Alternating multilinear map

In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (e.g., a bilinear form or a multilinear form) that is zero whenever any pair of arguments is equal. More generally, the vector space may be a module over a commutative ring.

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

DefinitionEdit

A multilinear map of the form   is said to be alternating if it satisfies any of the following equivalent conditions:

  1. whenever there exists   such that   then  .[1][2]
  2. whenever there exists   such that   then  .[1][3]
  3. if   are linearly dependent then  .

ExampleEdit

  • The determinant of a matrix is a multilinear alternating map of the rows or columns of the matrix.

PropertiesEdit

  • If any component xi of an alternating multilinear map is replaced by xi + c xj for any ji and c in the base ring R, then the value of that map is not changed.[3]
  • Every alternating multilinear map is antisymmetric.[4]
  • If n! is a unit in the base ring R, then every antisymmetric n-multilinear form is alternating.

AlternatizationEdit

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 n-multilinear alternating map is n! 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 alsoEdit

NotesEdit

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

ReferencesEdit

  • Bourbaki, N. (2007). Eléments de mathématique. Algèbre Chapitres 1 à 3 (reprint ed.). Springer.CS1 maint: ref=harv (link)
  • Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley.CS1 maint: ref=harv (link)
  • Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. 211 (revised 3rd ed.). Springer. ISBN 978-0-387-95385-4. OCLC 48176673.CS1 maint: ref=harv (link)
  • Rotman, Joseph J. (1995). An Introduction to the Theory of Groups. Graduate Texts in Mathematics. 148 (4th ed.). Springer. ISBN 0-387-94285-8. OCLC 30028913.CS1 maint: ref=harv (link)