17.3.1 Matrix Lie Groups
The finite groups studied so far (symmetric groups, dihedral groups, and so on) are discrete. Many important groups in mathematics and physics are continuous: the rotation group , the group of invertible matrices, and others. Lie theory studies such groups through linear algebra. We give its usual topology, in which when every entry converges.
Definition 17.3.1 (Matrix Lie group). A matrix Lie group is a subgroup that is closed in : if and , then .
A theorem of von Neumann and Cartan, which we do not prove, says that a matrix Lie group is a smooth submanifold of on which multiplication and inversion are smooth maps; that is, it is a Lie group.
Example 17.3.2 (The classical matrix Lie groups). Each of the following is closed in , being defined by equations in the matrix entries and their complex conjugates (see also Chapter 9).
- and , the general linear groups.
- and , the special linear groups.
- , the orthogonal group, and , the group of rotations.
- , the unitary group, and .
- , where , the symplectic group (Section 9.7).
Note. The dimension of a Lie group as a manifold can be guessed by counting equations: has dimension ; the single equation gives ; and consists of equations, since is symmetric, giving . The Lie algebras below confirm these counts.
17.3.2 The Matrix Exponential
Definition 17.3.3 (Matrix exponential). For , the matrix exponential is
The series converges absolutely for every : for the operator norm , where is the standard length on , we have , hence , so the norms of the terms are bounded by those of the series for .
Proposition 17.3.4 (Properties of the matrix exponential). Let .
- .
- If , then . In particular is invertible with inverse , and is a homomorphism .
- for .
- .
- ; in particular .
Proof (sketch). (i) is clear. For (ii), multiply the two absolutely convergent series and use the binomial theorem, which holds for commuting matrices. For (iii), conjugate the series term by term. For (iv), by (iii) and Theorem 7.5.2 we may assume is upper triangular with diagonal entries ; then is upper triangular with diagonal entries , so . For (v), differentiate the series term by term. ∎
17.3.3 Lie Algebras
A Lie group is a curved object, but its tangent space at the identity is a vector space that retains much of the structure of the group. The extra structure it carries is a Lie bracket.
Definition 17.3.5 (Lie algebra). A Lie algebra over a field is a -vector space with a bilinear map , the Lie bracket, such that:
- for all ;
- the Jacobi identity holds for all .
Expanding shows that (i) implies ; when , the two conditions are equivalent.
Note. On any associative algebra, such as or , the commutator is a Lie bracket: is clear, and the Jacobi identity follows by expanding the triple brackets, all twelve terms cancelling in pairs. With this bracket, is the Lie algebra , and is .
Definition 17.3.6 (Lie algebra of a matrix Lie group). For a matrix Lie group , the Lie algebra of is
We state without proof that is a real vector subspace of closed under the commutator bracket, so that it is a real Lie algebra, and that it is the tangent space of at the identity. By Proposition 17.3.4(v), each is the velocity at of the curve in .
Example 17.3.7 (Classical Lie algebras).
- , of dimension .
- , of dimension : by Proposition 17.3.4(iv), equals 1 for all exactly when .
- , the skew-symmetric matrices, of dimension ; differentiating at gives .
- , the traceless skew-Hermitian matrices, of real dimension .
- , of dimension : the condition says that is symmetric, and is a bijection.
The exponential map of is , .
Note. Near the identity, the exponential map translates between and . For near , the Baker–Campbell–Hausdorff formula holds, where the omitted terms are iterated brackets of and ; so multiplication in near the identity is determined by the bracket of . By the inverse function theorem, is a diffeomorphism from a neighborhood of onto a neighborhood of , since its derivative at is the identity. It need not be surjective: the matrix is not for any . For compact connected groups such as and , however, is surjective.
17.3.4 Homomorphisms and Representations
Definition 17.3.8 (Lie algebra homomorphism). A linear map between Lie algebras is a homomorphism if for all .
We state the next two facts without proof.
Proposition 17.3.9 (The differential of a homomorphism). Let be a continuous homomorphism of matrix Lie groups. There is a unique real-linear map with for all and , and is a Lie algebra homomorphism. Moreover and , so , is a functor (Definition 8.2.1) from matrix Lie groups to real Lie algebras.
Note. Conversely, if is connected and simply connected (every loop in can be contracted to a point), every Lie algebra homomorphism is for a unique . Together with Lie’s third theorem (every finite-dimensional real Lie algebra is the Lie algebra of a Lie group), this makes the category of simply connected Lie groups equivalent to that of finite-dimensional real Lie algebras. These statements concern Lie groups in general, which need not be matrix groups.
Definition 17.3.10 (Representation of a Lie algebra). A representation of a Lie algebra on a vector space is a Lie algebra homomorphism , that is, a linear map with .
By Proposition 17.3.9, a continuous representation of a matrix Lie group induces a representation of on , its differential.
Example 17.3.11 (The adjoint representation). Every Lie algebra has the adjoint representation , . It is a homomorphism because of the Jacobi identity: for , where the second equality is the Jacobi identity rewritten using antisymmetry.
17.3.5 Structure and Classification
The structure theory of Lie algebras parallels that of finite groups, with ideals in the role of normal subgroups. An ideal of a Lie algebra is a subspace with for all and . For subspaces , let be the span of the brackets with , . A Lie algebra is abelian if all brackets vanish.
Definition 17.3.12 (Solvable, semisimple, and simple Lie algebras). Let be a finite-dimensional Lie algebra. Its derived series is , .
- is solvable if for some .
- is semisimple if its only solvable ideal is .
- is simple if it is not abelian and its only ideals are and .
Note. We state the main structure theorems without proof. The sum of two solvable ideals is a solvable ideal, so has a largest solvable ideal , its radical, and is semisimple exactly when . Over a field of characteristic 0, as vector spaces for a semisimple subalgebra (the Levi decomposition), and every semisimple Lie algebra is, uniquely, a direct sum of simple ideals.
The simple Lie algebras over are classified by Dynkin diagrams:
- four infinite families, (), (), (), and ();
- five exceptional algebras, , , , , , of dimensions 14, 52, 78, 133, 248.
The classification proceeds through root systems. Given a semisimple and a Cartan subalgebra (a maximal abelian subalgebra consisting of semisimple elements), the adjoint action of decomposes into root spaces; the resulting set of roots is a root system, which is encoded by its Dynkin diagram. This is one of the great classification theorems of mathematics (Killing, 1888; Cartan, 1894).
Example 17.3.13 (, the simplest simple Lie algebra). The Lie algebra of traceless complex matrices has dimension 3, with standard basis satisfying , , and (Exercise 17.3(1)). It is the Lie algebra of type . Its finite-dimensional representations are completely known: for each there is, up to isomorphism, exactly one irreducible representation of dimension , realized on (Definition 9.10.3). The representation theory of is the foundation of the general theory.
Exercise 17.3.14 (Exercises).
- Verify the relations , , for the standard basis of .
- Show that , the real skew-symmetric matrices with the commutator bracket, is isomorphic as a Lie algebra to with the cross product as bracket.
- The Killing form of a finite-dimensional Lie algebra is . Show that for ,
Show solution
(1) Directly:
(2) A basis of is and a direct computation gives , , . The standard basis of satisfies , , . Let be the linear isomorphism with . Both brackets are bilinear and antisymmetric, so holds for all once it holds for pairs of basis elements, which is what we checked. So is an isomorphism of Lie algebras.
(3) For , the linear map on has trace : on the basis of matrix units , the coefficient of in is , and . Now The four terms have the form with , , , and , so For , the map preserves and kills , and ; so the Killing form of is . It is nondegenerate: if is traceless, so is , and . By Cartan’s criterion, which we do not prove (a finite-dimensional Lie algebra over is semisimple if and only if its Killing form is nondegenerate), is semisimple.