외대수(exterior algebra)와 다중선형대수(multilinear algebra)

개요

정의 \(\Lambda(V) := T(V)/I\)
\(\Lambda(V) = \Lambda^0(V)\oplus \Lambda^1(V) \oplus \Lambda^2(V) \oplus \cdots \oplus \Lambda^n(V)\)
\(\alpha\in \Lambda^k(V), \beta\in \Lambda^p(V)\) 에 대하여 \(\alpha\wedge\beta = (-1)^{kp}\beta\wedge\alpha\) 가 성립한다

\(\Lambda^k(V)^{*}\simeq\Lambda^k(V^{*})\)
\(v_1,\cdots, v_k \in V\), \(f_1,\cdots, f_k \in V^{*}\) 에 대하여, 다음과 같은 isomorphism \(\Lambda^k(V^{*})\to \Lambda^k(V)^{*}\)을 정의할 수 있다
\(\langle v_1\wedge\cdots\wedge v_k, f_1\wedge\cdots\wedge f_k\rangle=\det(\langle v_i,f_j\rangle)\)

외대수의 쌍대 공간을 생각하는 또다른 방식 \(\Lambda^k(V)^{*}\simeq A^k(V)\)
교대 겹선형 k-형식(k-alternating form)
\(f:V^k\to{\mathbb R},\qquad f(v_{\sigma(1),\cdots,v_{\sigma(k)}})=(\text{sgn}\sigma)f(v_1,\cdots,v_k) \quad\text{for all} \quad\sigma\in S_k.\)
\(A^k(V)\) : the set of k-alternating forms on V
\(A(V) = A^0(V)\oplus A^1(V) \oplus A^2(V) \oplus \cdots \oplus A^n(V)\)
wedge product
\(\omega\wedge\eta=\frac{(k+m)!}{k!\,m!}\operatorname{Alt}(\omega\otimes\eta)\)
여기서 \(\operatorname{Alt}(\omega)(x_1,\ldots,x_k)=\frac{1}{k!}\sum_{\sigma\in S_k}\operatorname{sgn}(\sigma)\,\omega(x_{\sigma(1)},\ldots,x_{\sigma(k)}).\)
(p,q)-shuffle 을 이용한 정의
\(\omega \wedge \eta(x_1,\ldots,x_{k+m}) = \sum_{\sigma \in Sh_{k,m}} \operatorname{sgn}(\sigma)\,\omega(x_{\sigma(1)}, \ldots, x_{\sigma(k)}) \eta(x_{\sigma(k+1)}, \ldots, x_{\sigma(k+m)}),\)