Math 410
Elements of Lie Theory

General Information

The information on this page is given tentatively and subject to changes during the term. The seminar is open to anyone with an interest in Lie Theory. Students taking the course for credit will be graded on their presentation(s) of the topics listed below.

Instructors

Pierre Clare and Angel Román
Office: 130 Jones Hall (PC) and 31 Boswell Hall (AR)
Office Hours: by chance (PC) and by appointment (PC & AR).

References

• Very Basic Lie Theory, by R. Howe
• Notes on Lie Algebras, by H. Samelson
• Introduction to Lie Groups and Lie Algebras, by A. Kirillov Jr.

Schedule

The seminar meets on Wednesdays at 6pm in Jones 302.

Date	Speaker	Topic
1/26	PC	Introduction: Symmetries and infinitesimal transformations
2/02	Boyang Li Sage Stanish	Lie algebras: definitions, first examples and basic properties Ideals and quotients
2/09	PC Paul Anderson	Generalities on representation theory Derivations, the adjoint representation
2/16	Cordelia Li	Solvable Lie algebras
2/23	Thomas Wang PC	Lie's theorem Nilpotent Lie algebras
3/02	PC	Engel's theorem, duality
3/09	John Hawthorne	Cartan's criterion for solvability
3/23	Henry Stratakis-Allen	Cartan's criterion for semisimplicity
3/30	Ryan Pesak PC	Semi-simple algebras More generalities on representation theory
4/06	Thomas Wang Paul Anderson	Representations of $\mathfrak{sl}(2,\mathbf{C})$ Cartan subalgebras
4/13	AR	Root space decomposition
4/20	Paul Anderson Cordelia Li Ryan Pesak	Abstract root systems
4/27	Paul Anderson Cordelia Li Ryan Pesak Thomas Wang	Dynkin classification
5/04	Sage Tanish Paul Anderson	Loose ends and Weyl groups

Academic Integrity

The Honor Code applies to all activities related to this course. The Honor System is responsible for resolving any suspected violations of the Honor Code. Do not hesitate to ask the instructor beforehand if you have any questions as to whether some action would be compatible with the Honor Code.

ADA Considerations

Students with disabilities who will be taking this course and may need disability-related accommodations are encouraged to make an appointment to see their instructor as soon as possible.

William & Mary accommodates students with disabilities in accordance with federal laws and university policy. Any student who feels they may need an accommodation based on the impact of a learning, psychiatric, physical, or chronic health diagnosis should contact Student Accessibility Services staff at 757-221-2512 or at sas@wm.edu to determine if accommodations are warranted and to obtain an official letter of accommodation. For more information, please visit www.wm.edu/sas.

Mental and Physical Well-Being

William & Mary recognizes that students juggle different responsibilities and can face challenges that make learning difficult. There are many resources available at W&M to help students navigate emotional/psychological, physical/medical, material/accessibility concerns. Asking for help is a sign of courage and strength. If you or someone you know is experiencing any of these challenges, we encourage you to reach out to the following offices:

• For psychological/emotional stress, please consider reaching out to the W&M Counseling Center; (757) 221-3620, 240 Gooch Dr., 2nd floor. Services are free and confidential.
• For physical/medical concerns, please consider reaching out to the W&M Health Center; (757) 221-4386, 240 Gooch Drive.
• For additional support or resources, please contact the Dean of Students by submitting a Care Report online or by calling 757-221-2510, or by emailing deanofstudents@wm.edu.

For a list of many other resources available to students, see Health and Wellness Resources for Students. As your professor, I also ask you to reach out to me if you are facing challenges inside or outside the classroom; I will guide you to ​appropriate resources on campus.