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 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.