Topics in Representation Theory

University of Southern Denmark, Fall 2026.

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Meeting time: Wednesdays and Fridays, 10 am – 12 pm.

Location: QM Lounge.

No meetings on Sept. 30, Oct. 2, Oct. 7, Oct. 9, Oct. 28, Oct. 30, Nov. 18 and Nov. 20 (I will be traveling).

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References

• J.-P. Serre, Complex Semisimple Lie Algebras, Springer Monographs in Mathematics, Springer, 2001.
• J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics 9, Springer, 1972.
• P. Etingof, Lie Groups and Lie Algebras, I, lecture notes for MIT 18.745.
• E. Frenkel and D. Ben-Zvi, Vertex Algebras and Algebraic Curves, 2nd ed., Mathematical Surveys and Monographs 88, AMS, 2004.

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Topics of Presentation

1. From Lie algebra representations to Lie group representations.
2. The Peter–Weyl theorem.
3. Representations of algebraic groups (SL_n, GL_n).
4. The Borel–Weil theorem for SL₂.
5. Category 𝒪 for 𝔰𝔩₂.
6. Representations of the symmetric group S_n and Schur–Weyl duality.
7. Quiver representations.
8. Quantum groups: representations of U_q(𝔰𝔩₂).
9. Lectures on vertex algebras.

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