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Difference between revisions of "Affinor"

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An [[Affine tensor|affine tensor]] of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011180/a0111801.png" />. Specifying an affinor with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011180/a0111802.png" /> is equivalent to specifying an endomorphism of the vector space according to the rule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011180/a0111803.png" />. To the identity endomorphism there corresponds a unique affinor. The correspondence by which the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011180/a0111804.png" /> is assigned to each affinor realizes an isomorphism between the algebra of affinors and the algebra of matrices. An affinor is sometimes defined in the literature as a general (affine) tensor.
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An [[Affine tensor|affine tensor]] of type  $  (1, 1) $.
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Specifying an affinor with components  $  f _ {j} ^ { i } $
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is equivalent to specifying an endomorphism of the vector space according to the rule  $  v  ^ {i} = f _ {s} ^ { i } v  ^ {s} $.
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To the identity endomorphism there corresponds a unique affinor. The correspondence by which the matrix  $  | f _ {j} ^ { i } | $
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is assigned to each affinor realizes an isomorphism between the algebra of affinors and the algebra of matrices. An affinor is sometimes defined in the literature as a general (affine) tensor.
  
 
====Comments====
 
====Comments====
I.e. one is concerned here with the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011180/a0111805.png" /> of linear algebra.
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I.e. one is concerned here with the isomorphism $  V \otimes V \simeq  \mathop{\rm End} ( V ) $
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of linear algebra.

Revision as of 16:09, 1 April 2020


An affine tensor of type $ (1, 1) $. Specifying an affinor with components $ f _ {j} ^ { i } $ is equivalent to specifying an endomorphism of the vector space according to the rule $ v ^ {i} = f _ {s} ^ { i } v ^ {s} $. To the identity endomorphism there corresponds a unique affinor. The correspondence by which the matrix $ | f _ {j} ^ { i } | $ is assigned to each affinor realizes an isomorphism between the algebra of affinors and the algebra of matrices. An affinor is sometimes defined in the literature as a general (affine) tensor.

Comments

I.e. one is concerned here with the isomorphism $ V \otimes V \simeq \mathop{\rm End} ( V ) $ of linear algebra.

How to Cite This Entry:
Affinor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affinor&oldid=45052
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article