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A Lie-Theoretic Approach to the Spin Groups

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A Lie-Theoretic Approach to the Spin Groups A Lie-Theoretic Approach to the Spin Groups Gardner R. Marshall submitted in partial fulfillment of the requirements for Honors in Mathematics at the University of Mary Washington Fredericksburg, Virginia April 2007 This thesis by Gardner R. Marshall is accepted in its present form as satisfying the thesis requirement for Honors in Mathematics. Date Approved Randall D. Helmstutler, Ph.D. thesis advisor Janusz Konieczny, Ph.D. committee member J. Larry Lehman, Ph.D. committee member Contents 1 Introduction 1 2 Manifolds and Lie Algebra 1 2.1 Basics of Manifolds . 2 2.2 Manifolds with Group Structure . 3 2.3 Matrix Groups and Useful Tools . 6 3 Spin Groups from Lie Theory 8 3.1 The Connection to Covering Spaces . 8 3.2 Revisiting Spin(3) . 10 3.3 Computing Spin(4) . 11 3.4 Conclusion . 13 References 14 Abstract We investigate a collection of topological groups known as spin groups from a Lie-theoretic viewpoint. From this analysis, we compute well-known groups isomorphic to both Spin(3) and Spin(4), and generate matrix representations of their elements. 1 Introduction The purpose of this paper is to provide an analysis of the spin groups through an alternative viewpoint to that taken in [2]. In addition to verifying our previous results regarding Spin(3) through new means, we will produce real matrix representations for the higher case of Spin(4) as well. The approach taken will be Lie-theoretic and will view the groups under consideration as manifolds. In [2] we do not arrive at matrices until near the end, when we are forced to seek matrices from the Clifford algebra in order to represent elements of Spin(3). However, by analyzing the spin groups from the perspective of manifold theory, we will be afforded easy access to matrices at the start of our efforts. In this paper we aim similarly to find matrix representations for the elements of Spin(3), but we go about it by means of a new approach. The spin group Spin(n) can be seen algebraically as a special kind of group equipped with a subgroup S of order two for which the quotient Spin(n)/S is the special orthogonal group SO(n). In other words, there is a special surjective two-to-one homomorphism that maps each spin group to its respective special orthogonal group. It should be noted, however, that finding Spin(n) is not as simple as trying such groups as SO(n) × Z2 as the spin group must be connected. This is because of the special characteristics that the quotient map must have. We will see later that for our case, it comes down to the additional requirement that the map be a local homeomorphism. To accomplish this task of uncovering spin groups, we employ Lie theory so that we may exploit the Inverse Function Theorem. The general paradigm goes as follows. Suppose we have a likely candidate X for the spin group Spin(n) along with a possible quotient map p : X → SO(n). Because of the uniqueness of such maps and spin groups, if we can show that p is a surjective, two-to-one homomorphism and local homeomorphism, then we know that X ≈ Spin(n) as a group and a space. Lie theory allows us to do this by checking that p is a local diffeomorphism at the identity. From the map p : X → SO(n), we obtain the new map dpI : L(X) → L(SO(n)) of the Lie algebras, which is the derivative of p evaluated at the identity. If dpI sends a basis of L(X) to a basis of L(SO(n)), then by the Inverse Function Theorem p is a local diffeomorphism and hence a local homeomorphism. This assures us that we have found Spin(n). We will begin with basic definitions of manifold theory, and will move from here into Lie algebra and its relation to covering spaces. With this point of view, we will have matrices available from the start, as matrix groups are in fact manifolds. We will then set about to compute Spin(4) through the same means. 2 Manifolds and Lie Algebra In this section we will run through much of the material needed to build up a working knowledge of Lie theory and apply it to the task of computing spin groups. Most of the material comes from 1 Kristopher Tapp’s book on matrix groups and will be explained here rather than proved, as it can be found in [4]. 2.1 Basics of Manifolds We begin by stipulating what we mean by matrix group. We will restrict this terminology to mean those groups of matrices that are closed in GLn as subspaces. The entries of matrices of these groups may be real, complex, or quaternionic. For the case of real-valued matrices, we may topologize n2 GLn as a subspace of R by thinking of matrix entries as coordinates in this Euclidean space. Complex and quaternionic counterparts are topologized similarly in higher-dimensional Euclidean spaces. In fact, with this topology, the general linear group is a topological group, and so too are all matrix groups. m n Definition 2.1. Let U ⊆ R be an open set. A map f : U → R is said to be smooth on U if its n rth order partial derivative exists and is continuous on U for all r ∈ N. Suppose f : X → R for an m arbitrary subset X ⊆ R . We simply call f smooth if for all x ∈ X there is some neighborhood m 0 n 0 Vx ⊆ R of x equipped with some map f : Vx → R which is smooth on Vx such that f and f agree on X ∩ Vx. m n Definition 2.2. Let X ⊆ R and Y ⊆ R .A diffeomorphism is a smooth bijection f : X → Y whose inverse is also smooth. If f : X → Y is a diffeomorphism, then X and Y are said to be diffeomorphic. It should be noted that every diffeomorphism is automatically a homeomorphism, making dif- feomorphism a stronger condition. Speaking loosely, it has the additional condition of “continued differentiability.” With this terminology, we turn our attention to manifolds. For our purposes we will take manifolds as being certain subsets of Euclidean spaces without viewing them from a purely m intrinsic point of view. In this spirit we define a manifold M as a subset of R such that for all n p ∈ M there is a neighborhood Vp of p that is diffeomorphic to some open set U ⊆ R . In this case, we say that the manifold M has dimension n. Because manifolds are locally Euclidean in a differentiable manner, it makes sense to talk about the derivative of a map between manifolds. As we shall see however, this only makes sense once we specify a particular point within the manifold. We first consider the generalized derivative for m n Euclidean spaces. For a smooth map f : R → R , the derivative is denoted df and is given by the Jacobian matrix of partial derivatives ∂f 1 ∂f 1 ∂f 1 ∂u1 ∂u2 ··· ∂um ∂f 2 ∂f 2 ∂f 2 1 2 ··· m df = ∂u ∂u ∂u . . .. . ∂f n ∂f n ∂f n ∂u1 ∂u2 ··· ∂um m Once the derivative is evaluated at a point p ∈ R , the entries become real numbers and the matrix represents a linear map. We now have a clear way of discussing the derivative of a map between Euclidean spaces. Since every point in a manifold lies in a neighborhood that is diffeomorphic to some open Euclidean subspace, this leads to a method for treating derivatives of maps between manifolds. 2 We treat the derivative of such a map at a point within some Euclidean neighborhood in a way purely analogous to our Euclidean treatment. However, we cannot speak solely about the derivative of a map between manifolds without first nailing it down at a point. This is because without doing so, we do not know which local neighborhood to use in our Euclidean treatment of the derivative. Further, we cannot continue without this information because the individual entry functions of the Jacobian matrix are liable to change abruptly to entirely different functions as we change Euclidean neighborhoods in the manifold. Suppose f : M1 → M2 is a smooth map between manifolds M1 and M2, which have dimension m and n respectively. To make sense of the derivative of f inside some Euclidean neighborhood Vp of p ∈ M1, what we are doing is moving through several steps. First we find a Euclidean neighborhood Vf(p) of f(p) ∈ M2, compatible with f. Next we move through the diffeomorphism m n φ1 that sends points in Vp to an open subset U1 of R . Next we move to an open subset U2 of R and lastly back through the inverse diffeomorphism of U2 into the Euclidean neighborhood Vf(p). It looks something like this: f ... M1 ⊇Vp ......................................................................................................................................................................................................................................................... Vf(p) ⊆ M2 ... ... ....... ....... ... ... −1 . −1 . φ . φ1 φ . φ2 1 . 2 . ....... ....... ..... f˜ . ..... m ................................................................................................................................................................................................................................................................... n R ⊇ U1 .... U2 ⊆ R where each φi is a diffeomorphism between the appropriate Euclidean neighborhood of Mi and Ui. In this picture, for all x ∈ Vp the map f˜ sends φ1(x) to φ2(f(x)) so that the diagram commutes. Because each φi is a diffeomorphism and f˜ is a map between purely Euclidean spaces, we may understand the derivative of f at p as the derivative of f˜. We write this as dfp, which is a linear m n map R → R . Suppose there exists a sequence of smooth maps between manifolds f g ..
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