A first course on linear Lie groups and symmetric cones

Abstract

This mini-course offers an accessible introduction to linear Lie groups and symmetric cones, two fundamental structures at the crossroads of geometry, algebra, and analysis. The first part focuses on matrix Lie groups, introducing their topological and smooth structures, the exponential and logarithm maps, Lie algebras, and basic structural properties such as compactness, solvability, and semisimplicity. Classical tools such as the polar decomposition and the Baker-Campbell-Hausdorff formula are presented. The second part is devoted to symmetric cones and their rich geometric and algebraic theory. After introducing convex and symmetric cones and their interpretation as Riemannian symmetric spaces, the course develops the fundamentals of Jordan algebras, with special emphasis on Euclidean Jordan algebras and their quadratic representations. The equivalence between symmetric cones and Euclidean Jordan algebras is explained, along with key results such as the Peirce decomposition, spectral theorem, and polar and Gauss decompositions.

Lectures

Part I. Linear Lie groups

March 7(Sat)

1. Lecture 1. The linear group: Topological groups, The linear group, Examples of subgroups, Polar decomposition, 10:00 – 11:00
2. Lecture 2. The exponential map: Exponential of a matrix, Logarithm of a matrix, 11:00 – 12:00
3. Lecture 3. Linear Lie groups: One parameter subgroups, Lie algebra of a linear group, Linear Lie groups are sub-manifolds, 14:00 – 15:00
4. Lecture 4. The Baker-Campbell-Hausdorff formula, 15:00 – 16:00
5. Lecture 5. Compactness, Connectedness, solvability and semi-simplicity, 16:00 – 17:00

Part II. Symmetric cones

March 8(Sun)

6. Lecture 6. Symmetric cones: Convex cones, Examples, The characteristic function, Symmetric cones as Riemannian symmetric spaces, 10:00 – 11:00
7. Lecture 7. Jordan algebras: Definitions, examples and first properties, The quadratic representation, Derivations and automorphisms, 11:00 – 12:00
8. Lecture 8. Euclidean Jordan algebras and symmetric cones: Euclidean Jordan algebras, The cone of squares, equivalence "Euclidean Jordan algebra - Symmetric cone", the group of the cone, 14:00 – 15:00
9. Lecture 9. The Peirce decomposition: Perice decomposition, systems of idempotents, spectral theorem, 15:00 – 16:00
10. Lecture 10. Polar and Gauss decompositions: Polar decomposition, Gauss decomposition, 16:00 – 17:00

Venue

• #213, College of Natural Sciences, Kyungpook National University, Daegu, Korea

Accommodation Suggestion

• Hotel Inter-Burgo Exco (호텔 인터불고 엑스코)
• Standard double room rate is about KRW140,000 – KRW200,000 as of Feb. 2026.
• Breakfast is not included.
• Distance from Venue : 40min. by walk (map), 30min. by bus and walk

• Browndot-KNU (브라운도트-경북대점)
• Standard double room rate is about KRW80,000 – KRW100,000 as of Feb. 2026.
• Breakfast is not included.
• Distance from Venue : 15min. by walk (map)

Contact

• Byung Hee An (anbyhee_at_knu.ac.kr)
• Hayoung Choi (hayoung.choi_at_knu.ac.kr)

Sponsored by

• National Research Foundation of Korea (2022R1A5A1033624 and RS-2024-00342939)

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