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Lie Groups, Representation Theory, and Branching Rules Lab
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A mathematics lab studying representations of Lie groups and their branching rules.
AboutAbout the lab
Studying group representations by restricting them to subgroups
Associate Professor Masatoshi Kitagawa studies the representation theory of Lie groups and Lie algebras.
The problem of studying how a representation of a group behaves when restricted to a subgroup is called the branching problem.
This problem is approached using the structure of the spaces on which representations are realized and the algebras that act on them.
Key terms
- Branching rules
- The problem of determining how an irreducible representation of a group G behaves when restricted to a subgroup G'.
ResearchResearch themes
The lab studies the branching problem of restricting representations of Lie groups and Lie algebras to subgroups, using the structure of the spaces on which the representations are realized and the algebras that act on them. It also aims to establish foundational results for large representations that are not irreducible.
*Illustrative imageCartan subalgebras of g-modules
The lab shows that when a larger algebra acts irreducibly on a g-module, the support of the action of Z(g) is, under certain conditions, a union of affine subspaces. Based on this, it proposes a definition of a Cartan subalgebra for such modules.
*Illustrative imageLarge representations and D-modules
The lab studies D-modules on the basic affine space G/U of a complex semisimple algebraic group G, and their global sections. The aim is to establish foundational results on large, non-irreducible representations for use in branching rules and harmonic analysis.
*Illustrative imageDiscretely decomposable restrictions and irreducible modules
In the branching problem for reductive Lie algebras, the lab investigates the conditions under which the U(g)^{G′}-module appearing in the space of intertwining operators is irreducible, for discretely decomposable restrictions that satisfy certain conditions. Using Zuckerman derived functors, it constructs families of finite-dimensional irreducible modules.
MembersFaculty
- Associate ProfessorMasatoshi Kitagawa
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Ho-Mg-Zn E8-5Cube” Materialscientist (original image) Jgmoxness (derivative work) / CC BY-SA 3.0 (cropped)
- Cartan subalgebras of g-modules: “Root system A2 with labels” Mark L MacDonald; derivative of work of User:Smithers888 and User:Maksim / CC BY-SA 3.0 (cropped)
- Large representations and D-modules: “A discourse concerning the residual analysis- a new branch of the algebraic art Fleuron T147723-6” Landen, John / Public domain (cropped)
- Discretely decomposable restrictions and irreducible modules: “Sphere symmetry group d3d” User:Tomruen / Public domain (cropped)
- Related lab (Lie Groups, Representation Theory, and Branching Laws Lab (Masatoshi Kitagawa)): “Representation of SU(3)” Arkadipta Sarkar at English Wikipedia / CC BY-SA 3.0 (cropped)
- Related lab (Social Mathematics and Computer Algebra Lab): “Mathics screenshot” Wikideas1 / CC0 (cropped)
- Related lab (Optimization and Mathematical Programming Lab): “Optimization Problem 2” Musical Inquisit / CC BY-SA 4.0 (cropped)
- Related lab (Optimization and Mathematical Programming Lab): “S asdf rot” Trabandt / CC0 (cropped)
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