Alternating multilinear map

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In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pair of its arguments is equal. This generalizes directly to a module over a commutative ring.

The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map of which all arguments belong to the same space.

Definition

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Let   be a commutative ring and  ,   be modules over  . A multilinear map of the form   is said to be alternating if it satisfies the following equivalent conditions:

  1. whenever there exists   such that   then  .[1][2]
  2. whenever there exists   such that   then  .[1][3]

Vector spaces

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Let   be vector spaces over the same field. Then a multilinear map of the form   is alternating if it satisfies the following condition:

  • if   are linearly dependent then  .

Example

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In a Lie algebra, the Lie bracket is an alternating bilinear map. The determinant of a matrix is a multilinear alternating map of the rows or columns of the matrix.

Properties

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If any component   of an alternating multilinear map is replaced by   for any   and   in the base ring  , then the value of that map is not changed.[3]

Every alternating multilinear map is antisymmetric,[4] meaning that[1]   or equivalently,   where   denotes the permutation group of degree   and   is the sign of  .[5] If   is a unit in the base ring  , then every antisymmetric  -multilinear form is alternating.

Alternatization

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Given a multilinear map of the form   the alternating multilinear map   defined by   is said to be the alternatization of  .

Properties

  • The alternatization of an  -multilinear alternating map is   times itself.
  • The alternatization of a symmetric map is zero.
  • The alternatization of a bilinear map is bilinear. Most notably, the alternatization of any cocycle is bilinear. This fact plays a crucial role in identifying the second cohomology group of a lattice with the group of alternating bilinear forms on a lattice.

See also

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Notes

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  1. ^ a b c Lang 2002, pp. 511–512
  2. ^ Bourbaki 2007, A III.80, §4
  3. ^ a b Dummit & Foote 2004, p. 436
  4. ^ Rotman 1995, p. 235
  5. ^ Tu 2011, p. 23

References

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  • Bourbaki, N. (2007). Eléments de mathématique. Vol. Algèbre Chapitres 1 à 3 (reprint ed.). Springer.
  • Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley.
  • Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211 (revised 3rd ed.). Springer. ISBN 978-0-387-95385-4. OCLC 48176673.
  • Rotman, Joseph J. (1995). An Introduction to the Theory of Groups. Graduate Texts in Mathematics. Vol. 148 (4th ed.). Springer. ISBN 0-387-94285-8. OCLC 30028913.
  • Tu, Loring W. (2011). An Introduction to Manifolds. Springer-Verlag New York. ISBN 978-1-4419-7400-6.