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 $\operatorname{SO}(n)$
  • 🔒(25/12/12)Generalized Orthogonal Group $\operatorname{O}(n, k)$
• Unitary Group $\mathrm{U}(n)$
  • Special Unitary Group $\operatorname{SU}(n)$
• 🔒(25/12/16)Symplectic Group $\operatorname{Sp}(n, \mathbb{R})$
• 🔒(25/12/24)Euclidean group $\operatorname{E}(n)$
• 🔒(25/12/26)Heisenberg Group $H$

Topological Properties

• Connected Lie Group
• 🔒(25/12/28)Compact Lie Group

Lie Algebras

• 🔒(26/01/29)Lie Algebra $\mathfrak{g}$
• 🔒(26/01/31)Li subalgebra
• 🔒(26/02/02)Lie Algebra Isomorphism
• 🔒(26/02/16)Adjoint map $\operatorname{ad}$

Representations

• Representation of Groups $\rho : G \to \operatorname{GL}(V)$
• Equivariant Map of Group Representations
• 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)