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Lecture Notes in Lie Groups

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Lecture Notes in Lie Groups Lecture Notes in Lie Groups Vladimir G. Ivancevic∗ Tijana T. Ivancevic† Abstract These lecture notes in Lie Groups are designed for a 1–semester third year or graduate course in mathematics, physics, engineering, chemistry or biology. This landmark theory of the 20th Century mathematics and physics gives a rigorous foundation to modern dynamics, as well as field and gauge theories in physics, engineering and biomechanics. We give both physical and medical examples of Lie groups. The only necessary background for comprehensive reading of these notes are advanced calculus and linear algebra. Contents 1 Preliminaries: Sets, Maps and Diagrams 3 1.1 Sets .................................... 3 1.2 Maps.................................... 4 1.3 CommutativeDiagrams ......................... 5 2 Groups 6 3 Manifolds 8 3.1 DefinitionofaManifold . .. .. .. .. .. .. .. 11 3.2 Formal Definition of a Smooth Manifold . 12 arXiv:1104.1106v2 [math.DG] 7 Apr 2011 3.3 Smooth Maps Between Smooth Manifolds . 13 3.4 Tangent Bundle and Lagrangian Dynamics . 14 3.5 Cotangent Bundle and Hamiltonian Dynamics . 17 ∗Land Operations Division, Defence Science & Technology Organisation, P.O. Box 1500, Edin- burgh SA 5111, Australia (e-mail: [email protected]) †Tesla Science Evolution Institute & QLIWW IP Pty Ltd., Adelaide, Australia (e-mail: [email protected]) 1 4 Lie Groups 18 4.1 DefinitionofaLieGroup ........................ 21 4.2 LieAlgebra ................................ 22 4.3 One-ParameterSubgroup . 24 4.4 ExponentialMap ............................. 24 4.5 AdjointRepresentation. 25 4.6 Actions of Lie Groups on Smooth Manifolds . 26 4.7 Basic Tables of Lie Groups and Their Lie Algebras . 28 4.8 RepresentationsofLiegroups . 31 4.9 RootSystemsandDynkinDiagrams . 32 4.9.1 Definitions ............................ 32 4.9.2 Classification ........................... 33 4.9.3 DynkinDiagrams......................... 34 4.9.4 IrreducibleRootSystems . 36 4.10 Simple and Semisimple Lie Groups and Algebras . 37 5 Some Classical Examples of Lie Groups 38 5.1 GalileiGroup ............................... 38 5.2 GeneralLinearGroup .......................... 39 5.3 Rotational Lie Groups in Human/Humanoid Biomechanics . 40 5.3.1 Uniaxial Group of Joint Rotations . 41 5.3.2 Three–Axial Group of Joint Rotations . 43 5.3.3 TheHeavyTop.......................... 45 5.4 Euclidean Groups of Rigid Body Motion . 45 5.4.1 Special Euclidean Group SE(2) in the Plane . 46 5.4.2 Special Euclidean Group SE(3) in the 3D Space . 47 5.5 BasicMechanicalExamples . 50 5.5.1 SE(2) Hovercraft ........................ 50 − 5.5.2 SO(3) Satellite ......................... 51 − 5.5.3 SE(3) Submarine ........................ 52 − 5.6 Newton–Euler SE(3) Dynamics .................... 52 − 5.6.1 SO(3) : Euler Equations of Rigid Rotations . 52 5.6.2 SE(3) : Coupled Newton–Euler Equations . 53 5.7 Symplectic Group in Hamiltonian Mechanics . 56 6 Medical Applications: Prediction of Injuries 57 6.1 General Theory of Musculo–Skeletal Injury Mechanics . ...... 57 6.2 Analytical Mechanics of Traumatic Brain Injury (TBI) . ...... 63 6.2.1 The SE(3) jolt:thecauseofTBI . 63 − 6.2.2 SE(3) group of brain’s micro–motions within the CSF . 64 − 2 6.2.3 Brain’s natural SE(3) dynamics................ 65 − 6.2.4 Brain’s traumatic dynamics: the SE(3) jolt ......... 68 − 6.2.5 Brain’s dislocations and disclinations caused by the SE(3) jolt 69 − 1 Preliminaries: Sets, Maps and Diagrams 1.1 Sets Given a map (or, a function) f : A B, the set A is called the domain of f, and → denoted Dom f. The set B is called the codomain of f, and denoted Cod f. The codomain is not to be confused with the range of f(A), which is in general only a subset of B (see [8, 9]). A map f : X Y is called injective, or 1–1, or an injection, iff for every y in the → codomain Y there is at most one x in the domain X with f(x)= y. Put another way, given x and x′ in X, if f(x)= f(x′), then it follows that x = x′. A map f : X Y → is called surjective, or onto, or a surjection, iff for every y in the codomain Cod f there is at least one x in the domain X with f(x)= y. Put another way, the range f(X) is equal to the codomain Y . A map is bijective iff it is both injective and surjective. Injective functions are called monomorphisms, and surjective functions are called epimorphisms in the category of sets (see below). Bijective functions are called isomorphisms. A relation is any subset of a Cartesian product (see below). By definition, an equivalence relation α on a set X is a relation which is reflexive, symmetrical and transitive, i.e., relation that satisfies the following three conditions: 1. Reflexivity: each element x X is equivalent to itself, i.e., xαx; ∈ 2. Symmetry: for any two elements a, b X, aαb implies bαa; and ∈ 3. Transitivity: aαb and bαc implies aαc. Similarly, a relation defines a partial order on a set S if it has the following ≤ properties: 1. Reflexivity: a a for all a S; ≤ ∈ 2. Antisymmetry: a b and b a implies a = b; and ≤ ≤ 3. Transitivity: a b and b c implies a c. ≤ ≤ ≤ A partially ordered set (or poset) is a set taken together with a partial order on it. Formally, a partially ordered set is defined as an ordered pair P = (X, ), where ≤ X is called the ground set of P and is the partial order of P . ≤ 3 1.2 Maps Let f and g be maps with domains A and B. Then the maps f + g, f g, fg, and − f/g are defined as follows (see [8, 9]): (f + g)(x) = f(x)+ g(x) domain = A B, ∩ (f g)(x) = f(x) g(x) domain = A B, − − ∩ (fg)(x) = f(x) g(x) domain = A B, ∩ f f(x) (x) = domain = x A B : g(x) = 0 . g g(x) { ∈ ∩ 6 } Given two maps f and g, the composite map f g, called the composition of f ◦ and g, is defined by (f g)(x)= f(g(x)). ◦ The (f g) machine is composed of the g machine (first) and then the f machine, ◦ − − − x [[g]] g(x) [[f]] f(g(x)). → → → → For example, suppose that y = f(u) = √u and u = g(x) = x2 + 1. Since y is a function of u and u is a function of x, it follows that y is ultimately a function of x. We calculate this by substitution y = f(u)= f g = f(g(x)) = f(x2 +1) = x2 + 1. ◦ If f and g are both differentiable (or smooth, i.e., C∞)p maps and h = f g is the ◦ composite map defined by h(x)= f(g(x)), then h is differentiable and h′ is given by the product: h′(x)= f ′(g(x)) g′(x). In Leibniz notation, if y = f(u) and u = g(x) are both differentiable maps, then dy dy du = . dx du dx The reason for the name chain rule becomes clear if we add another link to the chain. Suppose that we have one more differentiable map x = h(t). Then, to calculate the derivative of y with respect to t, we use the chain rule twice, dy dy du dx = . dt du dx dt 0 ∂(x,...) Given a 1–1 continuous (i.e., C ) map F with a nonzero Jacobian ∂(u,...) that maps a region S onto a region R, we have the following substitution formulas: 4 1. For a single integral, ∂x f(x) dx = f(x(u)) du; ∂u ZR ZS 2. For a double integral, ∂(x,y) f(x,y) dA = f(x(u, v),y(u, v)) dudv; ∂(u, v) ZZR ZZS 3. For a triple integral, ∂(x,y,z) f(x,y,z) dV = f(x(u, v, w),y(u, v, w), z(u, v, w)) dudvdw; ∂(u, v, w) ZZZR ZZZS 4. Generalization to n tuple integrals is obvious. − 1.3 Commutative Diagrams Many properties of mathematical systems can be unified and simplified by a presen- tation with commutative diagrams of arrows. Each arrow f : X Y represents a → function (i.e., a map, transformation, operator); that is, a source (domain) set X, a target (codomain) set Y , and a rule x f(x) which assigns to each element x X 7→ ∈ an element f(x) Y . A typical diagram of sets and functions is (see [8, 9]): ∈ f f X✲ Y X✲ f(X) ❅ ❅ h❅ g or h❅ g ❅ ❅ ❅❘ ❄ ❅❘ ❄ Z g(f(X)) This diagram is commutative iff h = g f, where g f is the usual composite function ◦ ◦ g f : X Z, defined by x g(f(x)). ◦ → 7→ Less formally, composing maps is like following directed paths from one object to another (e.g., from set to set). In general, a diagram is commutative iff any two paths along arrows that start at the same point and finish at the same point yield the same ‘homomorphism’ via compositions along successive arrows. Commutativity of the whole diagram follows from commutativity of its triangular components. Study of commutative diagrams is popularly called ‘diagram chasing’, and provides a powerful tool for mathematical thought. Many properties of mathematical constructions may be represented by universal properties of diagrams. Consider the Cartesian product X Y of two sets, consisting × as usual of all ordered pairs x,y of elements x X and y Y . The projections h i ∈ ∈ 5 x,y x, x,y y of the product on its ‘axes’ X and Y are functions p : h i 7→ h i 7→ X Y X, q : X Y Y . Any function h : W X Y from a third set W × → × → → × is uniquely determined by its composites p h and q h.
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