9.6 Symmetric and antisymmetric multilinear forms

9.6 Symmetric and antisymmetric multilinear forms

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The symmetric bilinear forms lie in $S^2(V^{\ast })\subset V^{\ast }\otimes V^{\ast }$ and correspond to symmetric matrices. Elements of $V^{\ast }$ give linear functions on $V,$ and one can get quadratic functions on $V$ from elements $B\in S^2(V^{\ast })$ by taking

$$v\in V\to B(v,v)=\mathbf{v}\cdot \mathbf{Bv}$$

Equivalently, in terms of tensor products, one gets quadratic functions as the product of linear functions by taking

$$({\alpha }_1,{\alpha }_2)\in V^{\ast }\times V^{\ast }\to \frac{1}{2}({\alpha }_1\otimes {\alpha }_2+{\alpha }_2\otimes {\alpha }_1)\in S^2(V^{\ast })$$

and then evaluating at $v\in V$ to get the number

$$\frac{1}{2}({\alpha }_1(v){\alpha }_2(v)+{\alpha }_2(v){\alpha }_1(v))={\alpha }_1(v){\alpha }_2(v)$$

This multiplication can be extended to a product on the space

$$S^{\ast }(V^{\ast })={\oplus }_n{S}^n(V^{\ast })$$

(called the space of symmetric multilinear forms) by defining

$$\begin{array}{rl}({\alpha }_1\otimes \cdots \otimes {\alpha }_j)({\alpha }_{j+1}\otimes \cdots \otimes {\alpha }_n)& =P^{+}({\alpha }_1\otimes \cdots \otimes {\alpha }_n)\\ & \equiv \frac{1}{n!}\sum _{\sigma \in S_n}{\alpha }_{\sigma (1)}\otimes \cdots \otimes {\alpha }_{\sigma (n)}\end{array}\text{\hspace{1em}(9.3)}$$

One can show that $S^{\ast }(V^{\ast })$ with this product is isomorphic to the algebra of polynomials on $V.$ . For a simple example of how this works, take $v_j\in V^{\ast }$ to be the jth coordinate function. Then the correspondence between monomials in $v_j$ and elements of $S^{\ast }(V^{\ast })$ is given by

$$v_j^n\leftrightarrow \underset{\text{ n - times }}{\underbrace{(v_j\otimes v_j\otimes \cdots \otimes v_j)}}\text{\hspace{1em}(9.4)}$$

Both sides can be thought of as the same function on $V_{;}$ , given by evaluating the jth coordinate of $v\in V$ and multiplying it by itself n-times.

We will later find useful the fact that $S^{\ast }(V^{\ast })$ and $(S^{\ast }(V){)}^{\ast }$ are isomorphic, with the tensor product

$$\left({\alpha }_1\otimes \cdots \otimes {\alpha }_j\right)\in S^{\ast }(V^{\ast })$$

corresponding to the linear map

$${\mathbf{e}}_{i_1}\otimes \cdots \otimes {\mathbf{e}}_{i_j}\to {\alpha }_1({\mathbf{e}}_{i_1})\dots {\alpha }_j({\mathbf{e}}_{i_j})$$

Antisymmetric bilinear forms lie in ${\Lambda }^2(V^{\ast })\subset V^{\ast }\otimes V^{\ast }$ and correspond to antisymmetric matrices. A multiplication (called the “wedge product”) can be defined on $V^{\ast }$ that takes values in ${\Lambda }^2(V^{\ast })$ by

$$({\alpha }_1,{\alpha }_2)\in V^{\ast }\times V^{\ast }\to {\alpha }_1\wedge {\alpha }_2=\frac{1}{2}({\alpha }_1\otimes {\alpha }_2-{\alpha }_2\otimes {\alpha }_1)\in {\Lambda }^2(V^{\ast })\text{\hspace{1em}(9.5)}$$

This multiplication can be extended to a product on the space

$${\Lambda }^{\ast }(V^{\ast })={\oplus }_n{\Lambda }^n(V^{\ast })$$

(called the space of antisymmetric multilinear forms) by defining

$$\begin{array}{rl}({\alpha }_1\otimes \cdots \otimes {\alpha }_j)\wedge ({\alpha }_{j+1}\otimes \cdots \otimes {\alpha }_n)& =P^{-}({\alpha }_1\otimes \cdots \otimes {\alpha }_n)\\ & \equiv \frac{1}{n!}\sum _{\sigma \in S_n}(-1{)}^{|\sigma |}{\alpha }_{\sigma (1)}\otimes \cdots \otimes {\alpha }_{\sigma (n)}\end{array}\text{\hspace{1em}(9.6)}$$

This can be used to get a product on the space of antisymmetric multilinear forms of diferent degrees, giving something in many ways analogous to the algebra of polynomials (although without a notion of evaluation at a point $v$). This plays a role in the description of fermions and will be considered in more detail in chapter 30. Much like in the symmetric case, there is an isomorphism between ${\Lambda }^{\ast }(V^{\ast })$ and $({\Lambda }^{\ast }(V){)}^{\ast }$

Previous section: 9.5 Bilinear forms and tensor products · Next section: 9.7 For further reading · 中文版本

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