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 map

Let S be a set, A be an abelian group, given a map $\alpha: S \times S \to A$, $\alpha$ is said to be an alternating map if

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

Alternating bilinear map

An alternating bilinear map is a bilinear map that is also an alternating map.

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 $\alpha: V \times V \to K$ be a bilinear form. Then $\alpha$ is said to be an alternating bilinear form if ^[1][2]

$\forall x \in V, \quad \alpha(x,x) = 0.$

Modules

Let M be a module over a ring R, and $\alpha: M \times M \to R$ be a bilinear form. Then $\alpha$ is said to be an alternating bilinear form if

$\forall x \in M, \quad \alpha(x,x) = 0.$

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 $\alpha: V \times V \times \ldots \times V \to K$ be a multilinear form. Then $\alpha$ is said to be an alternating multilinear form if

$\forall x_1,x_2,\ldots,x_n \in V, \quad \forall i \in \{1,2,\ldots,n-1\}, \quad x_i = x_{i+1} \, \implies \, \alpha(x_1,x_2,\ldots,x_n) = 0.$

Modules

Let M be a module over a ring R, and $\alpha: M \times M \times \ldots \times M \to R$ be a multilinear form. Then $\alpha$ is said to be an alternating multilinear form if ^[3]

$\forall x_1,x_2,\ldots,x_n \in M, \quad \forall i \in \{1,2,\ldots,n-1\}, \quad x_i = x_{i+1} \, \implies \, \alpha(x_1,x_2,\ldots,x_n) = 0.$

Alternatization of a bilinear map

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

$\beta: S \times S \to A$

$(x,y) \mapsto \alpha(x,y) - \alpha(y,x).$

Example

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

Properties

• Every alternating multilinear form is antisymmetric:^[4]

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

Proof for a bilinear form $\forall x,y \in V,$ $0 = \alpha(x+y,x+y)$    $= \alpha(x,x+y) + \alpha(y,x+y)$    $= \alpha(x,x) + \alpha(x,y) + \alpha(y,x) + \alpha(y,y)$    $= \alpha(x,y) + \alpha(y,x)$

• 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.
• 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.
• There may be non-bilinear maps whose alternatization is bilinear.

See also

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

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

• 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 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 148 (4th ed.). Springer. ISBN 0-387-94285-8. OCLC 30028913.