What is a Lie group? Define a unitary representation and state the Peter–Weyl theorem. What is the Lie algebra of a Lie group, the Jacobi identity, and the adjoint representation? What is the commutator of vector fields?
For a finite-dimensional complex representation of \(\ZZ\), when is it completely reducible, and what are its indecomposable modules?
Solution
1. A Lie group is a smooth manifold \(G\) equipped with a group structure for which multiplication and inversion are smooth.
Its Lie algebra is \[ \mathfrak g=T_eG, \] identified with the left-invariant vector fields on \(G\). The bracket is the commutator of vector fields: \[ [X,Y](f)=X(Yf)-Y(Xf). \] It is bilinear, alternating, and satisfies the Jacobi identity \[ [X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0. \]
The adjoint representation of the Lie algebra is \[ \operatorname{ad}:\mathfrak g\to\mathfrak{gl}(\mathfrak g), \qquad \operatorname{ad}_X(Y)=[X,Y]. \]
2. A unitary representation of a topological group \(G\) is a continuous homomorphism\[ \pi:G\to U(V) \]for a complex Hilbert space \(V\).
For a compact group, the Peter–Weyl theorem says that the matrix coefficients of finite-dimensional irreducible unitary representations are dense in \(C(G)\); equivalently, the left regular representation on \(L^2(G)\) decomposes as a Hilbert direct sum of finite-dimensional irreducibles, each occurring with multiplicity equal to its dimension.
3. A finite-dimensional complex representation of \(\ZZ\) is the same thing as an invertible linear operator.
A representation \[ \rho:\ZZ\to GL(V) \] is determined by \[ T=\rho(1)\in GL(V), \] and then \(\rho(n)=T^n\). Conversely every invertible \(T\) defines such a representation.
4. The representation is completely reducible if and only if \(T\) is diagonalizable over \(\CC\).
The irreducible complex representations of the abelian group \(\ZZ\) are one-dimensional characters \[ n\longmapsto\lambda^n, \qquad \lambda\in\CC^\times. \] Thus a direct sum of irreducibles is exactly a decomposition of \(V\) into eigenspaces of \(T\), i.e. diagonalizability.
5. The finite-dimensional indecomposable complex representations are the Jordan blocks\[ J_r(\lambda), \qquad r\ge1, \quad \lambda\in\CC^\times. \]
The group algebra is \[ \CC[\ZZ]\cong\CC[t,t^{-1}], \] a PID. Finite-dimensional modules are torsion modules, and the indecomposable primary cyclic modules are \[ \CC[t,t^{-1}]/((t-\lambda)^r), \qquad \lambda\ne0. \] Under the action of \(t\), these are exactly Jordan blocks \(J_r(\lambda)\).