- #1
Jakob1
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Hello.
I've just read about natural identifications of exterior powers with spaces of alternating maps, etc here: Some Natural Identifications
However, I have problems showing that the following operations give the same space:
\(\displaystyle V \rightarrow \Lambda_p V \rightarrow (\Lambda_p V)^* \cong \Lambda_p V^*\)
\(\displaystyle V \rightarrow V^* \rightarrow \Lambda_p V ^*\)
I know that \(\displaystyle (\Lambda_p V)^* \cong \Lambda_p V^*\), because \(\displaystyle (\Lambda_p V)^* \cong \mathcal{A}_p(V)\), and \(\displaystyle \mathcal{A}_p(V) \ni f \rightarrow L_f \in (\Lambda_p V)^*\), where \(\displaystyle L_f\) is the only linear map which makes the universal factorization diagram commute, is an isomorphism.
And \(\displaystyle \mathcal{A}_p(V) \cong \Lambda_p V^*\), and here we consider this map:
\(\displaystyle V^* \times ... \times V^* \ni (f_1, ..., f_p) \rightarrow (V^p \ni (v_1, ..., v_p) \rightarrow \det [f_i(v_j)] \in \mathbb{K}) \in \mathcal{A}_p(V)\) which is $p$-linear and antisymmetyric and together with the exterior power map \(\displaystyle V^* \times ... \times V^* \rightarrow \Lambda_p V\) we get the universal factorization diagram and
the isomorphism we are looking for is the only linear map which makes the diagram commute.
This is all I know at the moment.
Could you help me with this problem?
Thank you.
I've just read about natural identifications of exterior powers with spaces of alternating maps, etc here: Some Natural Identifications
However, I have problems showing that the following operations give the same space:
\(\displaystyle V \rightarrow \Lambda_p V \rightarrow (\Lambda_p V)^* \cong \Lambda_p V^*\)
\(\displaystyle V \rightarrow V^* \rightarrow \Lambda_p V ^*\)
I know that \(\displaystyle (\Lambda_p V)^* \cong \Lambda_p V^*\), because \(\displaystyle (\Lambda_p V)^* \cong \mathcal{A}_p(V)\), and \(\displaystyle \mathcal{A}_p(V) \ni f \rightarrow L_f \in (\Lambda_p V)^*\), where \(\displaystyle L_f\) is the only linear map which makes the universal factorization diagram commute, is an isomorphism.
And \(\displaystyle \mathcal{A}_p(V) \cong \Lambda_p V^*\), and here we consider this map:
\(\displaystyle V^* \times ... \times V^* \ni (f_1, ..., f_p) \rightarrow (V^p \ni (v_1, ..., v_p) \rightarrow \det [f_i(v_j)] \in \mathbb{K}) \in \mathcal{A}_p(V)\) which is $p$-linear and antisymmetyric and together with the exterior power map \(\displaystyle V^* \times ... \times V^* \rightarrow \Lambda_p V\) we get the universal factorization diagram and
the isomorphism we are looking for is the only linear map which makes the diagram commute.
This is all I know at the moment.
Could you help me with this problem?
Thank you.
Last edited: