral automorphisms in quantum logics, Internat. J. Theoret. Phys. 49 (2010), 3146–3152]_in an attempt to construct, in the abstract framework of orthomodular lattices, an analogue of the spectral theory in Hilbert spaces. We generalize spectral automorphisms to the framework of effect algebras with compression bases and study their properties. Characterizations of spectral automorphisms as well as necessary conditions for an automorphism to be spectral are given. An example of a spectral automorphism on the standard effect algebra of a finite-dimensional Hilbert space is discussed and the consequences of spectrality of an automorphism for the unitary Hilbert space operator that generates it are shown. The last section is devoted to spectral families of automorphisms and their properties, culminating with the formulation and proof of a Stone type theorem (in the sense of Stone’s theorem on strongly continuous one-parameter unitary groups — see, e.g. [REED, M.#x2014;SIMON, B.: Methods of Modern Mathematical Physics, Vol. I, Acad. Press, New York, 1975]) for a group of spectral automorphisms.     

Trialitarian automorphisms of lie algebras

Over an algebraically closed field of characteristic zero simple Lie algebras admit outer automorphisms of order 3 if and only if they are of type D₄. Moreover, thereare two conjugacy classes of such automorphisms. Among orthogonal Lie algebras over arbitrary fields of characteristic zero, only orthogonal Lie algebras relative to quadratic norm forms of Cayley algebras admit outer automorphisms of order 3. We give a complete list of conjugacy classes of outer automorphisms of order 3 for orthogonal Lie algebras over arbitrary fields of characteristic zero. For the norm form of a given Cayley algebra, one class is associated with the Cayley algebra and the others with central simple algebras of degree 3 with involution of the second kind such that the cohomological invariant of the involution is the norm form.