---- > [!definition] Definition. ([[algebra of alternating multilinear forms]]) >The **algebra of alternating multilinear forms** on a [[vector space]] $V$ is the [[graded algebra|graded]] [[algebra]] $\text{Alt}^{\bullet}(V)=\bigoplus_{\ell \geq 0} \text{Alt}^{\ell}(V),$ where $\text{Alt}^{\ell}(V)$ is the space of [[alternating multilinear map|alternating multilinear]] $\ell$-forms on $V$, with multiplication determined by the **wedge product of alternating multilinear forms**: if $\omega \in \text{Alt}^{p}(V)$ and $\eta\in \text{Alt}^{q}(V)$, then $\omega \wedge \eta$ is defined (up to scaling convention) by $\omega \wedge \eta = \frac{1}{k!\ell!}\text{Alternize}(\omega \otimes \eta).$ where $\text{Alternize}$ denotes [[the alternization function|the alternization function]]. > There is no confusion with the notion of 'wedge product of multivectors' found in [[exterior algebra]]: [[algebra of multilinear forms|recalling]] that there is a canonical [[isomorphism]] $(V^{*})^{\otimes \ell} \cong \text{Mult}^{\ell}(V)$ for each $\ell$, and indeed that $\mathbb{T}^{\bullet}(V^{*}) \cong \text{Mult}^{\bullet}(V)$, in [[quotient module|quotienting]] both sides by the [[group action|action]] of the [[alternating group]] one obtains a canonical identification $\Lambda^{\bullet}(V^{*}) \cong \text{Alt}^{\bullet}(V)$ under which the 'wedge products' defined in both places agree... [[the exterior algebra of a dual vector space is canonically identified with that of alternating multilinear forms|see here]] > Thus, the notation $\Lambda^{\ell}(V)$ is sometimes (but hopefully uncommonly in these notes) also used to denote the space $\text{Alt}^{\ell}(V)$ of alternating $\ell$-forms on $V$, and alternating $\ell$-forms are sometimes called **alternating tensors**. > > [!note] Note. > Because of the canonical identification with $\Lambda^{\bullet}(V^{*})$, many of the properties of $\text{Alt}^{\bullet}(V)$ are inherited from the discussions related to general [[exterior algebra|exterior algebras]]. So it is wise to check both spots when reviewing. ^note - [ ] but the 'see here' note needs to be cleaned up (good knowledge consolidation when i do that) > [!basicproperties] > - Note that many of the properties below are more clearly observed in the [[exterior algebra|abstract setting]] > - [[basic properties of the wedge product]] ^754c47 >- [[wedge product is associative]] ^properties ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```