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Matrix Lie Groups Maths Seminar 2007 MATRIX LIE GROUPS Claudiu C Remsing Dept of Mathematics (Pure and Applied) Rhodes University Grahamstown 6140 26 September 2007 RhodesUniv CCR 0 Maths Seminar 2007 TALK OUTLINE 1. What is a matrix Lie group ? 2. Matrices revisited. 3. Examples of matrix Lie groups. 4. Matrix Lie algebras. 5. A glimpse at elementary Lie theory. 6. Life beyond elementary Lie theory. RhodesUniv CCR 1 Maths Seminar 2007 1. What is a matrix Lie group ? Matrix Lie groups are groups of invertible • matrices that have desirable geometric features. So matrix Lie groups are simultaneously algebraic and geometric objects. Matrix Lie groups naturally arise in • – geometry (classical, algebraic, differential) – complex analyis – differential equations – Fourier analysis – algebra (group theory, ring theory) – number theory – combinatorics. RhodesUniv CCR 2 Maths Seminar 2007 Matrix Lie groups are encountered in many • applications in – physics (geometric mechanics, quantum con- trol) – engineering (motion control, robotics) – computational chemistry (molecular mo- tion) – computer science (computer animation, computer vision, quantum computation). “It turns out that matrix [Lie] groups • pop up in virtually any investigation of objects with symmetries, such as molecules in chemistry, particles in physics, and projective spaces in geometry”. (K. Tapp, 2005) RhodesUniv CCR 3 Maths Seminar 2007 EXAMPLE 1 : The Euclidean group E (2). • E (2) = F : R2 R2 F is an isometry . → | n o The vector space R2 is equipped with the standard Euclidean structure (the “dot product”) x y = x y + x y (x, y R2), • 1 1 2 2 ∈ hence with the Euclidean distance d (x, y) = (y x) (y x) (x, y R2). E − • − ∈ q RhodesUniv CCR 4 Maths Seminar 2007 Any isometry F of (the Euclidean plane) R2 has the form F (x) = Ax + c, where A R2 2 such that A A = I , ∈ × > 2 and c R2 = R2 1. ∈ × It turns out that the Euclidean group E (2) is (isomorphic to) the group of invertible matrices 1 0 2 2 2 A R × , A>A = I2, c R . ("c A# | ∈ ∈ ) The underlying set of this group of matri- ces consists of two components, each of which is (diffeomorphic to) the smooth submanifold R2 S1 R4. × ⊂ RhodesUniv CCR 5 Maths Seminar 2007 EXAMPLE 2 : The rotation group SO (2). • A non-identity isometry of (the Euclidean plane) R2 is exactly one of the following: – rotation – translation – reflection – glide reflection. The rotation group SO (2) consists of all isometries of R2 which – fix the origin (i.e., are linear transfor- mations) – preserve the orientation (of R2). RhodesUniv CCR 6 Maths Seminar 2007 It turns out that the rotation group SO (2) is (isomorphic to) a group of invertible matrices 2 2 SO (2) = A R A A = I2 ∈ × | > n o cos θ sin θ = − θ R . (" sin θ cos θ # | ∈ ) The underlying set of this group of ma- trices is (diffeomorphic to) the smooth submanifold S1 = z C z =1 C = R2. { ∈ | | | } ⊂ RhodesUniv CCR 7 Maths Seminar 2007 EXAMPLE 3 : The 3-sphere S3. • 3 2 2 2 2 4 S = (x0, x1, x2, x3) x + x + x + x = 1 R | 0 1 3 4 ⊂ is asmooth submanifold. It is also a group. Indeed, the vector space R R3, equipped with the prod- × uct p q = (p q p q, p q + q p + p q) , · 0 0 − • 0 0 × where p = (p , p), q = (q , q) R R3, is 0 0 ∈ × a (real) division algebra (or skew field), denoted by H ; its elements are called quaternions (or Hamilton numbers). S3 is exactly the group of unit quater- nions : S3 = x = (x , x) x =1 R R3. { 0 | | | } ⊂ × (The modulus of the quaternion x = 2 2 (x0, x) H is x = x0 + x .) ∈ | | k k p RhodesUniv CCR 8 Maths Seminar 2007 It turns out that the 3-sphere S3, as a group, is isomorphic to a group of invert- ible matrices 3 x0 + ix1 (x2 + ix3) 2 2 S = − x + x =1 x2 ix3 x0 ix1 0 − − | k k α β¯ 2 2 = − α + β =1 . ("β α¯ # | | | | | ) This group is the special unitary group SU (2); it can be viewed as representing all unitary transformations of the stan- dard Hermitian space C2, of determinant equal to 1. (The standard Hermitian product on C2 is defined by z w =¯z w +¯z w , 1 1 2 2 where z = (z ,z ), w = (w , w ) C2.) 1 2 1 2 ∈ RhodesUniv CCR 9 Maths Seminar 2007 DEFINITION : A matrix Lie group is • any subgroup G of the general linear group GL (n, k), k R, C ∈{ } (for some positive integer n) which is also a smooth submanifold of the matrix n n space k × . REMARK 1 : As their name suggests, • matrix Lie groups are (abstract) Lie groups. Though not all Lie groups are (isomor- phic to) matrix Lie groups, most of the interesting examples are. REMARK 2 : A matrix Lie group G is a • closed subset of GL (n, k) kn n. Quite ⊂ × remarkably - and this is an important re- sult in the theory of Lie groups - it turns out that any closed subgroup of GL (n, k) is a matrix Lie group. RhodesUniv CCR 10 Maths Seminar 2007 A subset S of the Euclidean space Rm • is called a smooth submanifold, of di- mension `, provided that one (and hence all) of the following equivalent conditions are satisfied : (a) For every x S, there exist a nbd U ∈ of x and a smooth diffeomorphism φ : U U˜ Rm such that → ⊆ φ(S U) = U˜ R`. ∩ ∩ (b) For every x S, there exist a nbd U ∈ of x and a smooth submersion F : U m ` → R − such that S U = F 1(0). ∩ − RhodesUniv CCR 11 Maths Seminar 2007 (c) For every x S, there exist a nbd U ∈ of x = (x1,...,xm), a nbd U 0 of x0 = (x1,...,x`) and smooth functions hi : U R, i =1,...,m ` such that, possi- 0 → − bly after a permutation of coordinates, S U = graph(H), ∩ m ` where H = (h1,...,hm `): U 0 R − . − → (d) For every x S, there exist a nbd U ∈ of x, a nbd V of 0 R` and a smooth ∈ embedding Φ: V Rm such that → S U = im (Φ). ∩ In (b) we think of a smooth submanifold • as the zero-set of a smooth submersion, in (c) as a graph of a smooth map, and in (d) as a parametrized set. All these are local descriptions. RhodesUniv CCR 12 Maths Seminar 2007 A (real) Lie group is a smooth manifold • which is also a group so that the group operations are smooth. In most of the literature, Lie groups are • defined to be real analytic. In fact, no generality is lost by this more restrictive definition. Smooth Lie groups always sup- port an analytic group structure, and some- thing even stronger is true. HILBERT’s 5th PROBLEM was to show that if G is only assumed to be a topological mani- fold with continuous group operations, then it is, in fact, a real analytic Lie group. (This was finally proved by the combined work of A. GLEASON, D. MONTGOMERY, L. ZIPPIN and H. YAMABE in 1952.) RhodesUniv CCR 13 Maths Seminar 2007 A Lie subgroup of a Lie group G is a Lie • group H that is an abstract subgroup and an immersed submanifold of G. This means that the inclusion map ι : H G is a one-to-one smooth immer- → sion; when ι is a smooth embedding, then the set H is closed in G. FACT : Any closed abstract subgroup H • of a Lie group G has a unique smooth structure which makes it into a Lie sub- group of G; in particular, H has the in- duced topology. Matrix Lie groups are closed Lie sub- groups of general linear groups. They are also known in literature as closed lin- ear (Lie) groups. RhodesUniv CCR 14 Maths Seminar 2007 2. Matrices revisited. Matrices (and groups of matrices) have • been introduced by – Arthur CAYLEY (1821-1895) – William Rowan HAMILTON (1805-1865) – James Joseph SYLVESTER (1814-1897) in the 1850s. RhodesUniv CCR 15 Maths Seminar 2007 About twenty years later, Sophus LIE (1842- • 1899) began his research on “continuous groups of transformations”, which gave rise to the mod- ern theory of the so-called Lie groups. In 1872, Felix KLEIN (1849-1925) delivered • his inaugural address in Erlangen in which he gave a very general view on what geometry should be regarded as; this lecture soon became known as “The Erlanger Programm”. KLEIN saw geometry as the study of invariants under a group of transformations. LIE’s and KLEIN’s research was to a certain ex- tent inspired by their deep interest in the theory of groups and in various aspects of the notion of symmetry . RhodesUniv CCR 16 Maths Seminar 2007 The ground field is k = R or k = C. Consider • the matrix space kn n of all n n ma- × × trices over k. n n The set k × has a certain amount of structure : n n – k × is a real vector space. n n – k × is a normed algebra. n n – k × is a Lie algebra. RhodesUniv CCR 17 Maths Seminar 2007 n n (a) k × is a real vector space. Firstly, the correspondence A = a a = (a , ,a ,a , ,ann) ij 7→ 11 · · · n1 12 · · · definesh ani isomorphism between (vector spaces n n n2 over k ) k × and k . Secondly, since any vector space over C can be viewed as a vector space over R (of twice 2 the dimension), we have that Cn , as a real 2 vector space, is isomorphic to R2n : 2 2 2 2 Cn = Rn i Rn = R2n .
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