Monday, August 31, 2015 at 3:00 PM in SAS 4201
Uladimir Shtukar, NC Central University
Subalgebras and reductive pairs for Lie algebras of small dimensions
This research is stimulated by many issues performed in the geometry of reductive homogeneous spaces G/H. Locally it means that we need to analyze a Lie algebra g and its subalgebra h that has a reductive complement m such that g=h+m, [h,m] contained in m. Well known Lie theory guarantees that global pair (G,H) is equivalent to local pair (g,h). Authors usually suppose that the global pair and its local equivalent satisfy some special conditions which are a good base for research. One of the conditions is reductive complement m in algebra g as described above. Unfortunately, the number of examples (the real situations for g,h,m) is very short. Just symmetric homogeneous spaces are described and locally classified for simple Lie algebras. For general Lie algebras we have nothing, we dont know how many subalgebras and reductive pairs exist. This is the first reason to find subalgebras of some Lie algebras.
Some subalgebras have been found for special Lie algebras of dimensions <=4 (see Journal of Mathematical Physics). Unfortunately, the corresponding classification is too big even up to inner automorphisms of Lie algebras. Evaluation details are not represented in the articles. This is the second reason to classify subalgebras of Lie algebras in some different way which is clear and well grounded. My analysis is done for two Lie algebras of dimension 3. The results will be demonstrated in the report. One Lie algebra of dimension 4 is under evaluation right now.
Monday, October 12, 2015 at 3:00 PM in SAS 4201
Natasha Rozhkovskaya, Kansas State University
Monday, October 19, 2015 at 3:00 PM in SAS 4201
David Penneys, UCLA
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