Even and odd exterior algebra

Definition

For a module $V$ , the exterior algebra has a canonical parity decomposition

\Lambda V=\Lambda^{\mathrm{even}}V\oplus\Lambda^{\mathrm{odd}}V, \qquad \Lambda^{\mathrm{even}}V=\bigoplus_{j\geq0}\Lambda^{2j}V, \quad \Lambda^{\mathrm{odd}}V=\bigoplus_{j\geq0}\Lambda^{2j+1}V.

This makes $\Lambda V$ a $\mathbb Z/2\mathbb Z$ -graded algebra: wedge multiplication adds parities modulo two.

Multiplication by parity

The multiplication rules are

\Lambda^{\mathrm{even}}V\wedge\Lambda^{\mathrm{even}}V \subseteq\Lambda^{\mathrm{even}}V, \qquad \Lambda^{\mathrm{even}}V\wedge\Lambda^{\mathrm{odd}}V \subseteq\Lambda^{\mathrm{odd}}V,

and

\Lambda^{\mathrm{odd}}V\wedge\Lambda^{\mathrm{odd}}V \subseteq\Lambda^{\mathrm{even}}V.

Consequently, the even part is a subalgebra containing the unit, whereas the odd part is generally not a subalgebra.

For homogeneous elements $a,b$ of parities $|a|,|b|\in\mathbb Z/2\mathbb Z$ , graded commutativity reads

a\wedge b=(-1)^{|a||b|}b\wedge a.

The full integer grading retains more information than parity, but parity is the grading used when $\Lambda V$ is viewed as a superalgebra.

Representations

An action on $V$ by linear maps extends to the exterior algebra. It preserves degree and hence preserves the even and odd summands. Thus

\Lambda^{\mathrm{even}}V \quad\text{and}\quad \Lambda^{\mathrm{odd}}V

are subrepresentations whenever $V$ is a representation.

This parity decomposition resembles the even/odd grading of a Clifford algebra, but the products differ: exterior generators square to zero, while Clifford generators square to the value prescribed by a quadratic form.

Characteristic two

Over a base ring of characteristic $2$ , the sign $(-1)^{|a||b|}$ becomes invisible. The exterior algebra is nevertheless defined using the alternating relations $v\wedge v=0$ ; parity still exists, but it cannot be recovered from signs alone.

References

Nicolas Bourbaki, Algebra I: Chapters 1–3, Springer, 1989, Chapter III. Publisher record.
Pierre Deligne and John W. Morgan, “Notes on supersymmetry (following Joseph Bernstein),” in Quantum Fields and Strings: A Course for Mathematicians, American Mathematical Society, 1999. Publisher record.