AppD.tex — March 5, 2000

P.J.M. Bongaarts Instituut Lorentz, University of Leiden bongaart@lorentz.leidenuniv.nl

Topics from 20^{th} century physics. An introductory course for students in mathematics

APPENDIX D: LIE GROUPS AND LIE ALGEBRAS

1. Groups 1.1. The definition of a group 1.2. Subgroups 1.3. Morphisms of groups 1.4. Examples of groups 1.5. Quotient, products and semidirect products of groups

2. Representations of groups 2.1. Definition of representation 2.2. Equivalence of representations 2.3. Constructing new representations from given ones 2.4. Reducible and irreducible representations 2.5. The central problem of group representation theory

3. Lie groups 3.1. Definition of a Lie group 3.2. Examples of Lie groups 3.3. Infinitesimal characterization of Lie groups

4. Lie algebras 4.1. Definition and basic properties 4.2. Constructing new Lie algebras from given ones 4.3. Examples of Lie algebras 4.4. Representations of Lie algebras

5. The relation of Lie groups to Lie algebras 5.1. The Lie algebra L(G) of a Lie group G 5.2. More about the relation between Lie groups and Lie algebras 5.3. Matrix Lie groups and matrix Lie algebras

1. GROUPS

1.1. The definition of a group DEFINITION: A nonempty set G is called a group iff it is provided with a multiplication , i.e. a map

G × G → G,

written as

(g_{1}, g_{2}) ↦ g_{1}g_{2},

which has the following properties:

1. It is associative , i.e. with g_{1}(g_{2}g_{3}) = (g_{1}g_{2})g_{3}, for all g_{1}, g_{2} and g_{3} in G.

2. It has a unit element , i.e. an element e such that ge = eg = g, for all g in G.

3. Each g ∈ G has an inverse , i.e. an element g^{−1} with g^{−1}g = gg^{−1} = e.

One shows easily that the unit element e and the inverse g^{−1} are unique.

A group G is called abelian or commutative iff g_{1}g_{2} = g_{2}g_{1}, for all g_{1} and g_{2}. An abelian group is often written additively, with the multiplication g_{1}g_{2} as g_{1} + g_{2}, with g^{−1} as −g and e as 0.

1.2. Subgroups DEFINITION: A subset H of G is a subgroup of G if it is closed under the multiplication (g_{1}, g_{2}) ↦ g_{1}g_{2} and contains with each g its inverse g^{−1}.

Note that this definition implies that H contains the unit element e of G, so the set H is a group in itself.

DEFINITION: A subgroup H of G is called an invariant subgroup or a normal subgroup if ghg^{−1} is in H for all h in H and all g in G.

In an abelian group every subgroup is clearly an invariant subgroup. In general non-abelian groups invariant subgroups are scarce; many groups have no (nontrivial) invariant subgroups. Such groups are called simple . A group that has no abelian invariant subgroups is called semisimple .

1.3. Morphisms of groups DEFINITION: A map Φ from a group G_{1} into a second group G_{2} is called a (group) homomorphism if it maps the identity element of G_{1} onto the identity element of G_{2} and respects the multiplication, i.e. has Φ(gg′) = Φ(g)Φ(g′), for all g and g′ in G_{1}.

Note that this definition implies that Φ(g^{−1}) = (Φ(g))^{−1}, for all g in G_{1}. The kernel of Φ is the subset of G_{1} mapped onto the unit element of G_{2}. It is an invariant subgroup of G_{1}, as one can check easily.

A homomorphism that is both injective and surjective is called an isomorphism . An automorphism of a group G is an injective and surjective homomorphism of G onto itself. The set of automorphisms of a group G is again a group; it is usually denoted as Aut(G).

1.4. Examples of groups Obvious examples of groups are transformation groups. Let X be a nonempty set, and G(X) the set of all invertible maps of X onto itself. G(X) is a group with the composition g_{1} ∘ g_{2} as multiplication and the identity transformation as unit element. One can show that an arbitrary group G is isomorphic to a group of transformations of some set X. Nevertheless, one should see in this general context groups in the first place as sets with a multiplication operation, satisfying certain requirements.

Examples of finite groups:

1. The symmetric groups S_{n}, the groups of permutations of n objects. S_{n} has n! elements and is nonabelian for n ≥ 4.

2. The groups Z_{n} of cyclic permutations . Z_{n} has n elements and is abelian for all n.

There are many other more complicated finite groups. In physics one uses for example the crystallographic groups for a systematic description of the properties of crystalls.

For this course the important groups are infinite groups , in particular groups of real or complex matrices, such as for example GL(n, R), the group of all invertible n × n real matrices, O(n), the group of n × n orthogonal matrices and SU(n), the group of unitary n × n matrices with determinant 1. As sets GL(n, R), O(n) and SU(n) carry the structure of a (real) differentiable manifold, of dimension n^{2} , respectively \frac{1}{2}n(n − 1) and n^{2} − 1; such groups are called ‘continuous’ groups, or – more properly – Lie groups , and will be treated more extensively further on. Remark : Matrix groups can of course be seen as groups of linear transformations in a finite dimensional vector space in which a basis has been chosen.

The n-dimensional translation groups , the spaces R^{n} with vector addition as group multiplication and the zero element as identity, are examples of abelian Lie groups. Elementary as these groups are, they are nevertheless of great importance in physics: R^{3} as the group of translations in space, R^{4} as translations in (relativistic) space-time.

1.5. Quotient, products and semidirect products of groups There are various ways of constructing new groups from given groups:

1. Let G a group and H an invariant subgroup of G. Define an equivalence relation in G: g_{1} ∼ g_{2} if and only if there is an element h in H such that g_{1} = hg_{2}. The space of equivalence classes is denoted as G/H. It is a group with respect to the obvious multiplication [g_{1}] [g_{2}] = [g_{1}g_{2}], for all g_{1} and g_{2} in G, and with [e] as unit element. G/H is called a quotient group . Note that if the subgroup H is not invariant, G/H is still defined as a space of equivalence classes, but is not a group.

2. Consider groups G_{1} and G_{2}. The cartesian product G_{1} × G_{2} has a natural group multiplication

(g_{1}, g_{2})(g′_{1}, g′_{2}) = (g_{1}g′_{1}, g_{2}g′_{2}),

for all g_{1}, g′_{1} in G_{1} and all g_{2}, g′_{2} in G_{2}, with (e_{G_{1}}, e_{G_{2}}) as unit element. G_{1} ×G_{2}, with this group structure, is called a (direct) product group .

3. There is a useful generalization of the notion of direct product group. Consider two groups G_{1} and G_{2} together with an homomorphism Φ from G_{2} into Aut(G_{1}), the group of all automorphisms of G_{1}. The product space G_{1} × G_{2} can be given a Φ-dependent group structure, in general different from the direct product group structure, by defining the multiplication

(g_{1}, g_{2})(g′_{1}, g′_{2}) = (g_{1}((Φ(g_{2}))g′_{1}), g_{2}g′_{2}),

for all g_{1}, g′_{1} in G_{1} and g_{2}, g′_{2} in G_{2}. The unit element is again (e_{G_{1}}, e_{G_{2}}). The group defined in this manner is called the semi-direct product of G_{1} and G_{2}, determined by Φ, and is usually denoted as G_{1} ×_{Φ} G_{2}. One verifies that the inverse of an element is given by the formula

(g_{1}, g_{2})^{−1} = ((Φ(g_{2}^{−1}))g_{1}^{−1}, g_{2}^{−1}).

The elements of the form (g_{1}, e_{G_{2}}), respectively (e_{G_{1}}, g_{2}), form a subgroup of G_{1} ×_{Φ} G_{2} isomorphic to G_{1}, respectively G_{2}. The first subgroup is invariant; the quotient of G_{1} ×_{Φ} G_{2} over it is isomorphic to G_{2}. One has (g_{1}, g_{2}) = (g_{1}, e_{G_{2}})(e_{G_{1}}, g_{2}), so an arbitrary element of G_{1} ×_{Φ} G_{2} can be written as a (unique) product of elements from both subgroups.

The semi-direct product multiplication rule is not very transparent at first sight. Its meaning can be better understood from the example of the group of inhomogeneous linear transformations of a vectorspace. Let x be an element of R^{n}, with coordinates x_{1}, …, x_{n}, a also an element of R^{n}, with coordinates a_{1}, …, a_{n}, and T = {T_{jk}} an n × n matrix from GL(n, R). An inhomogeneous linear transformations of R^{n} onto itself is defined as

x′_{j} = ∑_{k=1}^{n} T_{jk}x_{k} + a_{j}

Such transformations are clearly invertible and form a group. It is a semi-direct product G_{1} ×_{Φ} G_{2}, with G_{1} = R^{n}, G_{2} = GL(R, n), and the homomorphism Φ : G_{2} → G_{1} given by Φ(T)x = T x, i.e. by (T x)_{j} = ∑_{k=1}^{n} T_{jk}x_{k}. Writing such an inhomogeneous linear transformation as a pair (a, T), with a in R^{n} and T in GL(n, R), we have as multiplication rule for two such transformations

(a, T)(a′, T′) = (a + T a′, T T′).

The identity transformation is (0, 1) and the inverse transformation is written as

(a, T)^{−1} = (−T^{−1}a, T^{−1}).

2. REPRESENTATIONS OF GROUPS

2.1. Definition of representation DEFINITION: Let G be a group and V a real or complex vector space. A (linear) representation Π of G in V is a homomorphism Φ of G into the group GL(V) of invertible linear transformations of V onto itself. We can state this in a somewhat more explicit fashion. A representation Π of G in V assigns to every element g of G an invertible linear map Π(g) of V onto itself such that Π(e_{G}) = 1_{V} and Π(g_{1}g_{2}) = Π(g_{1})Π(g_{2}), for all g_{1} and g_{2} in G. Note that this implies that Π(g^{−1}) = (Π(g))^{−1}. If V is a vector space of finite dimension n, one can choose a basis. A representation Π then assigns n × n matrices to elements of G. The representation Π becomes in this manner a matrix representation of G.

DEFINITION: A representation Π of a group G in a complex Hilbert space is called unitary if the operators Π(g) are unitary, for all g in G.

2.2. Equivalence of representations A representation of a group is a homomorphism of that group onto a group of linear transformations. For two injective representations of the same group these groups of linear transformations are clearly isomorphic. They may however be different as representations . What it means for representations to be the same or different is expressed by the following definition:

DEFINITION: Two representations Π_{1} and Π_{2} of a group G, in vector spaces V_{1}, respectively V_{2}, are called equivalent iff there is an invertible linear map S from V_{1} onto V_{2} such that Π_{2}(g) = SΠ_{1}(g)S^{−1}, for all g in G. For unitary representations the linear map S is required to be unitary.

Two representations of a group in vector spaces of different dimensions are obviously inequivalent. However, representations in vector spaces with the same dimensions may be inequivalent.

2.3. Constructing new representations from given ones Let Π_{1} and Π_{2} be representations of the group G in vector spaces V_{1}, respectively V_{2}. There are two important ways of constructing a new representation from Π_{1} and Π_{2}.

1. Direct sum: Consider the vector space V_{1} ⊕ V_{2}, the direct sum of V_{1} and V_{2}. Define for each g in G the linear operator (Π_{1} ⊕ Π_{2})(g) in V_{1} ⊕ V_{2} as

((Π_{1} ⊕ Π_{2})(g))(ψ_{1}, ψ_{2}) : = ((Π_{1}(g))ψ_{1}, (Π_{2}(g))ψ_{2}),

for all ψ_{1} in V_{1} and all ψ_{2} in V_{2}. This defines the direct sum representation Π_{1} ⊕ Π_{2}.

2. Tensor product: Consider the vector space V_{1} ⊗ V_{2}, the tensor product of V_{1} and V_{2}. Define for each g in G the linear operator (Π_{1} ⊗ Π_{2})(g) in V_{1} ⊗ V_{2} by linear extension as

((Π_{1} ⊗ Π_{2})(g))(ψ_{1} ⊗ ψ_{2}) : = (Π_{1}(g))ψ_{1} ⊗ (Π_{2}(g))ψ_{2},

for all ψ_{1} in V_{1} and all ψ_{2} in V_{2}. This defines the tensor product representation Π_{1} ⊗ Π_{2}.

There are obvious generalizations to the notions of direct sum and tensor product representations of a finite number of given representations Π_{1}, …, Π_{n}. There is also a technically more subtle generalization of the notion of direct sum to certain infinite sets of representations: the direct integral .

2.4. Reducible and irreducible representations DEFINITION: A representation Π of a group G in V is called reducible iff there is a nontrivial linear subspace of V, i.e. not V itself and not {0}, which is invariant under Π(g) for all g in G. The representation Π is called irreducible if V has no such subspace.

There is a stronger notion than reducibility: DEFINITION: A representation Π of G in V is called completely reducible iff each invariant subspace has an invariant complement, i.e. if W is an invariant subspace, then there exists a second invariant subspace W′ such that V can be written as the direct sum V = W ⊕ W′.

Complete reducibility of a representation Π in a finite dimensional vector space V implies that it can be written as a direct sum representation Π = Π_{1}⊕…⊕Π_{p} in V = V_{1}⊕…⊕V_{p}, with the Π_{j} irreducible representations of G in the invariant subspaces V_{j}. This reduction is unique up to equivalence.

2.5. The central problem of group representation theory A central problem in the representation theory of groups is to find all representations of a given group, or class of groups, up to equivalence. The case of finite dimensional representations is the simplest. For many important groups all finite dimensional representations are completely reducible. In that case the irreducible representations are the building blocks of representation theory and the problem of finding all representations reduces to finding all irreducible representations. For large classes of groups this problem has been completely solved. The problem is more difficult for infinite dimensional representations; the notions of equivalence and irreducibility remain crucial in this case and much is known.

An important problem in the application of group theory to physics is that of finding explicitly the reduction of a given reducible representation into its irreducible components. A standard example of this problem is the reduction of a tensor product of two irreducible reprentations, which is in general reducible.

3. LIE GROUPS

3.1. Definition of a Lie group DEFINITION: A group G is called an n-dimensional Lie group iff it is a ndimensional (real) differentiable manifold, with the group structure such that the multiplication (g_{1}, g_{2}) ↦ g_{1}g_{2} is a differentiable map from G × G onto G and the map g ↦ g^{−1} a differentiable map from G onto itself. Note that this is the definition of a real Lie group. Complex Lie groups, with underlying complex manifolds will not be discussed in this course.

One can prove that a closed subgroup of a Lie groups is again a Lie group. The direct product G_{1} × G_{2} of two Lie groups G_{1} and G_{2} is Lie group, and so is the a semidirect product if appropriate conditions are put on the homomorhpism Φ : G_{2} → Aut(G_{1}).

A Lie group is called compact if the underlying manifold is compact as a topological space. The theory of compact Lie groups and their representations is much simpler than that of noncompact Lie groups, and is in many respect close to the theory of finite groups.

3.2. Examples of Lie groups For applications in physics the main examples of Lie groups are, apart from the translation groups R^{n}, groups of real or complex matrices. Note in this respect that a translation group can also be written as a matrix group, by writing an element a = (a_{1}, …, a_{n}) from R^{n} as the (n + 1) × (n + 1) matrix {A_{j,k}} with the diagonal elements equal to 1, A_{j,n+1} = a_{j}, for j = 1, …, n, and the other matrix elements equal to 0.

Groups of n×n matrices are subsets of R^{n^{2}} for real matrices, or R^{2n^{2}} for complex matrices. Each of the following standard matrix Lie groups is defined either by an inequality or as the null set of a system of simple algebraic expressions in R^{n^{2}} or R^{2n^{2}}, and has therefore in a natural way the structure of a (real) differentiable manifold.

• The general linear group GL(n, R), consisting of all invertible real n × n matrices. As a manifold it has clearly dimension n^{2}, because it is defined by restricting R^{n^{2}} by an inequality, n.l. the condition that the determinant of each matrix is nonzero.

• The general linear group GL(n, C), consisting of all invertible complex n × n matrices. As a real manifold it has dimension 2n^{2}.

• The special linear group SL(n, R), consisting of all real n × n matrices with determinant equal to 1. It has dimension n^{2} − 1.

• The special linear group SL(n, C), consisting of all complex n × n matrices with determinant equal to 1. As a manifold it has (real) dimension 2(n^{2} − 1).

In physics : SL(2, C) is closely related to the Lorentz group (see below) and is needed for the description of spin in relativistic physics.

• The orthogonal group O(n), consisting of the n × n orthogonal matrices. Remember that a (real) matrix is orthogonal iff its transverse is equal to its inverse. The determinant of an orthogonal matrix is equal to ±1. As a linear transformation in R^{n} it leaves the positive definite quadratic form x_{1}^{2} + … + x_{n}^{2} invariant. O(n) has dimension \frac{1}{2}n(n − 1).

In physics : O(3) is the group of rotations and reflections in 3-dimensional physical space.

• The pseudo-orthogonal group O(m, n), consisting of (m+n)×(m+n) matrices that, as linear operators in R^{n}, leave the indefinite quadratic form x_{1}^{2} + x_{2}^{2} + … + x_{m}^{2} − x_{m+1}^{2} − … − x_{m+n}^{2} invariant. Its dimension is \frac{1}{2}(m + n)(m + n − 1).

In physics : O(1, 3) is the Lorentz group , the basic group of Special Relativity.

• The special orthogonal group SO(n), consisting of orthogonal matrices with determinant equal to 1. Its dimension is the same as that of O(n).

In physics : SO(3) is the group of rotations 3-dimensional physical space and is as such the most important symmetry group in elementary quantum mechanics.

• The unitary group U(n), consisting of all (complex) unitary n × n matrices. Remember that a matrix is unitary iff its inverse is equal to its hermitian adjoint. Its dimension as a (real) manifold is n^{2}.

• The special unitary group SU(n), consisting of all unitary n×n matrices with the determinant equal to 1. It has real dimension n^{2} − 1.

In physics : SU(2) is closely related to SO(3) and is needed for the description of nonrelativistic spin in quantum mechanics. U(1), SU(2) and SU(3) play a role in elementary particle physics.

• The inhomogenous linear groups which can be obtained as semi-direct product from the translation groups R^{n} and the real matrix matrix groups, in the manner described in 1.5.

In physics : The semi-direct product of R^{4} and O(1, 3) is called the inhomogeneous Lorentz group or Poincaré group . It is important in Special Relativity.

3.3. Infinitesimal characterization of Lie groups A Lie group G can be generated from elements in a neighbourhood of the unit element e. Because G is an n-dimensional differentiable manifold, such a neighbourhood can be approximated in first order by T_{e}(G), the tangent space of G at e. The group multiplication in G induces a bracket structure on T_{e}(G), which makes it into an n-dimensional Lie algebra . This Lie algebra structe in turn characterizes the group G completely, up to a certain global topological property. A consequence of this is that for many purposes one can use the Lie algebra instead of the group itself. This may simplify matters because a Lie algebra is a linear object. Before explaining the relation between a Lie group and its associated Lie algebra, it is useful to discuss the properties of Lie algebras in a general way. This is the subject of the next chapter.

4. LIE ALGEBRAS

4.1. Definition and basic properties DEFINITION: A (real or complex) vector space L is called a (real or complex) Lie algebra iff it is provided with a Lie bracket , i.e. a bilinear map

L × L → L,

written as

(u, v) ↦ [u, v],

which has the following two properties:

1. It is antisymmetric , i.e. it has [u, v] = −[v, u], for all u and v in L.

2. It satisfies the Jacobi identity , i.e.

[u, [v, w]] + [v, [w, u]] + [w, [u, v]] = 0,

for all u, v and w in L.

If the [u, v] = 0 for all u and v in L, satisfying in this trivial way the requirements for a Lie bracket, L is called an abelian Lie algebra.

Let L be finite dimensional Lie algebra. Choose a basis u_{1}, …, u_{p} in L. The Lie bracket of two basis elements, u_{j} and u_{k}, can be written as [u_{j}, u_{k}] = ∑_{l=1}^{p} C_{jk}^{l}u_{l}. The numbers C_{jk}^{l}, called structure constants , determine the Lie algebra L. They are of course basis dependent. It is easy to derive necessary and sufficient conditions for a set of numbers {C_{jk}^{l}} to be the set of structure constants of a Lie group.

DEFINITION: A linear subspace K of the Lie algebra L is called a Lie subalgebra iff it is closed under the operation of taking the bracket of two elements. K is called an invariant Lie subalgebra or an ideal iff moreover [u, v] is in K, for all u in K and all v in L. A Lie algebra is called simple iff it has no nontrivial ideals, semisimple iff it has no nontrivial abelian ideals.

DEFINITION: A map ϕ from a Lie algebra L_{1} into a second Lie algebra L_{2} is called a (Lie algebra) homomorphism iff it is linear and connects the brackets, i.e. satisfies ϕ([u, v]_{1}) = [ϕ(u), ϕ(v)]_{2}, for all u and v in L_{1}. There are obvious definitions of the notions of isomorphism and automorphism . The automorphisms of a Lie algebra L form a group denoted as Aut(L).

4.2. Constructing new Lie algebras from given ones There are Lie algebra analogues for the group constructions discussed in 1.5:

1. For an ideal K in a Lie algebra L a quotient Lie algebra L/K is defined as the quotient as a vector space, provided with the obvious Lie bracket in terms of equivalence classes.

2. There is a direct sum L_{1} ⊕ L_{2} of two Lie algebras L_{1} and L_{2}. As a vector space it is the vector space direct sum. This is made into a Lie algebra by defining the Lie bracket as [(u_{1}, u_{2}), (v_{1}, v_{2})] := ([u_{1}, v_{1}], [u_{2}, v_{2}]), for all u_{1}, v_{1} in L_{1} and u_{2}, v_{2} in L_{2}. This direct sum is sometimes called direct product and then denoted as L_{1} × L_{2}.

3. There is also a semidirect product of two Lie algebras L_{1} and L_{2}, determined by the choice of an automorpism ϕ : L_{2} → Aut(L_{1}). The definition of this semidirect product is omitted.

4.3. Examples of Lie algebras An obvious example is the space of all linear operators in a vector space with the commutator [S, T] = ST − T S as Lie bracket. It may be denoted as gl(V). If V has finite dimension n, one can choose a basis in V. This leads to a matrix Lie algebra, denoted as gl(n, R) for real V and gl(n, C) for complex V. This notation is of course suggestive of the connection of Lie algebras with Lie groups, to be discussed further on. An example of an infinite dimensional Lie algebra is the space of vector fields on a smooth manifold, with as Lie bracket the bracket of vector fields. See for this Appendix A (Manifolds ). Another infinite dimensional Lie algebra is the space of smooth functions on a symplectic manifold, with as Lie bracket the Poisson bracket, defined in the context of classical mechanics in Part II, 4.1.

In this course the discussion will be restricted to real finite dimensional Lie algebras. In fact, all the Lie algebras needed in our presentation of quantum theory are matrix Lie algebras with the usual matrix commutator as Lie bracket. We have the following representative list:

• gl(n, R): all real n × n matrices; dimension n^{2}.

• sl(n, R): the real n × n matrices with trace 0; dimension n^{2} − 1.

• sl(n, C): the complex n × n matrices with trace 0; dimension as a real Lie algebra 2(n^{2} − 1).

• o(n) = so(n): the real antisymmetric n × n matrices; dimension \frac{1}{2}n(n − 1).

• u(n): the complex antihermitian n × n matrices; real dimension n^{2}.

• su(n): the complex antihermitian n × n matrices with trace 0; real dimension n^{2} − 1.

The notation for these Lie algebras are again suggestive for the relation with the Lie groups listed in 3.2.

4.4. Representations of Lie algebras DEFINITION: Let L be a Lie algebra and V a real or complex vector space. A representation π of L in V is a Lie algebra homomorphism of L into the Lie algebra of linear operators in V, with the operator commutator as Lie bracket. This can be stated in a more explicit fashion. A representation π of L in V assigns to every element u of L a linear map π(u) of V into itself such that π([u, v]) = [π(u), π(v)], for all u and v in L. The bracket on the left is the Lie bracket in L, that on the right the commutator for operators in V.

Let V be finite dimensional. With a choice of basis in V the operators π(u) become matrices. This leads to the notion of matrix representation of a Lie algebra.

DEFINITION: A representation π of the Lie algebra L in the Hilbert space H is called unitary iff, for each u in L, the operator π(u) is antihermitian , i.e. iff (π(u)ψ_{1}, ψ_{2}) = −(ψ_{1}, π(u)ψ_{2}), for all ψ_{1} and ψ_{2} in H. We leave aside here the additional technicalities connected with unbounded operators, required by this definition in case the Hilbert space H is infinite dimensional. It should also be remarked that the term ‘unitary’ looks a bit strange at this point. It will become more meaningful after the discussion of the relation between Lie groups and Lie algebras in the next chapter.

A representation π of a finite dimensional Lie algebra L is completely determined by the representatives π(u_{1}), …, π(u_{p}) of a basis u_{1}, …, u_{p} in L, because of the fact that the homomorphism π is a linear map. For this reason applying the representation theory of Lie algebras is much simpler than applying that of Lie groups.

Let π_{1} and π_{2} be representations of the Lie algebra L in V_{1}, respectively V_{2}. The definition of the direct sum representation is the same as in the group case, n.l. as

((π_{1} ⊕ π_{2})(u))(ψ_{1}, ψ_{2}) : = ((π_{1}(u))ψ_{1}, (π_{2}(u))ψ_{2}),

for all u in L and all (ψ_{1}, ψ_{2}) in V_{1} ⊕ V_{2}. The definition of the tensor product representation is slightly different. It is by linear extension of

((π_{1} ⊗ π_{2})(u))(ψ_{1} ⊗ ψ_{2}) : = (π_{1}(u))ψ_{1} ⊗ ψ_{2} + ψ_{1} ⊗ (π_{2}(u))ψ_{2},

for all ψ_{1} in V_{1} and ψ_{2} in V_{2}.

The definitions of irreducibility , reducibility and complete reducibility are the same as in the case of group representations.

5. THE RELATION OF LIE GROUPS TO LIE ALGEBRAS

5.1. The Lie algebra L(G) of a Lie group G Define, for a fixed element g_{0} of G, the map L_{g_{0}} : G → G as L_{g_{0}}g = g_{0}g. This map is called left translation over g_{0}. Because of the manifold properties of G, it is a diffeomorphism and induces therefore invertible linear maps between tangent spaces, the differentials (dL_{g_{0}})_{g} : T_{g}(G) → T_{g_{0}g}(G). Starting from a single given tangent vector at an initial point, for which we may take the unit element e, one can obtain unique tangent vectors in all other points g by using the differentials of the left translation maps L_{g}, for all g in G. In this way one gets a vector field on G. This vector field is a left invariant vector field , i.e. it satisfies the condition (dL_{g_{2}g_{1}^{−1}})_{g_{1}}X_{g_{1}} = X_{g_{2}}, for all g_{1} and g_{2} in G. All left invariant vector fields can be obtained in this manner. Restricting a left invariant vector field to e gives a tangent vector at e. This together establishes a linear correspondence between the tangent space T_{e}(G) and the space of left invariant vector fields.

The space of all vector fields on G is an infinite dimensional Lie algebra with respect to the commutator of vector fields. One can show that the commutator of two left invariant vector field is again left invariant. This means that the left invariant vector fields form a Lie subalgebra, of finite dimension, because of the one-to-one linear correspondence with T_{e}(G), a vector space of finite dimension. This linear correspondence induces a Lie algebra structure on T_{e}(G).

Conclusion : The tangent space at the identity element e of an n-dimensional Lie group G has the structure of a real n dimensional Lie algebra, with a bracket induced by the commutator of left invariant vector fields on G. The tangent space T_{e}(G) with this structure is called the Lie algebra of G. It will be denoted as L(G).

5.2. More about the relation between Lie groups and Lie algebras The procedure outlined in the preceding section assigns to every Lie group G a finite dimensional Lie algebra L(G). According to a classical theorem this has a converse: every finite dimensional Lie algebra is the Lie algebra L(G) of a Lie group G. A stronger statement holds if we restrict the global topological properties of G: every finite dimensional Lie algebra is the Lie algebra of a unique simply connected Lie group G. Remember that a manifold M is simply connectd iff every closed curve on M can be contracted to a point. A connected Lie group has a universal covering group , i.e. a simply connected Lie group together with a surjective homomorphism from this covering group onto the given group. This covering group is in a certain sense the ‘smallest’ Lie group with these properties. The Lie algebra of a Lie group is isomorphic with the Lie algebra of the universal covering group.

Conclusion : There is a one-to-one correspondence between finite dimensional Lie algebras and simply connected Lie groups.

Examples :

1. The 1-dimensional translation group R^{1} is simply connected; it is the universal covering group of U(1) = SO(2), the group of rotations in the plane, which is connected but not simply connected. The groups R^{1} and U(1) obviously have the same (trivial) 1-dimensional abelian Lie algebra.

2. O(3), the group of rotations and reflections in 3-dimensional Euclidean space is not connected. SO(3), the group of rotations only is connected but not simply connected; SU(2) is its simply connected universal covering group. The groups O(3), SO(3) and SU(2) have isomorphic 3-dimensional Lie algebras.

Let G_{1} and G_{2} be Lie groups, with unit elements e_{1} and e_{2}, and Φ : G_{1} → G_{2} a homomorphism. Note that in the definition of a homomorphism for Lie groups one has to require that is a diffeomorphism. It can be shown that the differential map, (dΦ)_{e_{1}} : T_{e_{1}}(G_{1}) → T_{e_{2}}(G_{2}) is consistent with the Lie brackets on these tangent spaces, i.e. (dΦ)_{e_{1}} is a Lie algebra homomorphism.

Conclusion : A homomorphism Φ from a Lie group G_{1} into a second Lie group G_{2} induces a Lie algebra homomrphism from L(G_{1}) into L(G_{2}).

This has important consequences for representation theory. A representation Π of a Lie group G in a finite dimensional vector space V is a homomorphism of G into GL(V), the group of invertible linear operators in V. It induces a homomorphism π from the Lie algebra L(G) into the Lie algebra L(GL(V)), the Lie algebra of all linear operators in V, i.e. it induces a representation π of L(G) in V. It will become clear in the next section that the original group representation Π can in general be recovered by exponentiation from the Lie algebra π. This means that the theory – and practice! – of representing Lie groups, can to a large extent be reduced to representing Lie algebras, a reduction of analysis and geometry to pure algebra.

5.3. Matrix Lie groups and matrix Lie algebras An element a of the Lie algebra L(G) is a tangent vector of G, at the unit element e. Tangent vectors at a point can be seen as equivalence classes of smooth curves γ(τ) through that point, with the same derivative at τ = 0. One can show that for each a in T_{e}(G) there is among the curves which have a as their tangent vector a unique curve γ(τ), defined for all τ in R^{1}, with the property γ(τ_{1} + τ_{2}) = γ(τ_{1})γ(τ_{2}), for all real τ_{1} and τ_{2}, i.e. such that γ(τ) is a 1-parameter subgroup of G. This curve is symbolically written as γ(τ) = e^{τa}.

In the case in which G is a group of n×n matrices, γ is a smooth matrix-valued function, which can be differentiated at τ = 0, giving (\frac{d}{dτ} γ(τ))_{τ=0} = a, with a an n × n matrix, in general not belonging to G, but of a different type. The formula γ(τ) = e^{τa} is no longer symbolic; it is a matrix formula with on the right hand side a matrix exponential function, well-defined for an arbitrary n×n matrix. The matrices a obtained in this manner form in fact the Lie algebra L(G), as a matrix Lie algebra, with the matrix commutator as Lie bracket.

With this in mind one may look at the examples of matrix Lie algebras given in Section 3.2:

• Take γ(τ) in GL(n, R) or GL(n, C). Differentiation in τ = 0 gives a general n × n real or complex matrix. So the corresponding Lie algebras are the Lie algebras of all n × n real or complex matrices, denoted as gl(n, R), respectively gl(n, C).

• For a finite dimensional matrix b one has the well-known formula det e^{b} = e^{tr b}. Using this one finds easily that differentiating a curve γ(τ) = e^{τa} in SL(n, R) at τ = 0 gives an n × n matrix with trace 0. So L(SL(n, R)) = sl(n, R), the Lie algebra of all n × n matrices with trace 0. Similarly one has L(SL(n, C)) = sl(n, C).

• An orthogonal matrix O has the property O^{T} O = OO^{T} = 1, with O^{T} the transverse of O. For a curve γ(τ) = e^{τa} with values in O(n), differentiating the left hand and the right hand side of the equation (γ(τ))^{T} γ(τ) = 1 at τ = 0 gives immediately a^{T} + a = 0, i.e. a is antisymmetric. So L(O(n)) is the algebra of allantisymmetric n×n matrices, which was denoted as o(n) in Section 3.2. Note that a smooth curve going through the identity element of O(n) has only values in the connected part of O(n) which is SO(n). Moreover an antisymmetric matrix has trace 0, so so(n) = o(n).

• Using the identity U^{∗}U = UU^{∗} = 1 for unitary matrices gives in a similar way a^{∗} + a = 0, for the matrix a in a curve γ(τ) = e^{τa} in U(n), so L(U(n)) is the Lie algebra of antihermitian matrices, which was denoted as u(n) in 3.2. An antihermitian matrix does in general not have trace 0, so su(n) ≠ u(n).

The remark about the importance of the relation between a representation Π of a Lie group G and the induced representation π of the Lie algebra L(G) made at the end of 3.2 can be expanded and made more explicit for the case in which G is a matrix Lie algebra, and therefore L(G) a matrix Lie algebra. For a representation in a finite dimensional vector space V one has the important and very useful relation

Π(γ(τ)) = Π(e^{τa}) = e^{τπ(a)}.

This shows us in a very explicit way how the problem of finding representations of a Lie group can be reduced to the purely algebraic problem of finding the representations of the corresponding Lie algebra.