ExteriorAlgebra

Exterior algebra, also known as the Grassmann algebra or the wedge algebra, is an algebraic system that extends the concept of vector spaces and provides a framework for studying multilinear algebra and differential forms. The exterior algebra is built upon the exterior product (or [[WedgeProduct|wedge product]]), which is an antisymmetric, bilinear operation that combines vectors or differential forms.

Given a vector space $V$ over a field $F$, the exterior algebra of $V$, denoted as $\Lambda(V)$, is constructed by taking the direct sum of all the exterior powers of $V$:

$$ \Lambda(V) = \Lambda^0(V) \oplus \Lambda^1(V) \oplus \Lambda^2(V) \oplus \cdots \oplus \Lambda^n(V) $$

Here, $n$ is the dimension of the vector space $V$, and $\Lambda^k(V)$ denotes the $k$-th exterior power of $V$. The elements of $\Lambda^k(V)$ are called $k$-vectors or $k$-forms. The exterior product of two $k$-forms is a $(k+1)$-form.

The exterior algebra has some important properties:

1. Antisymmetry: For any two elements $u$ and $v$ in $V$, their exterior product satisfies:

$$ u \wedge v = - (v \wedge u) $$

2. Bilinearity: The exterior product is linear in each argument, so for any scalar $a$, $b \in F$ and vectors $u$, $v$, $w \in V$, we have:

$$ (a * u + b * v) \wedge w = a * (u \wedge w) + b * (v \wedge w) $$

3. Graded Associativity: Although the exterior product is not associative in general, the exterior algebra still obeys a modified form of associativity called graded associativity:

$$ (u \wedge v) \wedge w = u \wedge (v \wedge w) $$

for any $u$, $v$, and $w$ in the exterior algebra.

Exterior algebras are widely used in various areas of mathematics, including differential geometry, topology, and mathematical physics.