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Hodge Theory Victor Guillemin Notes, Spring 1997 Contents 1 Definition and properties of ?. 1 2 Exterior and interior multiplication. 3 3 The case of B symmetric positive definite. 5 4 The case of B symplectic. 6 5 Graded sl(2). 7 6 Hermitian vector spaces. 10 7 Symplectic Hodge theory. 13 8 Excursus on sl(2) modules. 15 9 The strong Lefschetz property. 19 10 Riemannian Hodge theory. 22 11 Kaehler Hodge theory. 23 1 Definition and properties of ?. Let V be an n-dimensional vector space over R and B a bilinear form on V . B induces a bilinear form on ^pV , also denoted by B determined by its value on decomposable elements as B(µ, ν) := det(B(ui; vj )); µ = u1 ^ · · · up; ν = v1 ^ · · · vp: Suppose we also have fixed an element Ω 2 ^nV which identifies ^nV with R. Exterior multiplication then identifies ^n−pV with (^pV )∗ and B maps ^pV ! (^pV )∗. We thus get a composite map ? : ^pV ! ^n−pV 1 characterized by α ^ ?β = B(α, β)Ω: (1) Properties of ?. • Dependence on Ω. If Ω1 = λΩ then ?1 = λ? as follows immediately from the definition. • Dependence on B. Suppose that B1(v; w) = B(v; Jw); J 2 End V: Extend J to an element of ^V by J(v1 ^ · · · vp) := Jv1 ^ · · · ^ Jvp. Thus the extended bilinear forms are also related by B1(µ, ν) = B(µ, Jν) and hence ?1 = ? ◦ J: • Behavior under direct sums. Suppose V = V1 ⊕ V2; B = B1 ⊕ B2; Ω = Ω1 ^ Ω2 under the identification ^V = ^V1 ⊗ ^V2: r s Then for α1; β1 2 ^ V1; α1; β2 2 ^ V2 we have B(α1 ^ α2; β1 ^ β2) = B(α1; β1)B(α2; β2) and (α1 ^α2)^?(β1 ^β2) = B(α1 ^α2; β1 ^β2)Ω = B(α1; β1)Ω1 ^B(α2; β2)Ω2 while α1 ^ ?1β1 = B(α1; β1)Ω1; α2 ^ ?2β1 = B(α2; β2)Ω2: Hence n1−r)s r s ?(!1 ^ !2) = (−1) ?1 !1 ^ ?2!2 for !1 2 ^ V1; !2 2 ^ V2: (n1−r)(n2−s) Since ?1!1 ^ ?2!2 = (−1) ? 2!2 ^ ?1!1 we can rewrite the preceding equation as (n1−r)n2 ?(!1 ^ !2) = (−1) ?2 !2 ^ ?1!1: 2 In particular, if n2 is even we get the simpler looking formula ?(!1 ^ !2) = ?2!2 ^ ?1!1: So, by induction, if V = V1 ⊕ · · · ⊕ Vm is a direct sum of even dimensional subspaces and Ω = Ω1 ^ · · · ^ Ωm then ?(!1 ^ · · · ^ !m) = !m ^ · · · ^ !1; !i 2 ^(Vi): (2) 2 Exterior and interior multiplication. Suppose that B is non-degenerate. For u 2 V we let eu : ^V ! ^V denote ∗ exterior multiplication by u. For γ 2 V we let iγ : ^V ! ^V denote interior multiplication by γ. We can also consider the transposes of these operators with respect to B: y p p−1 ev : ^ V ! ^ V; defined by y p−1 p B(evα, β) = B(α, evβ); α 2 ^ V; β 2 ^ V and y p−1 p iγ : ^ V ! ^ V defined by y p+1 p B(iγα, β) = B(α, iγ β); α 2 ^ V; β 2 ^ V: We claim that y p−1 −1 ev = (−1) ? ev ? (3) and y p −1 iγ = (−1) ? iγ ? (4) on ^pV . Proof of (3). For α 2 ^p−1V; β 2 ^pV we have B(ev ^ α, β)Ω = evα ^ ?β = (−1)p−1α ^ v ^ ?β p−1 −1 = (−1) α ^ ? ? ev ? β p−1 −1 = (−1) B(α, ? ev ? β)Ω: Proof of (4). Let α 2 ^p+1V; β 2 ^pV so that α ^ ?β = 0: 3 We have 0 = iγ(α ^ ?β) p−1 = (iγα) ^ ?β + (−1) α ^ iγ ? β p−1 −1 = (iγα) ^ ?β + (−1) α ^ ?(? iγ?)β so p −1 B(iγα, β)Ω = (−1) B(α, ? iγ ? β)Ω: y y There are alternative formulas for ev and iγ which are useful, and involve dualities between V and V ∗ induced by B. We let h ; i denote the pairing of V and V ∗, so hv; `i denotes the value of the linear function, ` 2 V ∗ on v 2 V . Define the maps op op ∗ L = LB; and L = LB : V ! V by hv; Lwi = B(v; w); hv; Lopwi = B(w; v); v; w 2 V: (5) We claim that y ev = iLopv (6) y iLv = ev (7) Proof. We may suppose that v 6= 0 and extend it to a basis v1; : : : ; vn of V , with v1 = v. Let w1; : : : ; wn be the basis of V determined by B(vi; wj ) = δij : 1 n ∗ Let γ ; : : : ; γ be the basis of V dual to w1; : : : ; wn and set γ := γ1. Then op hwi; L vi = B(v; wi) = δ1i = hwi; γi so γ = Lopv: If J = (j1; : : : ; jp) and K = (k1; : : : ; kp+1 are (increasing) multi-indices then J K B(evv ; w ) = 0 unless k1 = 1 and kr+1 = ir; r = 1; : : : ; p, in which case J K B(evv ; w ) = 1: The same is true for J K B(v ; iγw ): Hence y ev = iγ 4 which is the content of (6). Similarly, let w = w1 and β = L(w) so that iβvj = B(vj ; w1) = δ1j : Then K J B(iβ(v ); w ) = 0 unless k1 = 1 and kr+1 = jr; r = 1; : : : ; p in which case K J B(iβ(v ); w ) = 1 and the same holds for B(vK ; w ^ wJ ). This proves (7). Combining (3) and (6) gives −1 p−1 ? ev? = (−1) iLopv; (8) while combining (4) and (7) gives −1 p ? iLv? = (−1) ev: (9) On any vector space, independent of any choice of bilinear form we always have the identity ∗ iγew + ewiγ = hw; γi; v 2 V; γ 2 V : If γ = Lopv, then hw; γi = B(v; w) so (3) implies y y evew + ewev = B(v; w)I: (10) 3 The case of B symmetric positive definite. In this case it is usual to choose Ω such that kΩk = 1. The only choice left is then of an orientation. Suppose we have fixed an orientation and so a choice of Ω. To compute ? it is enough to compute it on decomposable elements. So let U be a p-dimensional subspace of V and u1 ^· · ·^wp an orthonormal basis of U. Let W be the orthogonal complement of U and let w1; : : : ; wq be an orthonormal basis of W where q := n − p. Then u1 ^ · · · ^ up ^ w1 ^ · · · ^ wq = Ω: We claim that ?(u1 ^ · · · ^ up) = w1 ^ · · · ^ wq: We need only check that B(α, u1 ^ · · · ^ up)Ω = α ^ w1 ^ · · · ^ wq p for α 2 ^ V which are wedge products of ui and wj since u1; : : : ; up; w1; : : : ; wq form a basis of V . Now if any w's are involved in this product decomposition 5 both sides vanish. And if α =1 ^ · · · ^ up then this is the definition of the occurring in the formula. Suppose we have chosen both bases so that = +. Then ?(u1 ^ · · · ^ up) = w1 ^ · · · ^ wq while ?(w1 ^ · · · wq) = u1 ^ · · · up where is the sign of the permutation involved in moving all the w's past the u's. This sign is (−1)p(n−p). We conclude ?2 = (−1)p(n−p) on ^p V: (11) In particular ?2 = (−1)p on ^p V if n is even. (12) 4 The case of B symplectic. Suppose n = 2m and e1; : : : ; em; f1; : : : ; fm is a basis of V with B(ei; fj) = δij ; B(ei; ej ) = B(fi; fj) = 0: We take Ω := e1 ^ f1 ^ e2 ^ f2 · · · ^ em ^ fm which is clearly independent of the choice of basis with the above properties. If we let Vi denote the two dimensional space spanned by ei; fi with Bi the restriction of B to Vi and Ωi := ei ^ fi then we are in the direct sum situation and so can apply (2). So to compute ? in the symplectic situation it is enough to compute it for a two dimensional vector space with basis e; f satisfying B(e; f) = 1; e ^ f = Ω: Now B(e; e) = 0 = e ^ e; B(f; e)Ω = −Ω = f ^ e so ?e = e: Similarly ?f = f: On any vector space the \induced bilinear form" on ^0 is given by B(1; 1) = 1 so ?1 = Ω: 6 On the other hand, B(e; e) B(e; f) 0 1 B(e ^ f; e ^ f) = det = det = 1: B(f; e) B(f; f) −1 0 So ?(e ^ f) = 1: This computes ? is all cases for a two dimensional symplectic vector space. We conclude that ?2 = id (13) first for a two dimensional symplectic vector space and then, from (2), for all symplectic vector spaces. 5 Graded sl(2). We consider the three dimensional graded Lie algebra g = g−2 ⊕ g0 ⊕ g2 where each summand is one dimensional with basis F; H; E respectively and bracket relations [H; E] = 2E; [H; F ] = −2F; [E; F ] = H: For example, g = sl(2) with 0 1 0 0 1 0 E = ; F = ; H = : 0 0 1 0 0 −1 Let V be a symplectic vector space with symplectic form, B and symplectic basis u1; : : : ; um; v1; : : : ; vm so B(ui; uj) = 0; = B(vi; vj ); B(ui; vj ) = δij : Let ! := u1 ^ v1 + · · · + um ^ vm: This element is independent of the choice of symplectic basis. (It is the image in ^2V of B under the identification of ^2V ∗ with ^2V induced by B.) Let E(!) : ^V ! ^V denote the operation of exterior multiplication by !. So E(!) = eui evi : X Let F (!) := E(!)y 7 so y y F (!) = evi eui : X For α 2 ^pV we have, by (3), y y p−1 p−2 eveuα = (−1) (−1) ? veveu ? α = − ? eveu ? α = ?euev ? α so F (!) = ?E(!) ? : (14) 1 m 1 m ∗ Alternatively, if µ ; : : : ; µ ; ν ; : : : ; ν is the basis of V dual to u1; : : : ; vm then F (!) = iνj iµj : (15) X We now prove the Kaehler-Weil identity [E(!); F (!)]α = (p − m)α, α 2 ^pV: (16) Write E(!) = E1 + · · · + Em; Ej := euj evj and F (!) = F1 + · · · Fm; Fj = iνj iµj : Let Vj be the two dimensional space spanned by uj; vj and write α = α1 ^ · · · ^ αp; αj 2 ^Vj : Then Ei really only affects the i-th factor since we are multiplying by an even element: Ei(α) = α1 ^ · · · ^ Eiαi ^ · · · ^ αp and Fi annihilates all but the i-th factor: Fi(α) = α1 ^ · · · ^ Fiαi ^ · · · ^ αp: So if i < j EiFj (α) = Fj Ei(α) = α1 ^ · · · ^ Eiαi ^ · · · ^ Fj αj ^ · · · ^ αp: In other words, [Ei; Fj ] = 0; i 6= j: So [E(!); F (!)]α = α1 ^ · · · ^ [Ei; Fi]αi ^ · · · ^ αp X Since the sum of the degrees of the αi add up to p, it is sufficient to prove (16) for the case of a two dimensional symplectic vector space with symplectic basis u; v .
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    A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a b + ... +c d. The scalars, vectors and tensors are handled as invariant (independent⊗ from the choice⊗ of the coordinate system) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write a = aig , A = aibj + ... + cidj g g i i ⊗ j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k k ik ik a gk ∑ a gk, A bk ∑ A bk ≡ k=1 ≡ k=1 In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e.
  • The Tensor Product and Induced Modules

    The Tensor Product and Induced Modules

    The Tensor Product and Induced Modules Nayab Khalid The Tensor Product The Tensor Product A Construction (via the Universal Property) Properties Examples References Nayab Khalid PPS Talk University of St. Andrews October 16, 2015 The Tensor Product and Induced Modules Overview of the Presentation Nayab Khalid The Tensor Product A 1 The Tensor Product Construction Properties Examples 2 A Construction References 3 Properties 4 Examples 5 References The Tensor Product and Induced Modules The Tensor Product Nayab Khalid The Tensor Product A Construction Properties The tensor product of modules is a construction that allows Examples multilinear maps to be carried out in terms of linear maps. References The tensor product can be constructed in many ways, such as using the basis of free modules. However, the standard, more comprehensive, definition of the tensor product stems from category theory and the universal property. The Tensor Product and Induced Modules The Tensor Product (via the Nayab Khalid Universal Property) The Tensor Product A Construction Properties Examples Definition References Let R be a commutative ring. Let E1; E2 and F be modules over R. Consider bilinear maps of the form f : E1 × E2 ! F : The Tensor Product and Induced Modules The Tensor Product (via the Nayab Khalid Universal Property) The Tensor Product Definition A Construction The tensor product E1 ⊗ E2 is the module with a bilinear map Properties Examples λ: E1 × E2 ! E1 ⊗ E2 References such that there exists a unique homomorphism which makes the following diagram commute. The Tensor Product and Induced Modules A Construction of the Nayab Khalid Tensor Product The Tensor Product Here is a more constructive definition of the tensor product.