Alternating multilinear map

Definition

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

   Example

• 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

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

  Alternatization

• 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.