What to call this extension of a linear transformation to the whole exterior algebra, defined by $F(A\wedge B)=F(A)\wedge B+A\wedge F(B)$

Question

Given a linear transformation $f$ on a vector space $V$, we can extend it to the exterior algebra $\Lambda V$ by defining $F(A\wedge B)=F(A)\wedge F(B)$, or

$$F(1)=1$$

$$F(a)=f(a)$$

$$F(a\wedge b)=f(a)\wedge f(b)$$

$$F(a\wedge b\wedge c)=f(a)\wedge f(b)\wedge f(c)$$

$$\vdots$$

This is sometimes called the "outermorphism" of $f$; or its restriction to a particular grade, an "exterior power" of $f$.

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But some situations (such as the rotational inertia tensor) suggest a different extension, defined by $F(A\wedge B)=F(A)\wedge B+A\wedge F(B)$, or

$$F(1)=0$$

$$F(a)=f(a)$$

$$F(a\wedge b)=f(a)\wedge b+a\wedge f(b)$$

$$F(a\wedge b\wedge c)=f(a)\wedge b\wedge c+a\wedge f(b)\wedge c+a\wedge b\wedge f(c)$$

$$\vdots$$

Does this extension have a standard name? If not, what do you think it should be called?

If nothing better appears, I'm considering "additive extension", as opposed to the outermorphism which is the "multiplicative extension".

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Using the concepts from this answer of mine, the extension's restriction to grade $k$ can be written as

$$f\wedge\frac{(\wedge\text{id})^{k-1}}{(k-1)!},$$

the exterior product of $f$ with an exterior power of the identity.

Answer 1

Your construction appears in various places but doesn't seem to have a standard name. In Sergei Winitzki's book "Linear Algebra via Exterior Products" it is called the "1-linear extension of $f$ to $\Lambda(V)$" (see page 131 of his book). One can generalize it and define the $k$-linear extension of $f$ to $\Lambda^m(V)$ which acts on $v_1 \wedge \dots \wedge v_m$ as a sum of ${m \choose k}$ terms which are obtained from $v_1 \wedge \dots \wedge v_m$ by choosing a subset $I \subseteq [m]$ of size $k$ and replacing each $v_i$ for $i \in I$ with $f(v_i)$. For example, the $m - 1$-extension of $f$ to $\Lambda^m(V)$ acts by

$$ v_1 \wedge \dots \wedge v_m \mapsto \\v_1 \wedge f(v_2) \wedge \dots \wedge f(v_m) + f(v_1) \wedge v_2 \wedge f(v_3) \wedge \dots \wedge f(v_m) + \dots + f(v_1) \wedge \dots \wedge f(v_{m-1}) \wedge v_m. $$

Such expressions arise naturally by taking derivatives of the full exterior power map (which you call outermorphism). They are also discussed in Chapter 6 of Werner Greub's "Multilinear Algebra" under the framework of "mixed exterior algebra" which is $$\Lambda(V^{*},V) := \Lambda(V^{*}) \otimes \Lambda(V) \cong \Lambda(V)^{*} \otimes \Lambda(V) \cong \operatorname{Hom}(\Lambda(V), \Lambda(V)). $$

In Greub's notation, the action of your map on $\Lambda^m(V)$ is denoted by $$ f \square \underbrace{\operatorname{id} \square \cdots \square \operatorname{id}}_{m - 1 \textrm{ times}}. $$