# Alternatization

In mathematics, more specifically in multilinear algebra, the notion of alternatization (or alternatisation in British English) is used to pass from any map to an alternating map.

An alternating map is a multilinear map (e.g., a bilinear map or a multilinear form) that is equal to zero for every tuple with two adjacent elements that are equal.

## Definitions

### Alternating bilinear map

Let S be a set, A be an abelian group, and ${\displaystyle \alpha :S\times S\to A}$ be a bilinear map. Then ${\displaystyle \alpha }$ is said to be an alternating bilinear map if

${\displaystyle \forall x\in S,\quad \alpha (x,x)=0.}$

### Alternating bilinear form

An alternating bilinear form is a special case of alternating bilinear map. As bilinear forms can be defined as maps between vector spaces or modules, we distinguish two cases.

Vector spaces
Let V be a vector space over a field K, and ${\displaystyle \alpha :V\times V\to K}$ be a bilinear form. Then ${\displaystyle \alpha }$ is said to be an alternating bilinear form if [1][2]
${\displaystyle \forall x\in V,\quad \alpha (x,x)=0.}$
Modules
Let M be a module over a ring R, and ${\displaystyle \alpha :M\times M\to R}$ be a bilinear form. Then ${\displaystyle \alpha }$ is said to be an alternating bilinear form if
${\displaystyle \forall x\in M,\quad \alpha (x,x)=0.}$

### Alternating multilinear form

An alternating multilinear form generalizes the concept of alternating bilinear form to n dimensions. As multilinear forms can be defined as maps between vector spaces or modules, we distinguish two cases.

Vector spaces
Let V be a vector space over a field K, and ${\displaystyle \alpha :V\times V\times ...\times V\to K}$ be a multilinear form. Then ${\displaystyle \alpha }$ is said to be an alternating multilinear form if
${\displaystyle \forall x_{1},x_{2},...,x_{n}\in V,\quad \forall i\in \{1,2,...,n-1\},\quad x_{i}=x_{i+1}\,\implies \,\alpha (x_{1},x_{2},...,x_{n})=0.}$
Modules
Let M be a module over a ring R, and ${\displaystyle \alpha :M\times M\times ...\times M\to R}$ be a multilinear form. Then ${\displaystyle \alpha }$ is said to be an alternating multilinear form if [3]
${\displaystyle \forall x_{1},x_{2},...,x_{n}\in M,\quad \forall i\in \{1,2,...,n-1\},\quad x_{i}=x_{i+1}\,\implies \,\alpha (x_{1},x_{2},...,x_{n})=0.}$

### Alternatization of a bilinear map

Let S be a set, A be an abelian group, and ${\displaystyle \alpha :S\times S\to A}$ be a bilinear map. ${\displaystyle \forall x,y\in S,}$ the alternatization of the map ${\displaystyle \alpha }$ is the map

${\displaystyle \beta :S\times S\to A}$
${\displaystyle (x,y)\mapsto \alpha (x,y)-\alpha (y,x).}$

## Properties

${\displaystyle \forall x,y\in S,\quad \alpha (x,y)+\alpha (y,x)=0}$.
Proof for a bilinear form
${\displaystyle \forall x,y\in V,}$
${\displaystyle \alpha (x+y,x+y)=0}$
${\displaystyle =\alpha (x,x+y)+\alpha (y,x+y)}$
${\displaystyle =\alpha (x,x)+\alpha (x,y)+\alpha (y,x)+\alpha (y,y)}$
${\displaystyle =\alpha (x,y)+\alpha (y,x)}$

• If the characteristic of the ring R is not equal to 2, then every antisymmetric multilinear form is alternating.[5]
• The alternatization of an alternating map is its double.
• There may be non-bilinear maps whose alternatization is bilinear.

## Notes

1. Rotman 1995, page 235.
2. Cohn 2003, page 298.
3. Lang 2002, page 511.
4. Rotman 1995, page 235.
5. Rotman 1995, page 235.

## References

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