Graded representations of graded Lie algebras and generalized representations of Jordan algebras
(2005) Satellite Conference on Noncommutative Geometry and Representation Theory in Mathematical Physics In Noncommutative Geometry and Representation Theory in Mathematical Physics 391. p.167174 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 2graded irreducible infinite dimensional representations of the 3graded Lie algebra L(A) = U1 circle plus U0 circle plus U1). (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1410592
 author
 Kantor, Isaiah ^{LU} and Shpiz, G
 organization
 publishing date
 2005
 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
 02714132
 10983627
 ISBN
 0821837184
 9780821837184
 language
 English
 LU publication?
 yes
 id
 9df32b7b3dd144dd8404f1c9d7dd2d9e (old id 1410592)
 date added to LUP
 20090528 15:52:03
 date last changed
 20160415 19:49:20
@inproceedings{9df32b7b3dd144dd8404f1c9d7dd2d9e, 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 2graded irreducible infinite dimensional representations of the 3graded Lie algebra L(A) = U1 circle plus U0 circle plus U1).}, author = {Kantor, Isaiah and Shpiz, G}, booktitle = {Noncommutative Geometry and Representation Theory in Mathematical Physics}, isbn = {0821837184}, issn = {02714132}, language = {eng}, pages = {167174}, publisher = {American Mathematical Society (AMS)}, title = {Graded representations of graded Lie algebras and generalized representations of Jordan algebras}, volume = {391}, year = {2005}, }