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Survey on exterior algebra and differential forms Daniel Grieser 16. Mai 2013 Inhaltsverzeichnis 1 Exterior algebra for a vector space 1 1.1 Alternating forms, wedge and interior product . 1 1.2 A scalar product enters the stage . 3 1.3 Now add an orientation: Volume element, Hodge ∗ .................. 4 1.4 Formulas in an arbitrary basis . 5 1.5 Modifications for not positive definite inner product . 6 2 Differential forms 7 2.1 Pointwise ('tensorial') constructions . 7 2.2 Integration . 8 2.3 Derivative operations . 9 1 Exterior algebra for a vector space Let V be an n-dimensional real vector space. Whenever needed, we let e1; : : : ; en be a basis of V and e1; : : : ; en its dual basis. At first reading you may leave out the parts on Hodge ∗ and non-positive definite metrics. 1.1 Alternating forms, wedge and interior product k 1. Let k 2 N0.A k-multilinear form on V is a map ! : V ! R which is linear in each entry, i.e. 0 0 !(av1 + bv1; v2; : : : ; vk) = a !(v1; v2; : : : ; vk) + b !(v1; v2; : : : ; vk) 0 for all v1; v1; v2; : : : ; vk 2 V and a; b 2 R, and similarly for the other entries. The form is called alternating if it changes sign under interchange of any two vectors: !(v1; : : : ; vi; : : : ; vj; : : : ; vk) = −!(v1; : : : ; vj; : : : ; vi; : : : ; vk) Equivalent conditions (to alternating) are: !(v1; : : : ; vk) = 0 if any two of the vi are the same. Or: !(vσ(1); : : : ; vσ(k)) = sign(σ)!(v1; v2; : : : ; vk) for all permutations σ of f1; : : : ; kg. The space of alternating k-multilinear forms on V is denoted ΛkV ∗. This is a vector space with basis feI : jIj = kg, where I runs over subsets of f1; : : : ; ng with k elements and I i1 ik e := e ^ · · · ^ e for I = fi1 < i2 < ··· < ikg; with ^ defined below, or explicitly: I I e (ej1 ; : : : ; ejk ) = δJ for J = fj1 < ··· < jkg f1;2g 1 2 1 2 1 For example, e = e ^ e satisfies (e ^ e )(e1; e2) = 1, and this implies that (e ^ 2 1 2 P i P j e )(e2; e1) = −1 and all other (e ^ e )(ej1 ; ej2 ) are zero, hence for v = v ei; w = w ej we get (e1 ^ e2)(v; w) = v1w2 − v2w1. 1 k ∗ n n ∗ k ∗ It follows that Λ V has dimension k , in particular dim Λ V = 1, and Λ V = f0g if k > n. Also Λ1V ∗ = V ∗ and Λ0V ∗ = R.1 We also write k = deg ! if ! 2 ΛkV ∗ and call k the degree of the form !. 2. Wedge (or exterior) product: For ! 2 ΛkV ∗, ν 2 ΛlV ∗ define ! ^ ν 2 Λk+lV ∗ by X (! ^ ν)(v1; : : : ; vk+l) = sign(σ) !(vσ(1); : : : ; vσ(k))ν(vσ(k+1); : : : ; vσ(k+l)) σ where the sum runs over all permutations σ of f1; : : : ; k + lg preserving the order in the first and second 'block', i.e. satisfying σ(1) < ··· < σ(k), σ(k + 1) < ··· < σ(k + l). For example, for k = l = 1 (! ^ ν)(v; w) = !(v)ν(w) − !(w)ν(v) Rule: ! ^ ν = (−1)deg !·deg ν ν ^ ! For this property one says that ^ is 'graded commutative' (in the physics literature also 'super commutative'). Also, ^ is bilinear and associative. Remark (Relation to cross product in R3): The wedge product generalizes the cross product in the following sense. If V = R3 then dim Λ1R3 = dim Λ2R3 = 3. So we have identifications (isomorphisms) 1 3 3 1 2 3 Λ R ! R ;! = !1e + !2e + !3e 7! (!1;!2;!3) 2 3 3 2 3 3 1 1 2 Λ R ! R ; µ = µ1e ^ e + µ2e ^ e + µ3e ^ e 7! (µ1; µ2; µ3) 1 3 P i P j Now for !; ν 2 Λ R with ! = !ie , ν = νje we have 2 3 3 1 1 2 ! ^ ν = (!2ν3 − !3ν2)e ^ e + (!3ν1 − !1ν3)e ^ e + (!1ν2 − !2ν1)e ^ e so if !; ν are identified with vectors as in the first line, then ! ^ ν corresponds (as in the second line) to the cross product of these vectors. 3. Let v 2 V . The interior product with v is the linear operator k ∗ k−1 ∗ ιv :Λ V ! Λ V ;! 7! !(v; : : : ) that is, (ιv!)(v2; : : : ; vk) = !(v; v2; : : : ; vk) ('plug in v in the first slot'). Here k 2 N, but we 0 ∗ also define ιv = 0 on Λ V . Clearly, ιv depends linearly on v. With wedge products it behaves as follows (as a 'super derivation'): deg ! ιv(! ^ ν) = (ιv!) ^ ν + (−1) ! ^ (ιvν) (1) 3 P i For example, in R , if v = i v ei then 1 2 3 1 2 3 2 3 1 3 1 2 ιv(e ^ e ^ e ) = v e ^ e + v e ^ e + v e ^ e (2) (of course one could write −v2e1 ^ e3 for the middle term) 4. Behavior under maps: A linear map A : V ! W defines the pull-back map ∗ k ∗ k ∗ ∗ A :Λ W ! Λ V ; (A !)(v1; : : : ; vk) := !(Av1; : : : ; Avk) (3) 1 0 By definition, V = R, and a linear map R ! R is determined by its value at 1. 2 k ∗ where ! 2 Λ W , v1; : : : ; vk 2 V . For k = 1 this is also called the dual (or transpose) map A∗ : W ∗ ! V ∗. Pullback behaves naturally with wedge product A∗(! ^ ν) = A∗! ^ A∗ν ∗ ∗ and with interior product: ιv(A !) = A (ιA(v)!), as follows directly from the definitions. For V = W and k = n pullback relates to the determinant as follows: A∗ = (det A) Id on ΛnV ∗. Explicitly, this means !(Av1; : : : ; Avn) = (det A)!(v1; : : : ; vn) which follows directly from the facts that ! is multilinear and alternating, and the Leibniz formula for the determinant. 1.2 A scalar product enters the stage From now on assume that a scalar product is given on V , that is, a bilinear, symmetric, positive definite2 form g : V × V ! R. We also write hv; wi instead of g(v; w). This defines some more structures: 1. Basic geometry: The scalar product allows us to talk about lenghts of vectors and angles between non-zero vectors: g(v; w) jvj = pg(v; v); (v; w) = arccos \ jvj · jwj 2. Using the scalar product on V we get a map g# : V ! V ∗; v 7! g(v; ·) Since g is non-degenerate, this map is injective, hence bijective (since dim V = dim V ∗ < 1). The inverse of g# is called g[ : V ∗ ! V Therefore, we may identify vectors and linear forms (but we do this only when necessary).3 3. Using this identification, we get a scalar product on V ∗, which we also denote by h ; i: hα; βi := hg[(α); g[(β)i for α; β 2 V ∗. 4. More generally, we get a scalar product on ΛkV ∗ for each k. It is easiest to define it by the property: I k ∗ If e1; : : : ; en are orthonormal then the basis fe : jIj = kg of Λ V is orthonormal: P I P J P 1 k 1 k In other words, h I aI e ; J bJ e i := I aI bI . Then for arbitrary v ; : : : v ; w ; : : : ; w 2 V ∗ one has4 hv1 ^ · · · ^ vk; w1 ^ · · · ^ wki = det(hvi; wji) (4) This formula also shows that one obtains the same scalar product if one uses a different orthonormal basis in the definition. 2Everything can be done in the more general case that g is only non-degenerate, but one needs to be careful with the signs, see Section 1.5. 3The fact that g# is surjective, i.e. that every linear form on V can be represented by a vector using the scalar product, is sometimes called the Riesz lemma. It holds more generally when (V; g) is a Hilbert space, that is, if V is allowed to be infinite-dimensional but required to be complete with the norm defined by g. 4Proof: By definition this holds if all vi, wj are taken from the basis vectors e1; : : : ; en. Then it holds in general since both sides are multilinear in the 2k entries v1; : : : vk; w1; : : : ; wk. 3 1.3 Now add an orientation: Volume element, Hodge ∗ Now assume that on V a scalar product and an orientation is given. 1. The Hodge ∗ operator is the unique linear map (for each k) ∗ :ΛkV ∗ ! Λn−kV ∗ with the property that5 ∗ (e1 ^ · · · ^ ek) = ek+1 ^ · · · ^ en for any oONB, (5) 1 n that is, for any oriented orthonormal basis (oONB) e1; : : : ; en with dual basis e ; : : : ; e . Intuition: k-forms e1 ^ · · · ^ ek correspond to k-dimensional subspaces W = spanfe1; : : : ; ekg of V ∗. Then ∗(e1 ^ · · · ^ ek) corresponds to the orthogonal complement of W . Of course not every form can be written in this way, but using linearity ∗ is defined when it is defined on forms of this type. So one can say:6 • Alternating multilinear forms are a 'linear extension' of the notion of vector subspace. • Then ∗ corresponds to orthogonal complement. 2. Define the volume element (or volume form) of V as dvol = ∗1; dvol 2 ΛnV ∗: Why is this reasonable? Because for any oONB e1; : : : ; en we have, by definition of ∗, dvol = e1 ^ · · · ^ en and therefore dvol(e1; : : : ; en) = 1 for any oONB. (6) So the volume of a 'unit cube' is one, as it should be. 3. Properties of ∗: ! ^ ∗ν = h!; νi dvol for !; ν 2 ΛkV ∗ (7) Also, if ν is fixed then the validity of (7) for all ! defines ∗ν. (7) can easily be checked on basis elements, and then extends by linearity.7 ∗ (ei1 ^ · · · ^ eik ) = sign(σ)ej1 ^ · · · ^ ejn−k for oONB (8) where fj1; : : : ; jn−kg = f1; : : : ; ngnfi1; : : : ; ikg and σ is the permutation sending (1; : : : ; n) 7! 8 (i1; : : : ; ik; j1; : : : ; jn−k).
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  • CLIFFORD ALGEBRAS Property, Then There Is a Unique Isomorphism (V ) (V ) Intertwining the Two Inclusions of V

    CLIFFORD ALGEBRAS Property, Then There Is a Unique Isomorphism (V ) (V ) Intertwining the Two Inclusions of V

    CHAPTER 2 Clifford algebras 1. Exterior algebras 1.1. Definition. For any vector space V over a field K, let T (V ) = k k k Z T (V ) be the tensor algebra, with T (V ) = V V the k-fold tensor∈ product. The quotient of T (V ) by the two-sided⊗···⊗ ideal (V ) generated byL all v w + w v is the exterior algebra, denoted (V ).I The product in (V ) is usually⊗ denoted⊗ α α , although we will frequently∧ omit the wedge ∧ 1 ∧ 2 sign and just write α1α2. Since (V ) is a graded ideal, the exterior algebra inherits a grading I (V )= k(V ) ∧ ∧ k Z M∈ where k(V ) is the image of T k(V ) under the quotient map. Clearly, 0(V )∧ = K and 1(V ) = V so that we can think of V as a subspace of ∧(V ). We may thus∧ think of (V ) as the associative algebra linearly gener- ated∧ by V , subject to the relations∧ vw + wv = 0. We will write φ = k if φ k(V ). The exterior algebra is commutative | | ∈∧ (in the graded sense). That is, for φ k1 (V ) and φ k2 (V ), 1 ∈∧ 2 ∈∧ [φ , φ ] := φ φ + ( 1)k1k2 φ φ = 0. 1 2 1 2 − 2 1 k If V has finite dimension, with basis e1,...,en, the space (V ) has basis ∧ e = e e I i1 · · · ik for all ordered subsets I = i1,...,ik of 1,...,n . (If k = 0, we put { } k { n } e = 1.) In particular, we see that dim (V )= k , and ∅ ∧ n n dim (V )= = 2n.