In mathematics, more specifically in

multilinear algebra Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of '' ...

, an alternating multilinear map is a

multilinear map In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function :f\colon V_1 \times \cdots \times V_n \to W\text where V_1,\ldots,V_n and W ar ...

with all arguments belonging to the same vector space (for example, a

bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is lin ...

or a

multilinear form In abstract algebra and multilinear algebra, a multilinear form on a vector space V over a field K is a map :f\colon V^k \to K that is separately ''K''-linear in each of its ''k'' arguments. More generally, one can define multilinear forms on ...

) 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.

Definition

Let

R

be a commutative ring and

V, W

be modules over

R

. A multilinear map of the form

f\colon V^n \to W

is said to be alternating if it satisfies the following equivalent conditions: # whenever there exists

1 \leq i \leq n-1

such that

x_i = x_

then

f(x_1,\ldots,x_n) = 0.

. # whenever there exists

1 \leq i \neq j \leq n

such that

x_i = x_j

then

f(x_1,\ldots,x_n) = 0.

Vector spaces

Let

V, W

be vector spaces over the same field. Then a multilinear map of the form

f\colon V^n \to W

is alternating iff it satisfies the following condition: * if

x_1,\ldots,x_n

are linearly dependent then

f(x_1,\ldots,x_n) = 0

Example

In a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

, the

Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...

is an alternating bilinear map. The

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...

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

x_i + c x_j

for any

j \neq i

and

c

in the base ring

R,

then the value of that map is not changed. Every alternating multilinear map is antisymmetric, meaning that

f(\dots,x_i,x_,\dots)=-f(\dots,x_,x_i,\dots) \quad \text 1 \leq i \leq n-1,

or equivalently,

f(x_,\dots,x_) = (\sgn\sigma)f(x_1,\dots,x_n) \quad \text \sigma\in S_n,

where

S_n

denotes the

permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...

of order

n

and

\sgn\sigma

is the

sign A sign is an Physical object, object, quality (philosophy), quality, event, or Non-physical entity, entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to ...

\sigma.

n!

is a

unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...

in the base ring

R,

then every antisymmetric

n

-multilinear form is alternating.

Alternatization

Given a multilinear map of the form

f : V^n \to W,

the alternating multilinear map

g : V^n \to W

defined by

g(x_1, \ldots, x_n) \mathrel \sum_ \sgn(\sigma)f(x_, \ldots, x_)

is said to be the alternatization of

f.

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 In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, ...

is bilinear. Most notably, the alternatization of any

cocycle In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in group cohomology. In autonomous d ...

is bilinear. This fact plays a crucial role in identifying the second

cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

of a lattice with the group of alternating

s on a lattice.

Notes

References

* * * *{{Cite book , last = Rotman , first = Joseph J. , title = An Introduction to the Theory of Groups , publisher = Springer , series = Graduate Texts in Mathematics , volume = 148 , edition = 4th , year = 1995 , isbn = 0-387-94285-8 , oclc = 30028913 Functions and mappings Mathematical relations Multilinear algebra fr:Application multilinéaire#Application alternée