1 Classical Groups (Problems Sets 1 { 5) De¯nitions. A nonempty set G together with an operation ¤ is a group provided ² The set is closed under the operation. That is, if g and h belong to the set G, then so does g ¤ h. ² The operation is associative. That is, if g; h and k are any elements of G, then g ¤ (h ¤ k) = (g ¤ h) ¤ k. ² 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. ² 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 2 G. A nonempty subset H ⊆ G is a subgroup of the group G if H is itself a group with the same operation ¤ as G. That is, H is a subgroup provided ² H is closed under the operation. ² H contains the identity e. ² Every element of H has an inverse in H. 1.1 Groups of symmetries. 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 de¯ned by (f ± g)(x) = f(g(x)) for x in X. The identity element 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 ¯nite group. 2. The group of symmetries of the vector space 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 linear group. 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 orthogonal group 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 rotations 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 Lorentz group. (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 \dot product") on Rn is denoted O(n; R), the general orthogo- nal group. The subgroup of transformations of determinant 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. De¯nitions. A dot product (or form) on Rn is a rule that assigns a real number to v ¢ w for every v; w 2 Rn. The form is symmetric if v ¢ w = w ¢ v for all v; w in R2. The form is bilinear if it satis¯es (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 2 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, bilinear form. 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 isometry. 1.2 Matrix groups. The elements of a matrix group are invertible n by n matrices for some positive integer n, and the group operation is matrix multiplication. The identity element is the n by n identity matrix 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 general linear group 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) = fM 2 M(n; R) : det(M) 6= 0g: 2. The special linear group SL(n; R) is the subgroup of GL(n; R) consisting of matrices of determinant 1. We can write SL(n; R) = fM 2 M(n; R) : det(M) = 1g: 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(®) ¡ sin(®) cos(®) sin(®) or sin(®) cos(®) sin(®) ¡ cos(®) The ¯rst matrix represents a rotation through an angle ® (counterclockwise if ® > 0 and clockwise if ® < 0). The second matrix represents a reflection across a line passing through the origin and making an angle ®=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 basis: · ¸ 1 0 C = : 0 ¡1 We can write L(2; R) = fM 2 M(2; R): M T CM = Cg: The matrices in L are of four types. The ¯rst two: · ¸ · ¸ cosh(®) sinh(®) cosh(®) ¡ sinh(®) or sinh(®) cosh(®) sinh(®) ¡ cosh(®) are rotations and reflections of the ¯rst kind. The second two: · ¸ · ¸ ¡ cosh(®) sinh(®) ¡ cosh(®) ¡ sinh(®) or sinh(®) ¡ cosh(®) sinh(®) cosh(®) are rotations and reflections of the second kind. (This corresponds to example 4 in 1.1.) L has two subgroups of interest, L+, the group of rotations, and L++, the group of rotations of the ¯rst 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) = fM 2 M(2; R): M T CM = Cg: 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), 2 3 1 0 0 0 6 0 ¡1 0 0 7 C = 6 7 : 4 0 0 ¡1 0 5 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) = fM 2 O(n; R) : det(M) = 1g: Also see the de¯nitions of the unitary group and the special unitary group in the next subsection. 1.3 Complex numbers and quaternions The complex numbers C are an extension of the real numbers R, C = fa + bi : a; b 2 Rg; 4 where addition and multiplication are de¯ned 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 complex conjugate of z = a + bi is z = a ¡ bi: A standard way to draw a picture of a complex number is to represent a + bi by the point (or, equivalently, the vector) (a; b) 2 R2.