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\section*{exterior algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{superalgebra_and_supergeometry}{}\paragraph*{{Super-Algebra and Super-Geometry}}\label{superalgebra_and_supergeometry}

[[!include supergeometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{explicit_definition}{Explicit definition}\dotfill \pageref*{explicit_definition} \linebreak
\noindent\hyperlink{for_vector_spaces}{For vector spaces}\dotfill \pageref*{for_vector_spaces} \linebreak
\noindent\hyperlink{in_general}{In general}\dotfill \pageref*{in_general} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{over_a_super_vector_space}{Over a super vector space}\dotfill \pageref*{over_a_super_vector_space} \linebreak
\noindent\hyperlink{with_an_inner_product}{With an inner product}\dotfill \pageref*{with_an_inner_product} \linebreak
\noindent\hyperlink{semifree_dgalgebras}{Semi-free dg-algebras}\dotfill \pageref*{semifree_dgalgebras} \linebreak
\noindent\hyperlink{differential_forms__derham_complex}{Differential forms / deRham complex}\dotfill \pageref*{differential_forms__derham_complex} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \textbf{exterior algebra} $\Lambda V$ of a vector space is the [[free object|free]] [[graded-commutative algebra]] over $V$, where the elements of $V$ are taken to be of degree $1$. (That is, the [[forgetful functor]] takes a graded-commutative algebra to its vector space of degree-$1$ elements.)

This construction generalizes to [[group representations]], [[chain complexes]], [[vector bundles]], [[coherent sheaves]], and indeed objects in any [[symmetric monoidal category|symmetric monoidal]] [[linear category|linear categories]] with enough [[colimit|colimits]], where the tensor product distributes over those colimits (as in a [[2-rig]]).

\hypertarget{explicit_definition}{}\subsection*{{Explicit definition}}\label{explicit_definition}

We begin with the construction for vector spaces and then sketch how to generalize it.

\hypertarget{for_vector_spaces}{}\subsubsection*{{For vector spaces}}\label{for_vector_spaces}

Suppose $V$ is a [[vector space]] over a [[field]] $K$. Then the \textbf{exterior algebra} $\Lambda V$ is generated by the elements of $V$ using these operations:

\begin{itemize}%
\item addition and scalar multiplication
\item an associative binary operation $\wedge$ called the \textbf{exterior product} or \textbf{wedge product},

\end{itemize}
subject to these identities:

\begin{itemize}%
\item the identities necessary for $\Lambda V$ to be an [[associative algebra]]
\item the identity $v \wedge v = 0$ for all $v \in V$.

\end{itemize}
It then follows that $\Lambda V$ is a [[graded algebra]] where $\Lambda^p V$ is spanned by $p$-fold wedge products, that is, elements of the form

\begin{displaymath}
v_1 \wedge \cdots \wedge v_p
\end{displaymath}
where $v_1, \dots, v_p \in V$. It also follows that $\Lambda V$ is [[graded commutative algebra|graded commutative]]: that is, if $\omega \in \Lambda^p V$ and $\nu \in \Lambda^q V$, then

\begin{itemize}%
\item $\omega \wedge \nu = (-1)^{p q}\, \nu \wedge \omega$.

\end{itemize}
If $K$ is a field not of [[characteristic]] $2$, we may replace the relations

\begin{equation}
v \wedge v = 0
\label{alternating}\end{equation}
by the relations

\begin{equation}
v \wedge w = - w \wedge v
\label{antisymmetric}\end{equation}
for all $v, w \in V$ (\hyperlink{Grassmann1844}{Grassmann 1844, \S{}37, \S{}55}). If we can divide by $2$, then the relations \eqref{antisymmetric} imply \eqref{alternating}, while the converse holds in any characteristic.

The exterior algebra of a vector space is also called the \textbf{Grassmann algebra} or \textbf{alternating algebra}. It is also denoted $\bigwedge V$, $\bigwedge^\bullet V$, or $Alt V$.

\hypertarget{in_general}{}\subsubsection*{{In general}}\label{in_general}

More generally, suppose $C$ is any [[symmetric monoidal category]] and $V \in C$ is any object. Then we can form the [[tensor power|tensor powers]] $V^{\otimes n}$. If $C$ has countable coproducts we can form the [[coproduct]]

\begin{displaymath}
T V = \bigoplus_{n \ge 0} V^{\otimes n}
\end{displaymath}
(which we write here as a [[direct sum]]), and if the tensor product distributes over these coproducts, $T V$ becomes a [[monoid]] object in $C$, with multiplication given by the obvious maps

\begin{displaymath}
V^{\otimes p} \otimes V^{\otimes q} \to V^{\otimes (p+q)}
\end{displaymath}
This monoid object is called the [[tensor algebra]] of $V$.

The [[symmetric group]] $S_n$ acts on $V^{\otimes n}$, and if $C$ is a [[linear category]] over a [[field]] of [[characteristic]] zero, then we can form the antisymmetrization map

\begin{displaymath}
p_A : V^{\otimes n} \to V^{\otimes n}
\end{displaymath}
given by

\begin{displaymath}
p_A = \frac{1}{n!} \sum_{\sigma \in S_n} sgn(\sigma) \, \sigma
\end{displaymath}
The [[cokernel]] of $1 - p_A$ is called the $n$th \textbf{antisymmetric tensor power} or \textbf{alternating power} $\Lambda^n V$. The coproduct

\begin{displaymath}
\Lambda V = \bigoplus_{n \ge 0} \Lambda^n V
\end{displaymath}
becomes a monoid object called the \textbf{exterior algebra} of $V$.

If $C$ is a linear category over a field of positive characteristic (or more generally, over a [[commutative ring]] in which not every positive integer is invertible, that is which is not itself an [[associative algebra|algebra]] over the [[rational numbers]]), then we need a different construction of $\Lambda^n V$; we define \ldots{} (please complete this!).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{over_a_super_vector_space}{}\subsubsection*{{Over a super vector space}}\label{over_a_super_vector_space}

For $V$ a [[super vector space]], the exterior algebra $\Lambda V$ is often called the \textbf{Grassmann algebra} over $V$. This $\Lambda V$ or $\wedge^\bullet V$ is the [[free object|free]] [[supercommutative algebra|graded commutative]] [[superalgebra]] on $V$.

Explicitly, this is the quotient of the [[tensor algebra]] $T V$ by the [[ideal]] generated by elements of the form

\begin{displaymath}
v \otimes w + (-1)^{deg v \cdot deg w } w \otimes v \, .
\end{displaymath}
The product in this algebra is denoted with a wedge, and called the \textbf{wedge product}. It obeys the relation

\begin{displaymath}
v \wedge w = - (-1)^{deg v \cdot deg w} w \wedge v
  \,.
\end{displaymath}
\hypertarget{with_an_inner_product}{}\paragraph*{{With an inner product}}\label{with_an_inner_product}

If $V$ is equipped with a [[bilinear form]] then there is also the [[Clifford algebra]] on $V$. This reduces to the Grassmann algebra for vanishing bilinear form.

But sometimes it is useful to consider the Grassmann algebra even in the presence of a non-degenerate bilinear form, in which case the inner product still serves to induce identifications between elements of the Grassmann algebra in different degree.

Let $V$ be $\mathbf{R}^3$ equipped with its standard inner product. Then an element of $\Lambda^0 V$ is a scalar (a [[real number]]), an element of $\Lambda^1 V$ may be identified with a vector in the elementary sense, an element of $\Lambda^2 V$ may be identified with a [[bivector]] or [[pseudovector]], and an element of $\Lambda^2 V$ may be identified a [[pseudoscalar]].

More generally, let $V$ be $\mathbf{R}^n$, or indeed any real [[inner product space]]. Then an element of $\Lambda^p V$ is a $p$-vector as studied in [[geometric algebra]]. Using the inner product, we can identify $p$-vectors with $(n-p)$-pseudovectors.

On a [[manifold]] (or [[generalized smooth space]]) $X$, let $T^*X$ be the [[cotangent bundle]] of $X$. Then we may define $\Lambda T^*X$ using the abstract nonsense describe earlier, taking $C$ to be the category of [[vector bundle|vector bundles]] over $X$. Then a [[differential form]] on $X$ is a [[section]] of the vector bundle $\Lambda T^*X$. If $X$ is an oriented (semi)-[[Riemannian manifold]], then we can identify $p$-forms with $(n-p)$-forms using the [[Hodge star]] operator.

\hypertarget{semifree_dgalgebras}{}\subsubsection*{{Semi-free dg-algebras}}\label{semifree_dgalgebras}

A [[semifree dga|semi-free dg-algebra]] is a [[dg-algebra]] whose underlying graded commutative algebra is free, i.e. is an exterior algebra. Examples include in particular [[Chevalley-Eilenberg algebra]]s of [[Lie algebra]]s, of $L_\infty$-[[L-∞ algebra|algebras]] and [[Lie ∞-algebroids]].

\hypertarget{differential_forms__derham_complex}{}\paragraph*{{Differential forms / deRham complex}}\label{differential_forms__derham_complex}

For $X$ a [[manifold]] consider the category of [[module]]s over its [[ring]] of smooth functions $C^\infty(X)$. One such module is $\Omega^1(X) = \Gamma(T^* X)$, the space of smooth sections of the [[cotangent bundle]] of $X$.

The [[deRham complex]] of $X$ is the exterior algebra

\begin{displaymath}
\Omega^\bullet(X) = \bigwedge_{C^\infty(X)} \Gamma(T^* X)
  \,.
\end{displaymath}
This is really a special case of the previous class of examples, as $\Omega^\bullet(X)$ equipped with the [[deRham complex|deRham differential]] is the [[Chevalley-Eilenberg algebra]] of the [[tangent Lie algebroid]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[decomposable differential forms]]


\item [[body]]/[[soul]]


\item [[Fock space]], [[exponential modality]]


\item [[differential form]], [[de Rham complex]]


\item [[Clifford algebra]], [[spin geometry]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The concept originates in

\begin{itemize}%
\item [[Hermann Grassmann]], \emph{[[Ausdehnungslehre]]}, 1844

\end{itemize}
where the graded-commutativity of the exterior product appears in \S{}37, \S{}55.

For the case of [[modules]] over a [[commutative ring]], see

\begin{itemize}%
\item [[Stacks Project]], \href{http://stacks.math.columbia.edu/tag/00DM}{Tag 00DM}.


\item [[Bourbaki]], \emph{Alg\`e{}bre}, chap. III, \S{} 7.



\end{itemize}
Discussion of Grassmann algebras [[internalization|internal]] to any [[symmetric monoidal category]] is on p. 165 of

\begin{itemize}%
\item [[Pierre Deligne]], \emph{[[Catégories Tannakiennes]], [[Grothendieck Festschrift]], vol. II, Birkh\"a{}user Progress in Math. 87 (1990) pp.111-195.}

\end{itemize}
See also at \emph{[[signs in supergeometry]]}.

[[!redirects exterior algebra]] [[!redirects exterior algebras]] [[!redirects Grassmann algebra]] [[!redirects Grassmann algebras]] [[!redirects alternating power]] [[!redirects alternating powers]] [[!redirects exterior power]] [[!redirects exterior powers]] [[!redirects exterior product]] [[!redirects exterior products]] [[!redirects wedge product]] [[!redirects wedge products]]



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