III.48   Lie Theory

   Mark Ronan

1   Lie Groups

Why are groups important in mathematics? One major reason is that it is often possible to understand a mathematical structure by understanding its symmetries, and the symmetries of a given mathematical structure form a group. Some mathematical structures are so symmetrical that they have not just a finite number of symmetries, but a continuous family of them. When this is the case, we find ourselves in the realms of Lie groups and Lie theory.

One of the simplest “continuous” groups is the group SO(2), which consists of all rotations of the plane about the origin. With each element of SO(2) one can associate an angle θ: the angle of the rotation in question. If we write R_θ for the counterclockwise rotation by θ, then the group operation is given by R_θR = R_θ+, where R_2π is understood to equal R₀, the identity element of the group.

The group SO(2) is not just a continuous group, but also a Lie group. Roughly speaking, this means that it is a group in which one can meaningfully define the concept of a smooth curve (that is, a curve that is not just continuous but differentiable as well). Given any two elements R_θ and R of SO(2), one can easily define a smooth path from R_θ to R by smoothly modifying θ until it becomes . (The most obvious such path would be given in parametric form by R_(1-t)θ+t, as t goes from 0 to 1.) It is not always the case that every pair of points in a Lie group can be connected by a path: when they can, the Lie group is said to be connected. An example of a Lie group that is not connected is O(2), which consists of SO(2) together with all reflections of the plane about lines through the origin. Any two rotations can be linked by a path, as can any two reflections, but there is no continuous way of changing a rotation into a reflection.

Lie groups were introduced by SOPHUS LIE [VI.53] in order to create an analogue of GALOIS THEORY [V.21] for differential equations. Lie groups that consist of invertible linear transformations of or , like the examples above, are called linear Lie groups, and they are an important subclass. For linear Lie groups it is fairly easy to work out what terms such as “continuous,” “differentiable,” or “smooth” should mean. However, one can also consider more abstract Lie groups (both real and complex), with elements that are not given as linear transformations. In order to give a proper definition of Lie groups in their full generality, one needs the concept of a smooth MANIFOLD [I.3 §6.9]. However, for simplicity we shall mostly restrict attention to linear Lie groups.

A very common way to create a Lie group is to collect all transformations of a given space that preserve one or more specified geometric structures. For instance, the general linear group GL_n () is defined to be the group of all invertible linear transformations from to . Inside this group is the special linear group SL_n (), in which we retain only those linear transformations that preserve volume and orientation (or equivalently those with DETERMINANT [III.15] equal to 1). If instead we retain the linear transformations that preserve distance, then we obtain the orthogonal group O(n); if we retain linear transformations that preserve both distance and orientation we obtain the special orthogonal group SO(n), which is easily seen to equal SL_n () ∩ O(n). The Euclidean group E(n) of rigid motions of (that is, all transformations that preserve distances and angles, such as rotations, reflections, and translations) is generated by the orthogonal group o(n), together with the group of translations (which is isomorphic to ). There are analogues of all of the above groups in which the real numbers are replaced by the complex numbers . For instance, GL_n() is the group of all invertible complex-linear transformations of , and the complex analogue of the orthogonal group O(n) is the unitary group U(n). There are also the symplectic groups Sp(2n), which are analogues of O(n) and U(n) over the QUATERNIONS [III.76]. These are all manifestly linear Lie groups except for E(n), and in fact it is not difficult to describe a linear Lie group that is isomorphic to E(n) as well.

Many important examples of Lie groups are finite dimensional, which roughly means that they can be described using a finite number of continuous parameters (Infinite-dimensional Lie groups, while important, are more difficult to handle and will not be discussed in detail here.) For example, the group SO(3), of rotations of that fix the origin, is three dimensional. Each rotation can be specified using three parameters, which could, for instance, be taken as rotations around the x-axis, y-axis, and z-axis. These particular parameters are known to airline pilots as roll, pitch, and yaw, where the x-axis is in the direction of the airplane. Another way of specifying each rotation is by its axis and angle of rotation. Two parameters are needed to specify the axis (using spherical coordinates for example), and one parameter is needed to specify the angle of rotation. Let us take this angle to be between 0 and π (a rotation by an angle greater than π has the same effect as a rotation by an angle less than π from the opposite direction).

We can represent SO(3) geometrically as follows. Let B be a ball of radius π centered at the origin. Given any noncentral point P of B, associate with it the rotation of about the axis OP (where O is the origin) through an angle that is given in radians by the distance from O to P. With O itself we associate the identity map, so the only ambiguity is that a rotation through π radians is associated with two opposite points P and P′ on the surface of B. We can remove this ambiguity by gluing all such pairs of points together. This tells us what SO(3) looks like as a TOPOLOGICAL SPACE [III.90]: it is equivalent to the three-dimensional PROJECTIVE SPACE [I.3 §6.7] . The group SO(2), by comparison, is much simpler, and is topologically equivalent to a circle.

Lie groups arise naturally in any subject that involves continuous motion. For instance, they appear in applied topics such as the design of guidance systems and also in very pure topics such as geometry or differential equations. Lie groups, and the closely related Lie algebras discussed below, also frequently arise in many types of algebra, particularly in the algebraic structures that appear in quantum mechanics and other related branches of physics.

2   Lie Algebras

As the examples above show, Lie groups are often “curved” and have some nontrivial topology. However, one can profitably analyze a Lie group by associating with it a flat space known as a Lie algebra. This idea is similar to the idea of studying a symmetric object such as a sphere by first studying its relationship to one of its tangent planes. The Lie algebra uses the tangent space to the Lie group at the identity element, and one can view it as a “logarithm” of the Lie group.

To see how Lie algebras arise, let us consider a linear Lie group. The elements of the group can be viewed as linear transformations on a vector space, or equivalently (when we have selected a coordinate basis) as square matrices. In general, two matrices A and B do not commute (that is, AB does not have to equal BA), but the situation becomes simpler if one looks at matrices that are very close to the identity matrix I. If A = I + ∈X and B = I + ∈Y for some very small positive ∈ and two fixed matrices X and Y, then

AB = I + ∈(X + Y) + ∈²XY

and

BA = I + ∈(X + Y) + ∈²YX.

Thus, if we ignore the terms containing ∈², we see that A and B “almost commute,” and that multiplication of A and B “almost corresponds to” addition of X and Y: indeed, one can view X and Y as analogous to “logarithms” of A and B respectively.

Let us now informally define the Lie algebra g of a linear Lie group G to be the space of all matrices X such that, for sufficiently small ∈, the matrix I + ∈X lies in G, up to errors of size ∈². For example, the Lie algebra gl_n() of the general linear group GL_n() is the space of all n × n complex matrices. One can view the Lie algebra as describing all possible instantaneous directions and speeds within the group G, and a more precise definition is the collection of all derivatives of smooth curves ∈ R_∈ in G that pass through the identity element R₀. This definition can also be extended to more abstract Lie groups without much difficulty. (To return to the example of the airplane pilot, an element of the Lie group SO(3) could be used to describe the current orientation of the aircraft with respect to a fixed coordinate system, whereas an element of the Lie algebra so(3) could be used to describe the current rate of roll, pitch, and yaw that the pilot is applying to the aircraft to smoothly change its orientation.)

As we have just seen, the Lie algebra gl_n() of the general linear group GL_n() is the space of all n × n complex matrices. The Lie algebra sl_n() of the special linear group SL_n() is the subspace of all matrices with trace zero. This is because det(I + ∈X) = 1 + ∈ tr X, up to errors of size ∈², so if ∈ I + ∈X is a path in the group, then tr X = 0. The Lie algebra so(n) of SO(n) is equal to the Lie algebra o(n) of O(n), and both are equal to the space of all antisymmetric matrices. Similarly, both the Lie algebra su(n) of SU(n) and the Lie algebra u(n) of U(n) are equal to the space of skew-Hermitian matrices. (A matrix is skew-Hermitian if it equals minus the complex conjugate of its transpose.)

The fact that a Lie group is closed under multiplication can be used to show that its Lie algebra is closed under addition. Thus, a Lie algebra is a (real) vector space. However, it has some additional structure that makes it far more than just a vector space. For instance, let A and B be two elements of the Lie group G that are very close to the identity. Then we can write A ≈ I + ∈X and B ≈ I + ∈Y for some very small ∈ and some elements X and Y of the Lie algebra g. A little matrix algebra shows that the commutator ABA^-1 B^-1 of A and B, which is the element of G that measures the extent to which A and B fail to commute, can be approximated by I + ∈² [X, Y], where [X, Y] = XY - YX. This quantity [X, Y] is called the Lie bracket of X and Y. Informally, it represents the net direction of motion if one first moves an infinitesimal amount in the X direction, then in the Y direction, then back in the X direction and back in the Y direction, in that order. The resulting new direction may be quite different from the original directions X and Y.

The Lie bracket obeys a number of nice identities, such as the antisymmetric identity [X, Y] = -[Y, X] and the Jacobi identity

[[X, Y],Z] + [[Y, Z],X] + [[Z, X],Y] = 0.

One can in fact use such identities to define Lie algebras in a completely abstract fashion, without any reference to matrices or Lie groups, in much the same way that other algebraic objects such as groups, rings, and fields can be defined using a handful of algebraic identities as axioms, but we shall not focus on the abstract approach to Lie algebras here. A familiar example of a Lie algebra is with the Lie bracket [x, y] defined to be the cross-product x × y. Notice that the Lie bracket does not satisfy the associative law (unless it is trivial).

We have seen that a linear Lie group G naturally generates the bracket operation [· , ·] on its Lie algebra g. Conversely, if the Lie group is connected, one can almost reconstruct it from the Lie algebra, with its addition, scalar multiplication, and Lie bracket operation. More precisely, every element A of the Lie group can be written as an EXPONENTIAL [III.25] exp(X) of an element X of the Lie algebra. For example, if the Lie group is SO(2), then we can identify it with the unit circle in . The tangent to this circle at 1 is a vertical line, so we can identify the Lie algebra with the set i of purely imaginary numbers. (Normally, however, we would just say that the Lie algebra is .) The rotation through an angle θ can then be written as exp(iθ). Note that this representation is not unique, since exp(iθ) = exp(i(θ + 2π)). It is not hard to see that the Lie group also has as its Lie algebra (to make sense of this it helps to replace by the multiplicative group of positive real numbers, which is isomorphic to ), and that in this case the representation of a group element as an exponential is unique. In general, if two connected Lie groups have the same Lie algebra, then those Lie groups share the same universal cover, and are therefore closely related to one another.

In the case of linear Lie groups, the exponential can be described by the familiar formula

For more abstract Lie groups, the exponential is best described in the language of ordinary differential equations,¹ using a suitable generalization of the identity

from single-variable calculus. However, owing to the noncommutativity of the Lie group, it is not quite true that exp(X + Y) equals exp(X) exp(Y); instead, the correct identity is the Baker-Campbell-Hausdorff formula

exp(X) exp(Y) = exp(X + Y + [X, Y] + · · ·),

where the missing terms consist of a moderately complicated infinite series involving the Lie bracket. The exponential map that connects Lie algebras and Lie groups is closely related to the Lie bracket, and because of this it is possible to study and classify Lie groups by first studying and classifying Lie algebras with their Lie bracket operation.

3   Classification

It is always of interest when a mathematical structure can be classified, but especially so if the structure is important and the classification is not straightforward. By these criteria, the results that have been obtained concerning the classification of Lie algebras are undeniably interesting, and they are regarded as one of the great mathematical achievements from around the turn of the twentieth century.

It turns out to be easier to classify complex Lie algebras: that is, Lie algebras such as sl_n() that have the structure of a complex vector space. Each real Lie algebra embeds in a complex Lie algebra of twice the (real) dimension, known as the complexification of the original algebra. However, a complex Lie algebra may arise as the complexification of several different real Lie algebras (known as real forms of the complex Lie algebra).

In classifying Lie groups and Lie algebras, the first step is to restrict attention to simple Lie groups and Lie algebras; these are analogous to prime numbers in the sense that they cannot be “factored” into smaller components. For instance, the Euclidean group E(n) contains the translation group as a connected normal subgroup. If we factor out this group, then we obtain the orthogonal group O(n), so E(n) is not simple. More formally, a Lie group is simple if it contains no proper connected normal subgroups, and a Lie algebra is simple if it contains no proper IDEALS [III.81 §2]. In this sense, the Lie group SL_n() and its Lie algebra sl_n() are simple for every n. Finite-dimensional, complex, simple Lie algebras were classified by Wilhelm Killing and Élie CARTAN [VI.69] in 1888-94.

This classification is often placed in the context of so-called semisimple Lie algebras, which can be factored in a unique way (up to rearrangement) as a direct sum of simple Lie algebras, just as a natural number can be factored uniquely as a product of prime numbers. Furthermore, a theorem of Levi shows that a general finite-dimensional Lie algebra g can be expressed as a combination (or, more precisely, a “semidirect product”) of a semisimple algebra (called a Levi subalgebra of g) and a solvable subalgebra (known as the radical of g). Solvable Lie algebras, which are related to the concept of a SOLVABLE GROUP [V.21] in group theory, are difficult to classify, but in many applications one can restrict attention to semisimple Lie algebras, and hence to simple Lie algebras.

A simple Lie algebra g splits into smaller subalgebras, which are not ideals but which are related to one another in particularly nice ways. The case of sl_n+1 is typical and we shall use it to explain the general theory. It comprises all (n + 1) × (n + 1) matrices of trace zero, and splits as a direct sum in the following way:

sl_n+1 = n₊ ⊕ h ⊕ n_-,

where h is the set of diagonal matrices of trace zero, and n₊ and n_- are, respectively, the sets of upper and lower triangular matrices with 0s on the diagonal. Two diagonal matrices X and Y commute with one another, so their Lie bracket [X, Y] = XY - YX is 0. In other words, if X and Y belong to h, then [X, Y] = 0. A Lie algebra in which [X, Y] = 0 for any two elements X and Y is called Abelian.

Each simple Lie algebra g has a similar decomposition where the subspace h is a maximal Abelian subalgebra called a Cartan subalgebra. (For Lie algebras that are not simple, the definition of Cartan subalgebras is more complicated.) Cartan subalgebras are important because their action on the rest of the Lie algebra can be simultaneously diagonalized. What this means is that a complement to h can be split up into one-dimensional components g_α, known as root spaces, that are invariant under the action of h. To put this another way, if X belongs to h, and Y belongs to a root space, then [X, Y] is a scalar multiple of Y. (The diagonalization requires THE FUNDAMENTAL THEOREM OF ALGEBRA [V.13], which is why we need to work with complex Lie algebras.)

For sl_n+1, this works as follows. Each root space g_ij is the one-dimensional space of matrices whose entries are 0 except for a single entry in the ith row and jth column. If X ∈ h (that is, if X is a diagonal matrix of trace zero) and Y ∈ g_ij, then it is not hard to check that [X, Y] also lies in g_ij. In fact,

[X, Y] = (X_ii - X_jj)Y.

If we identify the diagonal matrix X with the vector whose n coordinates appear down its diagonal, and if we write e_i for the vector that is 1 in the ith position and 0 elsewhere, then X_ii - X_jj can be rewritten

Figure 1 Dynkin diagrams.

as 〈e_i - e_j, X〉. We refer to the vectors e_i - e_j as root vectors.

In general, a complex semisimple Lie algebra g can be completely described by its root vectors α and corresponding root spaces g_α. The rank of g equals the dimension of the Cartan subalgebra h, and also equals the dimension of the vector space spanned by the root vectors. For example, sl_n+1 has rank n, and its root vectors are the vectors e_i - e_j, as we have just seen. Sets of root vectors are far from arbitrary: they must obey some simple but quite restrictive geometric properties. For instance, if a root vector α is reflected in the hyper-plane perpendicular to another root vector β, the result must be a third root vector s_β(α), where s_β is the reflection concerned. (To make the notion of “perpendicular” precise, one needs to define a special inner product on the Cartan subalgebra, known as the Killing form, but we shall not discuss this here.) The group generated by these reflections is called the Weyl group of the Lie algebra.

The root vectors form what is called a root system, and the geometric properties mentioned above allow one to classify all root systems, and hence all complex semisimple Lie algebras. This classification is given by some very simple diagrams called Dynkin diagrams, which are shown in figure 1.

The nodes of the diagram correspond to so-called simple roots. Every root is a linear combination of simple roots with coefficients that are either all nonnegative or all nonpositive. The nature of the bond (or lack thereof) between two nodes determines the inner product of the corresponding simple roots. If there is no bond, then the inner product is 0; if there is a single bond, then the root vectors have the same length and the angle between them is 120°. In diagrams that have only single bonds, the root vectors span a set of lines in in which the angle between any two lines is either 90° or 60°. In the diagrams B_n, C_n, F₄, and G₂ there are arrows between certain pairs of nodes. The direction of an arrow is from a long root to a short root: the ratio of the root lengths is in the first three cases and in the case of G₂. In these cases there are exactly two root lengths, but in the single-bond cases all roots have the same length.

The A_n diagram is the one for sl_n+1. The simple roots are e_i - e_i+l for 1 ≤ i ≤ n, going from left to right on the diagram. Notice that the inner product of two simple roots is 0 unless they are adjacent on the diagram, in which case it is -1. Each root e_i - e_j is a sum of simple roots with coefficients all 1 or all -1 on a connected segment of the diagram.

The four infinite families A_n B_n, C_n, and D_n correspond to the classical Lie algebras, of which sl_n+1(), so(2n + 1), sp(2n), and so(2n) are real forms. These are the algebras associated with the classical Lie groups SLn+1(), SO(2n+1), Sp(2n), and SO(2n), respectively.

As mentioned earlier, a simple Lie algebra g of rank n decomposes as the direct sum of a Cartan subalgebra of dimension n plus a set of one-dimensional root spaces, one for each root. It follows that

dim g = the rank of g + the number of roots.

Here are the dimensions of the simple Lie algebras:

dimA_n = n + n(n + 1) = n(n + 2),         

dimB_n = n + 2n² = n(2n + 1),              

dimC_n = n + 2n² = n(2n + 1),             

dimD_n = n + 2n(n - 1) = n(2n - 1),    

dimG₂ = 2 + 12 = 14,                         

dimF₄ = 4 + 48 = 52,                        

dimE₆ = 6 + 72 = 78,                       

dimE₇ = 7 + 126 = 133,                  

dimE₈ = 8 + 240 = 248.                 

Each node of the diagram corresponds to a simple root, and hence to a reflection across the hyperplane perpendicular to that root. This set of reflections generates the Weyl group W in a particularly elegant way. If s_i denotes the reflection corresponding to node i, then W is generated by elements s_i of order 2, subject only to the relations

(s_is_j)^m_ij = 1,

where m_ij is the order of s_is_j (see [IV.10 §2] for a discussion of generators and relations). These orders are determined by the diagram according to the following rules:

   (i) s_is_j has order 2 if there is no bond;

  (ii) s_is_j has order 3 if there is a single bond;

 (iii) s_is_j has order 4 if there is a double bond; and

 (iv) s_is_j has order 6 if there is a triple bond.

For example, the Weyl group of type A_n is isomorphic to the SYMMETRIC GROUP [III.68] S_n+1, and one can take s₁ , . . . , s_n to be the transpositions (1 2), (2 3), . . . , (n n + 1). Notice that the Dynkin diagrams for the B_n and C_n root systems yield the same Weyl group.

In principle, this classification of root systems leads to a classification of all semisimple finite-dimensional Lie algebras and Lie groups. However, there are many fundamental questions about simple Lie algebras and Lie groups that remain only partly understood. For instance, one particularly important aim of Lie theory is to understand the linear representations of a given Lie group or Lie algebra; roughly speaking, a linear representation is a way of interpreting an abstract Lie group or Lie algebra as a linear Lie group or Lie algebra by assigning a matrix to each of its elements. While the representations of all the simple Lie algebras and Lie groups have been classified and described explicitly, these descriptions are not always easy to work with, and answering basic questions (such as how a given representation decomposes into simpler representations) often requires some sophisticated tools from algebraic combinatorics.

The theory of root systems outlined above can also be extended to an important class of infinite-dimensional Lie algebras, namely the Kac-Moody algebras. Such algebras arise in several areas of physics (such as are described in VERTEX OPERATOR ALGEBRAS [IV.17]) and algebraic combinatorics.

1. Indeed, Lie groups and Lie algebras are an excellent tool for describing the algebraic aspects of ordinary and partial differential equations; the evolution of such equations through time can be modeled using a Lie group, and the differential operators used to describe an equation can be modeled on the associated Lie algebra. However, we will not discuss this important connection between Lie theory and differential equations here.