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1 Classical Groups (Problems Sets 1 – 5)

Definitions. A nonempty set G together with an operation ∗ is a provided

• The set is closed under the operation. That is, if g and h belong to the set G, then so does g ∗ h.

• The operation is associative. That is, if g, h and k are any elements of G, then g ∗ (h ∗ k) = (g ∗ h) ∗ k.

• There is an element e of G which is an identity for the operation. That is, if g is any element of G, then g ∗ e = e ∗ g = g.

• Every element of G has an inverse in G. That is, if g is in G then there is an element of G denoted g−1 so that g ∗ g−1 = g−1 ∗ g = e.

The group G is abelian if g ∗ h = h ∗ g for all g, h ∈ G. A nonempty subset H ⊆ G is a of the group G if H is itself a group with the same operation ∗ as G. That is, H is a subgroup provided

• H is closed under the operation.

• H contains the identity e.

• Every element of H has an inverse in H.

1.1 Groups of . Symmetries are invertible functions from some set to itself preserving some feature of the set (shape, distance, interval, ...). A set of symmetries of a set can form a group using the operation composition of functions. If f and g are functions from a set X to itself, then the composition of f and g is denoted f ◦ g, and it is defined by (f ◦ g)(x) = f(g(x)) for x in X. The for composition of functions is the function I that “doesn’t do anything”: I(x) = x for every x in X. Composition of functions is always associative, so to determine whether a set of functions forms a group under composition it is only necessary to verify that the set is closed under composition of functions, that the set contains the identity function I, and that the set contains the inverse of each of its elements.

1. The set of all symmetries preserving a particular geometric shape (e.g., a poly- gon, polyhedron, etc.) is a group. Often, this is a finite group.

2. The group of symmetries of the Rn consists of all invertible linear transformations from the Rn to itself—linear transformations preserve the vec- tor space operations of vector addition and scalar multiplication. The group is denoted GL(n, R), the general .

1 3. The group of symmetries of the Euclidean plane R2 —that is, the linear trans- formations of R2 which preserve Euclidean distance—is denoted O(2, R), the in the Euclidean case. Equivalently, the symmetries are the linear transformations preserving the usual inner product on R2. The group O(2, R) consists of about the origin and reflections across lines through the origin. (See the parable of the surveyors.) 4. The group of symmetries of the Lorentz plane R2 —that is, the linear trans- formations of R2 which preserve the Lorentz product and hence the interval—is denoted L, the . (See the story of the laboratory observer and the rocket observer.) 5. The group of symmetries of Rn preserving a symmetric, non-degenerate bilin- ear form (or “”) on Rn is denoted O(n, R), the general orthogo- nal group. The subgroup of transformations of one is denoted SO(n, R), the special orthogonal group. (This generalizes examples 3 and 4.) When the dot product is the usual one, we say it is the Euclidean case. Definitions. A dot product (or form) on Rn is a rule that assigns a to v · w for every v, w ∈ Rn. The form is symmetric if v · w = w · v for all v, w in R2. The form is bilinear if it satisfies (u + v) · w = u · w + v · w u · (v + w) = u · v + u · w (av) · w = a(v · w) = v · (aw) for all u, v, w in Rn, and all a in R. Finally, the form is non-degenerate if it has the following property: If v ∈ Rn is a particular vector satisfying v · w = 0 for every w in Rn then v is the zero vector v = (0, 0,..., 0). The usual (Euclidean) dot product on Rn is a non-degenerate, symmetric, . The Lorentz product (on R2 or on R4) is also a non-degenerate, symmetric, bilinear form. The vector space Rn together with a “nice” dot product is called a metric vector space. A linear transformation T : Rn → Rn that preserves the dot product is called an .

1.2 groups. The elements of a matrix group are invertible n by n matrices for some positive n, and the group operation is . The identity element is the n by n In. Matrix multiplication is always associative, so to determine whether a set of n by n matrices forms a group under matrix multiplication, it is only necessary to verify that the set is closed under matrix multiplication, that the set contains In, that matrices in the set are invertible, and the set contains the inverse of each of its elements. These groups provide alternative descriptions of the groups listed in 1.1. In this section, it is useful to be able to refer to the set of all n by n matrices with real number entries. We’ll denote this set M(n, R).

2 1. The GL(n, R) is the group of invertible n by n matrices. In other words, an n by n matrix M is in GL(n, R) if and only if det(M) 6= 0. (This corresponds to example 2 in the previous subsection.) More compactly, we can write GL(n, R) = {M ∈ M(n, R) : det(M) 6= 0}. 2. The SL(n, R) is the subgroup of GL(n, R) consisting of matrices of determinant 1. We can write SL(n, R) = {M ∈ M(n, R) : det(M) = 1}.

3. The orthogonal group O(2, R) in the Euclidean case consists of 2 by 2 T matrices M satisfying M M = I2. These matrices have the form · ¸ · ¸ cos(α) − sin(α) cos(α) sin(α) or sin(α) cos(α) sin(α) − cos(α) The first matrix represents a through an α (counterclockwise if α > 0 and clockwise if α < 0). The second matrix represents a reflection across a line passing through the origin and making an angle α/2 with the positive x axis. (This corresponds to example 3 in 1.1). 4. The two-dimensional Lorentz group L = L(2, R) consists of 2 by 2 matrices M satisfying M T CM = C, where the matrix C describes the Lorentz dot product with respect to the standard : · ¸ 1 0 C = . 0 −1 We can write L(2, R) = {M ∈ M(2, R): M T CM = C}. The matrices in L are of four types. The first two: · ¸ · ¸ cosh(α) sinh(α) cosh(α) − sinh(α) or sinh(α) cosh(α) sinh(α) − cosh(α) are rotations and reflections of the first kind. The second two: · ¸ · ¸ − cosh(α) sinh(α) − cosh(α) − sinh(α) or sinh(α) − cosh(α) sinh(α) cosh(α) are rotations and reflections of the second kind. (This corresponds to example 4 in 1.1.) L has two of interest, L+, the group of rotations, and L++, the group of rotations of the first kind.

3 5. The orthogonal group O(n, R) consists of n by n matrices M satisfying

M T CM = C,

where the matrix C has special properties:

[omitted for now, since this is the answer to (14*) on Problem Set 4]. (This corresponds to example 5 in 1.1.) We can write

O(n, R) = {M ∈ M(2, R): M T CM = C}.

In the Euclidean case, C = In, the identity matrix. In the case of L(2, R), · ¸ 1 0 C = . 0 −1

In the case of L(4, R),   1 0 0 0  0 −1 0 0  C =   .  0 0 −1 0  0 0 0 −1

By Sylvester’s Theorem, every real vector space with a non-degenerate symmet- ric bilinear form has a basis (in general not the standard basis) with respect to which the matrix C is diagonal, with diagonal entries ±1. The number r of +1s is called the signature of the form. The corresponding orthogonal group is often denoted O(r, n − r, R). So, for example, the Lorentz group L(4, R) would be denoted O(1, 3, R).

6. The rotation subgroup or special orthogonal group SO(n, R) consists of those elements of the orthogonal group O(n, R) which have determinant 1. The rotation subgroup is also sometimes denoted O+(n, R).

SO(n, R) = {M ∈ O(n, R) : det(M) = 1}.

Also see the definitions of the and the in the next subsection.

1.3 Complex numbers and The complex numbers C are an extension of the real numbers R,

C = {a + bi : a, b ∈ R},

4 where addition and multiplication are defined by i2 = −1 and the usual rules in R. That is,

(a + bi) + (a0 + b0i) = (a + a0) + (b + b0)i (a + bi)(a0 + b0i) = (aa0 − bb0) + (ab0 + ba0)i.

Another “operation” on C is complex conjugation, where the of z = a + bi is z = a − bi. A standard way to draw a picture of a is to represent a + bi by the point (or, equivalently, the vector) (a, b) ∈ R2. In this representation, the horizontal axis is the real axis, and the√ vertical is imaginary. The length of the vector corresponding to z = a + bi is then zz. A complex number of “length” 1 has the form eiα = cos(α) + i sin(α) for some real number α. In a natural way, each complex number eiα can be identified with a point of the S1 in R2. The unit circle is called a 1- — hence the exponent 1 in S1 — because it is 1-dimensional, a curve. Similarly, each point of the unit circle can be identified with a complex number of the form eiα:

S1 = {eiα : 0 ≤ α < 2π} = {z ∈ C : |z| = 1} = {z ∈ C : zz = 1}.

We can extend complex conjugation to matrices and vectors by defining A = (aij) for a matrix A with complex entries aij and v = va,..., vn) for a vector with complex entries vj. The most important groups of matrices with complex entries are the unitary group T U(n, C) = {M ∈ M(n, C): M M = In } and its subgroup the special unitary group

SU(n, C) = {M ∈ U(n, C) : det(M) = 1 }.

In Part 5 we’ll see that U(n, C) can be regarded as the group of of a complex vector space with a “Hermitian” form, which generalizes the Euclidean bilinear form. The quaternions H are a generalization of the complex numbers:

H = {a + bi + cj + dk : a, b, c, d ∈ R},

where i2 = j2 = k2 = −1, ij = k = −ji, jk = i = −kj, and ki = j = −ik. Otherwise, addition and multiplication follow the usual rules for R: for z = a + bi + cj + dk and z0 = a0 + b0i + c0j + d0k,

z + z0 = (a + a0) + (b + b0)i + (c + c0)j + (d + d0)k zz0 = (aa0 − bb0 − cc0 − dd0) + (ab0 + ba0 + cd0 − dc0)i +(ac0 + ca0 + da0 − ad0)j + (ad0 + da0 + bc0 − cb0)k

5 Addition and multiplication are associative and satisfy the distributive law. Addition is commutative, but multiplication is not. An analogue of complex conjugation is also defined for H: z = a − bi − bj − bk. When z = a + bi + cj + dk ∈ H is identified with the point (a, b, c, d) ∈ R4, the 3-sphere S3 in R4 is identified with the quaternions of “length” 1:

S3 = {z ∈ H : zz = 1}

by analogy to the 1-sphere S1 in R2. The 2-sphere in R3 is

S2 = {(x, y, z) ∈ R3 : x2 + y2 + z2 = 1 }.

The 2-sphere is the surface of a “regular” unit sphere in R3. Unlike S1 and S3, S2 does not have a natural group structure.

1.4 Theorems

The proofs of the following theorems are embedded in Problem Sets 1–5. Theorem 1.4.1. For any positive integer n, GL(n, R) is a group under matrix multiplication, as is SL(n, R). Theorem 1.4.2. Assume T : R2 → R2 is a linear transformation and that M is the matrix of T with respect to the standard basis of R2. Then the following conditions are equivalent. a. T preserves distance; that is, for any points P and Q in R2, the distance between P and Q is the same as the distance between T (P ) and T (Q). b. T preserves the (Euclidean) dot product; that is, for any vectors v, w in R2, v · w = T (v) · T (w). T c. M M = I2. d. T is either a rotation or a reflection, and M has one of the two forms in 1.2.3. Theorem 1.4.3. Assume T : R2 → R2 is a linear transformation and that M is the matrix of T with respect to the standard basis of R2. Then the following conditions are equivalent. a. T preserves the interval; that is, for any events P and Q in R2, the interval between P and Q is the same as the interval between T (P ) and T (Q). b. T preserves the Lorentz dot product; that is, for any vectors v, w in R2, v · w = T (v) · T (w). c. M T CM = C for the 2 by 2 matrix C in 1.2.5. d. T is either a rotation or a reflection and of the first or second kind, and M has one of the four forms in 1.2.4. Theorem 1.4.4. Assume T : R3 → R3 is a linear transformation and that M is the matrix of T with respect to the standard basis of R3. Then the following three conditions are equivalent.

6 a. T preserves distance; that is, for any points P and Q in R3, the distance between P and Q is the same as the distance between T (P ) and T (Q). b. T preserves the (Euclidean) dot product; that is, for any vectors v, w in R3, v · w = T (v) · T (w). T c. M M = I3. In addition, T satisfies (a)–(c) and also has determinant 1 if and only if T is a rotation about a line through the origin. In the special case when the axis of the rotation is the z-axis, there is some real number α for which   cos(α) − sin(α) 0 M =  sin(α) cos(α) 0  0 0 1

Theorem 1.4.5. Assume T : Rn → Rn is a linear transformation, and assume M is the matrix of T with respect to the standard basis. Assume also that v · w denotes a non-degenerate, on Rn and that C is the n by n matrix for which v · w = vT Cw. Then the following are equivalent. a. T preserves the bilinear form; that is, for any vectors v, w in Rn, v · w = T (v) · T (w). b. M T CM = C. Theorem 1.4.6. (a) Transformation version. O(n, R) is a group under composition of functions, and SO(n, R) is a subgroup of O(n, R). (b) Matrix version. O(n, R) is a group under matrix multiplication, and SO(n, R) is a subgroup of O(n, R). Theorem 1.4.7.

(a) Complex conjugation satisfies z1 + z2 = z1 + z2 and z1z2 = z1z2. (b) C is an under addition, and the nonzero elements of C form an abelian group under multiplication. The 1-sphere S1 is a subgroup of the latter group. Theorem 1.4.8. U(n, C) is a group under matrix multiplication, and SU(n, C) is a subgroup. In the special case that n = 2, every matrix in SU(2, C) has the form · ¸ a −b M = b a

for complex numbers a and b satisfying aa + bb = 1. Theorem 1.4.9.

(a) Quaternionic conjugation satisfies z1 + z2 = z1 + z2 and z1z2 = z2z1. (b) H is an abelian group under addition, and the nonzero elements of H form a non-abelian group under multiplication. The 3-sphere S3 is a subgroup of the latter group.

7 1.5 Lie groups So, what is a “”? The formal definition of a Lie group uses ideas from analysis and , beyond the prerequisites for this course. Very roughly speaking, a Lie group is a group on which we can do calculus. (More formally, a Lie group has the structure of a “differentiable .”) Further, the group operation ( (M1,M2) 7→ −1 M1M2) and inversion ( M 7→ M ) are required to be differentiable functions from this calculus point of view. We’re pretty much sticking to groups of matrices — initially, subgroups of GL(n, R). Many Lie groups are matrix groups, but not all. (A major theorem, called the Peter- Weyl theorem, has as a corollary that every “compact” Lie group is a matrix group.) The Lie groups with important applications in science are matrix groups. Also, not every matrix group is a Lie group. Only certain subgroups of GL(n, R) qualify as Lie groups, namely the ones that are “closed sets” in the topology on the group. A closed set is analogous to a closed interval on the . More formally, a set is closed if it contains the limit of any convergent sequence formed of elements of the set. The classical matrix groups GL(n, R), SL(n, R), O(n, R) (all cases), and SO(n, R) are all Lie groups. So are R, Rn, C, and H under addition. We can think of Q as a group of matrices G, where ½· ¸ ¾ 1 a G = : a ∈ Q . 0 1 Using language we’ll introduce in Part 2, G with operation matrix multiplication and Q with operation addition are isomorphic groups, and therefore essentially the same. But Q is not a Lie group because it’s not a manifold, so G isn’t either. Also Q is not a closed subgroup of R, because there are sequences of rational numbers that converge to irrational numbers. (Can you think of an example?) So G isn’t a closed subgroup of GL(2, R) either. Even with our restriction to matrix groups, we’ll be able to work with many of the key ideas in the theory and applications of Lie groups.

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