mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, more specifically in

multilinear algebra Multilinear algebra is the study of Function (mathematics), functions with multiple vector space, vector-valued Argument of a function, arguments, with the functions being Linear map, linear maps with respect to each argument. It involves concept ...

, an alternating multilinear map is a

multilinear map Multilinear may refer to: * Multilinear form, a type of mathematical function from a vector space to the underlying field * Multilinear map, a type of mathematical function between vector spaces * Multilinear algebra, a field of mathematics ...

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 linea ...

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 a mo ...

) that is zero whenever any pair of its arguments is equal. This generalizes directly to a module over a

commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...

. The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map of which all arguments belong to the same space.

Definition

Let

R

be a commutative ring and ,

W

be modules over

R

. A multilinear map of the form

f: 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 . # whenever there exists

1 \leq i \neq j \leq n

such that

x_i = x_j

then .

Vector spaces

Let

V, W

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

f: V^n \to W

is alternating if it satisfies the following condition: * if

x_1,\ldots,x_n

are

linearly dependent In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts ...

then .

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 ident ...

, the Lie bracket is an alternating bilinear map. The

determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

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 , 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 \mathrm_n,

where

\mathrm_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 degree

n

and

\sgn\sigma

is the

sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...

of . If

n!

is a unit in the base ring , 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 . 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. A bilinear map can also be defined for ...

is bilinear. Most notably, the alternatization of any

cocycle In mathematics a cocycle is a closed cochain (algebraic topology), 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 gr ...

is bilinear. This fact plays a crucial role in identifying the second cohomology group of a lattice with the group of alternating

s on a lattice.

Notes

References

* * * * * {{cite book , last = Tu , first = Loring W. , year = 2011 , title = An Introduction to Manifolds , publisher = Springer-Verlag New York , isbn = 978-1-4419-7400-6 Functions and mappings Mathematical relations Multilinear algebra fr:Application multilinéaire#Application alternée