I will discuss what the universal enveloping algebra of a semisimple Lie algebra (over C) is, and how to find the "right" basis to get an integral structure. This integral structure is the so-called Kostant-form. This is important as it will allow us to define the Chevalley groups over arbitrary fields.
There is a problem/example session as usual on Tuesday from 4:05-4:55pm in Math 350. The starting point for our session this week is a worksheet discussing the construction of g_2 from its root diagram. If you would like a copy of the worksheet, please email me at michael.woodbury@colorado.edu.
The universal enveloping algebra and Kostant's Z-form
Mar. 03, 2022 11:15am (Math …
Noncomm Geometry
Maxim Braverman (Northeastern University)
X
We show that the (graded) spectral flow of a family of Toeplitz operators on a complete Riemannian manifold is equal to the index of a certain Callias-type operator. When the dimension of the manifold is even this leads to a cohomological formula for the spectral flow. As an application, we compute the spectral flow of a family of Toeplitz operators on a strongly pseudoconvex domain in . This result is similar to the Boutet de Monvel's computation of the index of a single Toeplitz operator on a strongly pseudoconvex domain. As an application, we show that the bulk-boundary correspondence in a tight-binding model of topological insulators is a special case of our results. I then discuss a project of extending the Graf-Porta model for Z_2 topological insulators via the Katsnelson-Nazaikinskii construction of the "partial spectral flow". In the end, I will explain KK-theoretical extension of our main theorem to families of Toeplitz operators parametrized by an arbitrary compact manifold, obtained by Koen van den Dungen.
Spectral Flow of Toeplitz operators and bulk-edge correspondence
Mar. 03, 2022 2:30pm (MATH 3…
Functional Analysis
Rachel Chaiser (CU Boulder)
X
I will introduce the group KK(A,B) in the non-equivariant case and where the C*-algebras A and B have the trivial grading. I will be following Section 2 of Kasparov's paper, primarily focusing on pages 155-156. There will be some review of Hilbert C*-modules and discussion of basic examples of KK(A,B) (e.g., where A or B is the complex numbers).