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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeJun 2nd 2017
   • (edited Jun 2nd 2017)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI made some changes to [[bivector]]. While the idea section is correct (and should be strictly adhered to!) but the previous definition is wrong in general! The previous definition is consistent and used in wikipedia but it misses both the direct relation of bivectors, trivectors and general polyvectors to determinants as well as the standard nontrivial usage of bivectors in analytic geometry wher bivectors define equivalence classes of parallelograms and in particular with a point in space given define an affine plane. If we adhere to wikipedia and not to standard treatments in geometry (e.g. [[M M Postnikov]], Analytic geometry) then we miss the nontriviality of the notion of bivector and its meaning which is more precise than that of a general element in the second exterior power. Bivector in a vector space $V$ is not any element in the second exterior power, but a DECOMPOSABLE vector in the second tensor power -- in general dimension just such elements in $\Lambda^2 V$ have the intended geometric meaning and define vector 2-subspaces and of course affine 2-subspaces if a point in the 2-subspace is given. It is true that every bivector in 2-d or in 3-d space is decomposable, but in dimension 4 this is already not true. Thus the bivectors form a vector space just in the dimensions up to $3$. Similarly, trivectors form a vector space just in the dimensions up to $4$. In the context of differential graded algebras, polyvector fields are usually taken as arbitrary elements in the exterior powers of vector fields.

   I made some changes to bivector. While the idea section is correct (and should be strictly adhered to!) but the previous definition is wrong in general! The previous definition is consistent and used in wikipedia but it misses both the direct relation of bivectors, trivectors and general polyvectors to determinants as well as the standard nontrivial usage of bivectors in analytic geometry wher bivectors define equivalence classes of parallelograms and in particular with a point in space given define an affine plane. If we adhere to wikipedia and not to standard treatments in geometry (e.g. M M Postnikov, Analytic geometry) then we miss the nontriviality of the notion of bivector and its meaning which is more precise than that of a general element in the second exterior power.

   Bivector in a vector space $V$ is not any element in the second exterior power, but a DECOMPOSABLE vector in the second tensor power – in general dimension just such elements in $\Lambda^2 V$ have the intended geometric meaning and define vector 2-subspaces and of course affine 2-subspaces if a point in the 2-subspace is given. It is true that every bivector in 2-d or in 3-d space is decomposable, but in dimension 4 this is already not true. Thus the bivectors form a vector space just in the dimensions up to $3$. Similarly, trivectors form a vector space just in the dimensions up to $4$. In the context of differential graded algebras, polyvector fields are usually taken as arbitrary elements in the exterior powers of vector fields.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 2nd 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexThere should be disambiguation in the entry that the term is used in two senses, one more general, one more restrictive. Both are in use in different contexts.

   There should be disambiguation in the entry that the term is used in two senses, one more general, one more restrictive. Both are in use in different contexts.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 8th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded missing cross-link with _[[multivector]]_ <a href="https://ncatlab.org/nlab/revision/diff/bivector/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/bivector/7">v7</a>, <a href="https://ncatlab.org/nlab/show/bivector">current</a>

   added missing cross-link with multivector

   diff, v7, current