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テンソル積の成分計算

\{ e_i \}_{i=1}^m \subset V a basis set of V
V = \left \{ \sum_{i=1}^m x^i e_i | x^i \in \mathbb{R} \right \} m-dim vector space
V \cong \{ (x^1, \cdots, x^m ) | x^i \in \mathbb{R}\} =: \mathbb{R}^m

V^* := \{ \alpha : V \to \mathbb{R} \ linear \} dual space of V
e^i( e_j ) := \delta_{ij} dual basis vector of V^*
V^* = \left \{ \sum_{i=1}^m \alpha_i e^i : V \to \mathbb{R} | x_i \in \mathbb{R} \right \}
  i.e.  \sum_{i=1}^m \alpha_i e^i \left( \sum_{j=1}^m x^j e_j \right) = \sum_{i=1}^m \sum_{j=1}^m \alpha_i x^j e^i \left( e_j \right) = \sum_{i=1}^m \alpha_i x^i \in \mathbb{R}
  i.e. \begin{pmatrix} \alpha_1 \ \cdots \ \alpha_m \end{pmatrix} \begin{pmatrix} x^1 \\ \vdots \\ x^m \end{pmatrix}

V = \mathrm{Span}\{ e_i \}_{i=1}^m, W = \mathrm{Span}\{ f_j \}_{j=1}^n m, n-dim vector space
\mathrm{Hom}(V, W) := \{ \phi : V \to W \ linear\}
\phi_{ij}( e_k ) := \delta_{ik}f_j; \ e_i \mapsto f_j, \mbox{ otherwise } 0  canonical basis
\mathrm{Hom}(V, W) = \mathrm{Span}\{ \phi_{ij} \} \cong \mathrm{Mat}(m, n)
  i.e. \sum_{i,j} \alpha^{ij} \phi_{ij} \left( \sum_k x^k e_k \right) = \sum_{i,j,k} \alpha^{ij} x^k \phi_{ij} \left( e_k \right) = \sum_{i,j} \alpha^{ij} x^i f_j
  i.e. \begin{pmatrix} y^1 \\ \vdots \\ y^n \end{pmatrix} = \begin{pmatrix} \alpha^{11} \ \cdots \ \alpha^{1m} \\ \vdots \ \ddots \ \vdots \\ \alpha^{n1} \ \cdots \ \alpha^{nm} \end{pmatrix} \begin{pmatrix} x^1 \\ \vdots \\ x^m \end{pmatrix}
  Rem. e_i, f_j \mbox{ canonical} \Rightarrow \phi_{ij} \cong \mathbb{I}_{ij} = ( \delta_{ij} )\in \mathrm{Mat}(m,n)

V \otimes W := \mathrm{Hom}(V^*, W) tensor product of V and W
v \otimes w : V^* \to W; \alpha \mapsto \alpha(v)w 
  i.e. v = \sum v^i e_i \in V, \quad w = \sum w^j f_j \in W, \quad \alpha = \sum \alpha_k e^k \in V^*
       v \otimes w \left( \alpha \right) = \alpha( v ) w = \left \{ \sum \alpha_k e^k \left( \sum v^i e_i \right) \right \} \sum w^j f_j = \sum \alpha_i v^i w^j f_j
  i.e. \begin{pmatrix} w^1 \\ \vdots \\ w^n \end{pmatrix} \begin{pmatrix} v^1 \ \cdots \ v^m \end{pmatrix} \begin{pmatrix} \alpha_1 \\ \vdots \\ \alpha_m \end{pmatrix}
V \otimes W = \mathrm{Span} \{ e_i \otimes f_j \}
  i.e.  e_i \otimes f_j \left( \sum \alpha_k e^k \right) = \sum \alpha_k e^k( e_i ) f_j = \alpha_i f_j
  i.e.  e_i \otimes f_j \left( e^k \right) = e^k( e_i ) f_j = \delta_{ki} f_j

\mathrm{Hom}(V, W) \cong V^* \otimes W
  i.e. \phi_{ij} \cong e^i \otimes f_j; \alpha^{ij} \cong w^i v^j

L(V^* \times W^*) := \{ \phi : V^* \times W^* \to \mathbb{R} \ bilinear \}
  with \phi_{v w}( v^*, w^* ) := v^*(v) w^*(w) = w^*( v \otimes w( v^* ) ) \mbox{ for } v \in V, w \in W
  i.e. \beta^* w v^* \alpha suppose each vectors are col. orderd
L(V^* \times W^*) = \mathrm{Span}\{ \phi_{ e_i f_j } \}
  i.e. \phi_{ e_i f_j }( e^k, f^l ) = \delta_{ik} \delta_{jl}
  i.e. \phi_{ e_i f_j } \left( \sum \alpha_k e^k, \sum \beta_l f^l \right) = \sum \alpha_k \beta_l \phi_{ e_i f_j } \left( e^k, f^l \right) = \alpha_i \beta_j
L(V^* \times W^*) \cong V \otimes W
  as \phi_{ e_i f_j } \cong e_i \otimes f_j

T_s^r(V) := V^{ \otimes r } \otimes  \left( V^* \right) ^{ \otimes s } := \{ \phi : \left( V^* \right)^r \times V^s \to \mathbb{R} \ multilinear\} (r,s)-tensor space on V
  with v_1 \otimes \cdots \otimes v_r \otimes \beta_1 \otimes \cdots \otimes \beta_r( \alpha_1, \cdots, \alpha_s, w_1, \cdots, w_s ) := \prod \alpha_i(v_i) \beta_j(w_j)
T_s^r(V) = \mathrm{Span} \{ e_{i_1} \otimes \cdots \otimes e_{i_r} \otimes e^{j_1} \otimes \cdots \otimes e^{j_s}\}

T(V) := \bigoplus_{ r, s \geq 0 } T_s^r(V) is algebra
D : T(V) \to T(W) \ linear  is derivative when
(i) T(V)の型を保ち，縮約と可換
(ii) D( t \otimes s ) = Dt \otimes s + t \otimes Ds Leibniz rule
\mathcal{D} := \{ D : derivative \} is a vector space
and a lie-algebra with product [D, D'] = DD' - D'D

\Lambda^k(V) := \{ \alpha : V^k \to \mathbb{R} | skewsymmetric \ multilinear\} k-th exterior power, its elem. called k-form
  where skew-symmetric means : \alpha( x_{ \sigma(1) }, \cdots, x_{ \sigma(k) } ) = \mathrm{sgn} \, \sigma \, \alpha( x_1, \cdots, x_k ) \mbox{ for } \sigma \in S_k
  with \alpha_1 \wedge \cdots \wedge \alpha_k( v_1, \cdots, v_k ) := \det( \alpha_i( v_j ) )
\Lambda^k(V) = \mathrm{Span}\{ e^{i_1} \wedge \cdots \wedge e^{i_k} : i_1 < \cdots < i_k \leq m = \dim V \}
\Lambda^*(V) := \bigoplus_{k=0}^m \Lambda^k(V) is an algebra, called exterior algebra on V
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最終更新：2011年05月06日 16:47
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