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Graded representations of graded Lie algebras and generalized representations of Jordan algebras

Kantor, Isaiah LU and Shpiz, G (2005) Satellite Conference on Noncommutative Geometry and Representation Theory in Mathematical Physics In Noncommutative Geometry and Representation Theory in Mathematical Physics 391. p.167-174
Abstract
We introduce and discuss a connection between representations of a certain class of graded Lie algebras and representations of Jordan algebras. This connection is stimulating in both directions. On the one hand it allows to produce an unified point of view on ordinary and Jacobson representations of Jordan algebras and formulate a notion of a generalized representation of a Jordan algebra, which includes ordinary and Jacobson representations as very special cases. The classification of irreducible generalized representations of simple Jordan algebras is given. On the other hand we prove that there are no infinite dimensional irreducible finitely graded representations of graded semisimple Lie algebras and classify the finite dimensional... (More)
We introduce and discuss a connection between representations of a certain class of graded Lie algebras and representations of Jordan algebras. This connection is stimulating in both directions. On the one hand it allows to produce an unified point of view on ordinary and Jacobson representations of Jordan algebras and formulate a notion of a generalized representation of a Jordan algebra, which includes ordinary and Jacobson representations as very special cases. The classification of irreducible generalized representations of simple Jordan algebras is given. On the other hand we prove that there are no infinite dimensional irreducible finitely graded representations of graded semisimple Lie algebras and classify the finite dimensional representations of this kind. The theorem about nonexistence of infinite dimensional irreducible finitely graded representations of graded semisimple Lie algebras has in fact as motivation the theorem about the absence of infinite dimensional irreducible representations of the semisimple finite dimensional Jordan algebra A (what is under considered connection can be formulated as the absence of 2-graded irreducible infinite dimensional representations of the 3-graded Lie algebra L(A) = U-1 circle plus U-0 circle plus U-1). (Less)
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author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
in
Noncommutative Geometry and Representation Theory in Mathematical Physics
volume
391
pages
167 - 174
publisher
American Mathematical Society (AMS)
conference name
Satellite Conference on Noncommutative Geometry and Representation Theory in Mathematical Physics
external identifiers
  • wos:000234854100017
ISSN
0271-4132
1098-3627
ISBN
0821837184
978-0-8218-3718-4
language
English
LU publication?
yes
id
9df32b7b-3dd1-44dd-8404-f1c9d7dd2d9e (old id 1410592)
date added to LUP
2009-05-28 15:52:03
date last changed
2016-04-15 19:49:20
@inproceedings{9df32b7b-3dd1-44dd-8404-f1c9d7dd2d9e,
  abstract     = {We introduce and discuss a connection between representations of a certain class of graded Lie algebras and representations of Jordan algebras. This connection is stimulating in both directions. On the one hand it allows to produce an unified point of view on ordinary and Jacobson representations of Jordan algebras and formulate a notion of a generalized representation of a Jordan algebra, which includes ordinary and Jacobson representations as very special cases. The classification of irreducible generalized representations of simple Jordan algebras is given. On the other hand we prove that there are no infinite dimensional irreducible finitely graded representations of graded semisimple Lie algebras and classify the finite dimensional representations of this kind. The theorem about nonexistence of infinite dimensional irreducible finitely graded representations of graded semisimple Lie algebras has in fact as motivation the theorem about the absence of infinite dimensional irreducible representations of the semisimple finite dimensional Jordan algebra A (what is under considered connection can be formulated as the absence of 2-graded irreducible infinite dimensional representations of the 3-graded Lie algebra L(A) = U-1 circle plus U-0 circle plus U-1).},
  author       = {Kantor, Isaiah and Shpiz, G},
  booktitle    = {Noncommutative Geometry and Representation Theory in Mathematical Physics},
  isbn         = {0821837184},
  issn         = {0271-4132},
  language     = {eng},
  pages        = {167--174},
  publisher    = {American Mathematical Society (AMS)},
  title        = {Graded representations of graded Lie algebras and generalized representations of Jordan algebras},
  volume       = {391},
  year         = {2005},
}