線形代数の暗黙の同一視

本編ネタ その他代数

\newcommand{\con}[1]{\mathrm{#1}} % constant \newcommand{\hom}{\mathrm{hom}} % internal hom \newcommand{\In}{\text{ in } } % \newcommand{\R}{ {\bf R } } %

  1. U(\hom(V, W)) = {\bf Vect}(V, W) \In {\bf Set}
  2. \con{Linz}_{(V_1, \cdots, V_n), W}: {\bf MultiVect}( (V_1, \cdots, V_n), W) \to {\bf Vect}(\bigotimes(V_1, \cdots, V_n), W) \In {\bf Set}
  3. \con{tensrep}_{V, W} : \hom(V, W) \to W\otimes V^* \In {\bf Vect}
  4. \con{swtensrep}_{V, W} : \hom(V, W) \to V^*\otimes W \In {\bf Vect}
hom internal hom
Linz linearlization
MultiVec multilinear map of vec. spaces
tensrep tensor representation
swtensrep swapped tensor representation

使う射、すべて \In {\bf Vect} 。

  1. \con{sw} : A \otimes B \to B \otimes A
  2. \con{ev} : A^* \otimes A \to \R
  3. \con{coev} : \R \to A \otimes A^*
  4. \con{swev} : A \otimes A^* \to \R
  5. \con{swcoev} : \R \to A^* \otimes A
  6. \con{icomp} : \hom(V, W)\otimes \hom(W, Z) \to \hom(V, Z)
  7. \con{iid} : \R \to \hom(V, V)
  8. \con{tenscomp} : (Z\otimes W^*) \otimes (W\otimes V^* ) \to Z\otimes V^*
  9. \con{tensid} : \R \to V\otimes V^*
  10. \con{swtenscomp} : (V^* \otimes W) \otimes (W^* \otimes Z) \to V^*\otimes Z
  11. \con{swtensid} : \R \to V^*\otimes V

別名・別記法

sw \sigma swap
ev \varepsilon evaluation
coev \eta coevaluation
icomp ;, \cdot internal composition
iid 1, I internal identity
tenscomp \cdot tensor composition
tensid 1, I tensor identity
swtenscomp \cdot swapped tensor composition
swtensid 1, I swapped tensor identity

関係性

  1. swtensrep = tensrep;sw
  2. swev = sw;ev
  3. swcoev = coev;sw
  4. swtenscomp = (sw \otimes sw); sw ; tenscomp; sw
  5. swtensid = tensid;sw
  6. tensid = iid; tensrep;sw
  7. tensid = coev