Quantum Mechanics for Mathematicians

algebras aregl(n,R) =M(n,R) andgl(n,C) =M(n,C). These are

the vector spaces of allnbynmatrices, with Lie bracket the matrix

commutator.

Other Lie groups will be subgroups of these, with Lie algebras sub-Lie

algebras of these Lie algebras.

• The special linear groupsSL(n,R) andSL(n,C) are the groups of in-
  vertible matrices with determinant one. Their Lie algebrassl(n,R) and
  sl(n,C) are the Lie algebras of allnbynmatrices with zero trace.


• The orthogonal groupO(n)⊂GL(n,R) is the group ofnbynreal ma-
  trices Ω satisfying ΩT= Ω−^1. Its Lie algebrao(n) is the Lie algebra ofn
  bynreal matricesXsatisfyingXT=−X.


• The special orthogonal groupSO(n)⊂SL(n,R) is the subgroup ofO(n)
  with determinant one. It has the same Lie algebra asO(n):so(n) =o(n).


• The unitary groupU(n)⊂GL(n,C) is the group ofnbyncomplex
  matrices Ω satisfying Ω†= Ω−^1. Its Lie algebrau(n) is the Lie algebra of
  nbynskew-Hermitian matricesX, those satisfyingX†=−X.


• The special unitary groupSU(n)⊂SL(n,C) is the subgroup ofU(n) of
  matrices of determinant one. Its Lie algebrasu(n) is the Lie algebra ofn
  bynskew-Hermitian matricesXwith trace zero.
  In later chapters we’ll encounter some other examples of matrix Lie groups,
  including the symplectic groupSp(2d,R) (see chapter 16) and the pseudo-
  orthogonal groupsO(r,s) (see chapter 29).

5.4 Lie algebra representations

We have defined a group representation as a homomorphism (a map of groups
preserving group multiplication)

π:G→GL(n,C)

We can similarly define a Lie algebra representation as a map of Lie algebras
preserving the Lie bracket:

Definition(Lie algebra representation).A (complex) Lie algebra representation
(φ,V)of a Lie algebragon an n dimensional complex vector spaceV is given
by a real-linear map

φ:X∈g→φ(X)∈gl(n,C) =M(n,C)

satisfying
φ([X,Y]) = [φ(X),φ(Y)]

Such a representation is called unitary if its image is inu(n), i.e., if it satisfies

φ(X)†=−φ(X)