Representation Theory

Lie Groups

• Group Algebra $\mathbb{C}[G]$
• Topological Group
• Lie group

Matrix Lie Groups

• Matrix Lie Groups
• General Linear Group $\operatorname{GL}(n, \mathbb{R})$
  • Special Linear Group $\mathrm{SL}(n, \mathbb{R})$
• Orthogonal Group $\operatorname{O}(n)$
  • Special Orthogonal Group $\operatorname{SO}(n)$
  • Generalized Orthogonal Group $\operatorname{O}(n, k)$
• Unitary Group $\mathrm{U}(n)$
  • Special Unitary Group $\operatorname{SU}(n)$
• Symplectic Group $\operatorname{Sp}(n, \mathbb{R})$
  • Compact Symplectic Group $\operatorname{Sp}(n)$
• Euclidean group $\operatorname{E}(n)$
• Heisenberg Group $H$

Topological Properties

• Compact Lie Group
• Connected Lie Group
• Simply Connected Lie Groups

Lie Algebras

• Algebra on Fields
• Lie Algebra $\mathfrak{g}$
  • Lie subalgebra
  • Lie Algebra Isomorphism
• 🔒(26/09/30)리 대수의 표현 $\pi : \mathfrak{g} \to \mathfrak{gl}(V)$
• Structure Constants of Lie Algebras
• Direct Sum of Lie Algebras $\mathfrak{g} = \mathfrak{g}_{1} \oplus \mathfrak{g}_{2}$
• Adjoint map $\operatorname{ad}$
• 🔒(26/10/04)킬링 폼 $B(X, Y) = \trace(\ad_{X} \circ \ad_{Y})$
• 🔒(26/10/02)리 대수의 사다리 연산자
• Lie Algebra of the Lie Group of Matrices
  • 🔒(26/09/04)일반 선형 리 대수 $\mathfrak{gl}(n, \mathbb{C})$
  • Special Linear Lie Algebra $\mathfrak{sl}(n, \mathbb{C})$
    • 🔒(26/09/22)2차 특수 선형 리 대수 $\mathfrak{sl}(2, \mathbb{C})$
  • 🔒(26/09/12)특수 직교 리 대수 $\mathfrak{so}(n)$
  • 🔒(26/09/08)유니터리 리 대수 $\mathfrak{u}(n)$
  • 🔒(26/09/06)특수 유니터리 리 대수 $\mathfrak{su}(n)$
    • 🔒(26/09/10)Special Unitary Lie Algebra of Degree 2: $\mathfrak{su}(2)$

Representations

• Representation of Groups $\rho : G \to \operatorname{GL}(V)$
• Equivariant Map of Group Representations
  • 🔒(26/10/10)얽힘 사상
• Irreducible Representations of Groups
• Direct Sum of Group Representations $\rho = \rho_{1} \oplus \rho_{2}$

Main References

• William Fulton and Joe Harris. Representation Theory: A First Course (2004)
• Brian C. Hall. Lie Groups, Lie Algebras, and Representations (2nd)