Элементы теории симметрии. Часть I. Кирсанов А.А. - 87 стр.

Òåíçîðíàÿ àëãåáðà íàä êîìïëåêñíûì åâêëèäîâûì ïðîñòðàíñòâîì                                                                     87


     T ( p , q ) = Tβ1 ...1 β q p ⋅ Ψα11...αqp
                                     α ...α             β ...β


èìååì

     T ( p − 1, q − 1) = Tβ1 ...1 βq p Spki e α1 ⊗ ... ⊗ e αi ⊗ ... ⊗ e α p ⊗
                                                                  1                        i                 p
                                                  α ...α


        β           β           β
     ⊗~
      e 1 ⊗ ... ⊗ ~
                  e k ⊗ ... ⊗ ~
                              e q =
            1                            k                    q

                                     1                     i −1          i +1                         p
                α ...α
     = Tβ1 ...1 β q p δβαki e α1 ⊗ ... ⊗ e αi −1 ⊗ e αi +1 ⊗ ... ⊗ e α p ⊗
        β           β        β              β
     ⊗~
      e 1 ⊗ ... ⊗ ~
                  e k −1 ⊗ ~
                           e k +1 ⊗ ... ⊗ ~
                                          e q=
            1                        k −1              k +1                     q

                                              1                   i −1              i +1                    p
                α ...α       σα      ...α
     = Tβ1 ...1 βk −i1−σβ
                       1    i +1
                          k +1 ...β q
                                      p
                                        e α1 ⊗ ... ⊗ e αi −1 ⊗ e αi +1 ⊗ ... ⊗ e α p ⊗
        β           β        β              β
     ⊗~
      e 1 ⊗ ... ⊗ ~
                  e k −1 ⊗ ~
                           e k +1 ⊗ ... ⊗ ~
                                          e q,                                                                       (3.6.7)
            1                        k −1              k +1                     q

ãäå â ïðàâîé ÷àñòè ïðîèçâîäèòñÿ ñóììèðîâàíèå ïî                                                           σ , ñîãëàñíî ïðàâèëó
Ýéíøòåéíà. Îïåðàöèÿ                           Spki íàçûâàåòñÿ ñâ¸ðòûâàíèåì.
     Òàêèì                    îáðàçîì,                        êîîðäèíàòû                          ñâ¸ðíóòîãî         òåíçîðà
T ′ = T ( p − 1, q − 1) âûðàæàþòñÿ ÷åðåç êîîðäèíàòû èñõîäíîãî òåíçîðà
T ( p , q ) ïî ôîðìóëå
            α ...α α          ...α                     α ...α             α ...α σα            ...α
     Tβ′1 ...1βk −1iβ−1k +1i +...1βq p = δβαki Tβ1 ...1 βq p = Tβ1 ...1 βk −i1−σβ
                                                                               1    i +1
                                                                                  k +1 ...β q
                                                                                              p
                                                                                                .                    (3.6.8)




        §3.7. Ñèììåòðè÷åñêèå è àíòèñèììåòðè÷åñêèå òåíçîðû

     Ðàññìîòðèì òåíçîð                                T ( p ,0 ) âèäà
             1           2                        p
     ∑ x j ⊗ x j ⊗ ... ⊗ x j ,
        j
                                                                                                                     (3.7.1)