If \({G}\) and \({H}\) are groups or Lie groups with representations on vector spaces \({V}\) and \({W}\), we can define the direct sum of the representations as the representation of \({G\times H}\) on \({V\oplus W}\) defined by \({\left(g,h\right)\left(v,w\right)\equiv\left(g\left(v\right),h\left(w\right)\right)}\). The Lie algebra of \({G\times H}\) is \({\mathfrak{g}\oplus\mathfrak{h}}\), and it then has a representation on \({V\oplus W}\) similarly given by \({\left(A,B\right)\left(v,w\right)\equiv\left(A\left(v\right),B\left(w\right)\right)}\).

Since every linear transformation leaves the origin invariant, no linear representation is transitive. However, one can ask that at least no vector subspace be invariant. An **irreducible linear representation** (AKA irrep) on \({V}\) is defined as a group or algebra representation that has no non-trivial **invariant subspace** (AKA subrepresentation, or submodule if an algebra rep) \({0\subset W\subset V}\) such that \({gW\subset W\:\forall g\in G}\). A representation is **completely reducible** (AKA decomposable) if the orthogonal complement of every invariant subspace is also invariant; any finite-dimensional completely reducible representation can then be written as a direct sum of irreps. Referring back to the figure on group actions, the action of \({SO(2)}\) on \({\mathbb{R}^{3}}\) is completely reducible, and can be written as the direct sum of the identity irrep on the axis of rotation \({\mathbb{R}^{1}}\) and the rotation irrep on the plane \({\mathbb{R}^{2}}\) orthogonal to it.

Note that a representation can be reducible but not completely reducible, i.e. can have an invariant subspace and yet not be a direct sum of irreps. However, most representations of interest are either irreducible or completely reducible:

- Every representation of a finite group is completely reducible
- Every representation of a compact Lie group is completely reducible
- Every unitary representation is completely reducible
**Weyl’s theorem:**every representation of a Lie algebra is completely reducible iff the Lie algebra is semisimple (semisimple will be defined when we cover compact Lie groups)- Every representation of a connected semisimple Lie group is completely reducible

Again considering groups or Lie groups \({G}\) and \({H}\) with representations on \({V}\) and \({W}\), we can define the tensor product of the representations as the representation of \({G\times H}\) on \({V\otimes W}\) defined by \({(g,h)(v\otimes w)\equiv g(v)\otimes h(w)}\). In this case the representation of the Lie algebra \({\mathfrak{g}\oplus\mathfrak{h}}\) is given by \({\left(A,B\right)\left(v\otimes w\right)=A\left(v\right)\otimes I+I\otimes B\left(w\right)}\), in order to make it linear on \({V\otimes W}\).

The tensor product of two representations of the same group \({G}\) can be viewed as a new representation of \({G}\) on the vector space \({V\otimes W}\) given by \({g\left(v\otimes w\right)\equiv g\left(v\right)\otimes g\left(w\right)}\). Even if the two original representations are irreducible, this new tensor product representation may not be; decomposing it into a direct sum of irreps is called **Clebsch-Gordan** theory.

By noting that the kernel and image of an intertwiner are invariant subspaces, one arrives at **Schur’s Lemma**, which states that any intertwiner between irreps is either zero or an isomorphism. This has several immediate consequences, which are sometimes referred to themselves as Schur’s Lemma:

- Any self-intertwiner of a finite-dimensional complex irrep is a multiple of the identity map
- Any two intertwiners between finite-dimensional complex irreps differ by only a complex constant multiple
- Any matrix in the center of the image of a complex irrep is a multiple of the identity matrix
- A complex irrep maps any element in the center of a Lie group to a multiple of the identity transformation
- Any irrep of an abelian Lie group is one-dimensional (as a manifold)

For a real Lie algebra \({\mathfrak{g}}\), we can consider its complexification \({\mathfrak{g}_{\mathbb{C}}}\) as a complex Lie algebra. We can then ask, if \({\mathfrak{g}_{\mathbb{C}}}\) has an irrep on \({\mathbb{C}^{n}}\) (i.e. as an algebra with complex matrices in \({\mathbb{C}(n)}\) as vectors and scalars in \({\mathbb{C}}\)), does this correspond to an irrep of \({\mathfrak{g}}\) on \({\mathbb{C}^{n}}\) (i.e. as an algebra of complex matrices in \({\mathbb{C}(n)}\) as vectors and scalars in \({\mathbb{R}}\))? The answer is yes; the irreps on \({\mathbb{C}^{n}}\) of \({\mathfrak{g}}\) are one to one with those of \({\mathfrak{g}_{\mathbb{C}}}\) as a complex Lie algebra.