Alternating multilinear map

In mathematics, more specifically in multilinear algebra, an alternating map is a multilinear map (e.g., a bilinear map or a multilinear form) that is zero whenever any two adjacent arguments are equal. Equivalently, an alternating map is one whose value changes under permutation of its arguments by the sign of said permutation.

The notion of alternatization (or alternatisation in British English) is used to derive an alternating map from any multilinear map.

Definitions

An $n$-multilinear form $\alpha \colon V\times \cdots \times V\to K$ is said to be alternating if $\alpha (x_{1},x_{2},\ldots ,x_{n})=0$ whenever there exists ${\textstyle 1\leq i\leq n-1}$ such that $x_{i}=x_{i+1}.$

Given a bilinear form $\beta \colon V\times V\to K$, its alternatization is the form defined by $(x,y)\mapsto \beta (x,y)-\beta (y,x).$

Example

• In a Lie algebra, the multiplication is an alternating bilinear map called the Lie bracket.

Properties

• If any distinct pair of components are equal, an alternating multilinear map on them is zero.
• Exchanging any distinct pair of components changes the sign of the value of an alternating multilinear mapping.
• If any component v_i is replaced by v_i + cv_j for any j ≠ i and c in the base ring R, the value of an alternating mapping is not changed.
• Every alternating multilinear mapping is antisymmetric:

  $\forall x,y\in S,\quad \alpha (x,y)+\alpha (y,x)=0.$

Proof for a bilinear form $\forall x,y\in V,$ ${\begin{aligned}0&=\alpha (x+y,x+y)\\&=\alpha (x,x+y)+\alpha (y,x+y)\\&=\alpha (x,x)+\alpha (x,y)+\alpha (y,x)+\alpha (y,y)\\&=\alpha (x,y)+\alpha (y,x)\end{aligned}}$

• If n! is a unit in the base ring R, then every antisymmetric n-multilinear form is alternating.
• The alternatization of an n-multilinear alternating mapping 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 also

• Alternating algebra
• Bilinear map
• Map (mathematics)
• Multilinear algebra
• Multilinear map
• Symmetrization

References

• Cohn, P.M. (2003). Basic Algebra: Groups, Rings and Fields. Springer. ISBN 1-85233-587-4. OCLC 248833275.
• 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.