superalgebra and (synthetic ) supergeometry
The exterior algebra $\Lambda V$ of a vector space is the 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 linear categories with enough colimits, where the tensor product distributes over those colimits (as in a 2-rig).
We begin with the construction for vector spaces and then sketch how to generalize it.
Suppose $V$ is a vector space over a field $K$. Then the exterior algebra $\Lambda V$ is generated by the elements of $V$ using these operations:
subject to these identities:
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
where $v_1, \dots, v_p \in V$. It also follows that $\Lambda V$ is graded commutative: that is, if $\omega \in \Lambda^p V$ and $\nu \in \Lambda^q V$, then
If $K$ is a field not of characteristic $2$, we may replace the relations
by the relations
for all $v, w \in V$ (Grassmann 1844, §37, §55). If we can divide by $2$, then the relations (2) imply (1), while the converse holds in any characteristic.
The exterior algebra of a vector space is also called the Grassmann algebra or alternating algebra. It is also denoted $\bigwedge V$, $\bigwedge^\bullet V$, or $Alt V$.
More generally, suppose $C$ is any symmetric monoidal category and $V \in C$ is any object. Then we can form the tensor powers $V^{\otimes n}$. If $C$ has countable coproducts we can form the coproduct
(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
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 symmetric monoidal $\mathbb{K}$-linear category over a field $\mathbb{K}$ of characteristic zero, then we can form the antisymmetrization map
given by
The cokernel of $1 - p_A$ is called the $n$th antisymmetric tensor power or alternating power $\Lambda^n V$. The coproduct
becomes a monoid object called the 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 algebra over the rational numbers), then we need a different construction of $\Lambda^n V$; we define … (please complete this!).
For $V$ a super vector space, the exterior algebra $\Lambda V$ is often called the Grassmann algebra over $V$. This $\Lambda V$ or $\wedge^\bullet V$ is the free 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
The product in this algebra is denoted with a wedge, and called the wedge product. It obeys the relation
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 $\Lambda^3 V$ 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 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.
A 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 algebras of Lie algebras, of $L_\infty$-algebras and Lie ∞-algebroids.
For $X$ a manifold consider the category of modules 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
This is really a special case of the previous class of examples, as $\Omega^\bullet(X)$ equipped with the deRham differential is the Chevalley-Eilenberg algebra of the tangent Lie algebroid.
The concept originates in
where the graded-commutativity of the exterior product appears in §37, §55.
Textbook account:
For the case of modules over a commutative ring, see
Bourbaki, Algèbre, chap. III, § 7.
Discussion of Grassmann algebras internal to any symmetric monoidal category is on p. 165 of
See also at signs in supergeometry.
Last revised on September 5, 2023 at 15:05:45. See the history of this page for a list of all contributions to it.