Projects / Programmes
January 1, 2004
- December 31, 2008
| Code |
Science |
Field |
Subfield |
| 1.01.00 |
Natural sciences and mathematics |
Mathematics |
|
| Code |
Science |
Field |
| P120 |
Natural sciences and mathematics |
Number theory, field theory, algebraic geometry, algebra, group theory |
algebra, ring, Banach algebra, operator algebra, Lie algebra, Jordan algebra, operator, functional identities, linear preservers, derivation, automorphism.
Researchers (15)
Organisations (1)
Abstract
The major part of this research program is devoted to the study of mappings on rings and algebras. The methods that are used are primarily algebraic ones, though many of our results are functional analytic and operator theoretic. Namely, algebraic methods that we have developed in the past have proved to be useful in the analytic setting.
One of our main goals is to continue the study of functional identities. Roughly speaking, a functional identity can be described as an identical relation holding for all elements from a ring (algebra) that involves arbitrary (unknown) functions. The goal is to describe the form of these functions. Sometimes this is not possible, but in that case one can characterize the structure of the ring (algebra); often these exceptional cases occur in low dimensional algebras. It seems that the general theory of functional identities is close to its completion. One of our goals is to give a significant contribution to this project in which several mathematicians from different countries are involved. The results on functional identities have turned out to be applicable to several mathematical areas, for example to the theory of Lie algebras (Lie isomorphisms, Lie derivations, Lie-admissible algebras), the theory of Jordan algebras (Jordan mappings), the theory of (generalized) polynomial identities, linear preserver problems (commutativity preservers, normal preservers), automatic continuity theory, mathematical physics, etc. It seems reasonable to conjecture that there are still many possibilities for further applications, and this is one of our main goals. In this context some functional identities appear that are not covered by the general theory. The treatment of such special identities is also one of our intentions.
The theory of derivations and automorphisms (and related mappings) is a classical area in both algebra and functional analysis. Some important old problems are still open (for example, the noncommutative Singer-Wermer conjecture). We intend to continue working in this area. A special challenge for us is to find applications of algebraic methods in solving analytic problems.
Significance for science
Very large number of citations and other responses on the work of the members of the programm show that the programm is important for the development of science.
Significance for the country
The research work within the research programm P1-0288 is in the area of pure mathematics, therefore the results of the research do not have a direct impact on the development of Slovenia. However, the programm is very important for the development of the university curriculums (especially postgraduate ones). It also enables scientist from Slovenia to work on the most up to date research problems at the same time as researchers from all over the world.
Most important scientific results
Final report,
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Most important socioeconomically and culturally relevant results
Final report,
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