Tensor product of modules: Difference between revisions

Revision as of 23:05, 22 December 2020

Revision as of 12:32, 8 January 2021

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To lighten the notation, put <math>E = \Gamma(M, T M)</math> and so <math>E^* = \Gamma(M, T^* M)</math>.<ref>This is actually the ''definition'' of differential one-forms, global sections of <math>T^*M</math>, in Helgason, but is equivalent to the usual definition that does not use module theory.</ref> When ''p'', ''q'' ≥ 1, for each (''k'', ''l'') with 1 ≤ ''k'' ≤ ''p'', 1 ≤ ''l'' ≤ ''q'', there is an ''R''-multilinear map:

:<math>E^p \times {E^*}^q \to E\mathfrak{T}^{p-1} \times {E^*}^_{q-1}, \, (X_1, \dots, X_p, \omega_1, \dots, \omega_q) \mapsto \langle X_k, \omega_l \rangle (X_1,\otimes \dots,cdots\otimes \widehat{X_l},\otimes \dots,cdots\otimes X_p, \otimes \omega_1,\otimes \dotscdots, \widehat{\omega_l},\otimes \dots,cdots\otimes \omega_q)</math>

where <math>E^p</math> means <math>\prod_1^p E</math> and the hat means a term is omitted. By the universal property, it corresponds to a unique ''R''-linear map: