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I've got a question form Gille and Szamuely's "Central Simple Algebras' and it's about vector spaces equipped with tensors of certain types.

Let $V$ be a $k$-vector space. For a field extension $K/k$, let $V_K= V\otimes_k K$. Define a tensor $\Phi$ of type $(p,q)$ as an element of $V^{\otimes p}\otimes_k (V^*)^{\otimes q}$ where $p$ and $q$ are non-negative integers and $V^*={\rm Hom}_k(V,k)$ is the dual space of $V$. We let $\Phi_K=\Phi\otimes 1$ be the tensor induced on $V_K$ by $\Phi$.

A $k$-isomorphism between pairs $(V,\Phi)$ and $(W,\Psi)$ of $k$-vector spaces equipped with a tensor of fixed type $(p,q)$ is an isomorphism $f: V\xrightarrow{\sim} W$ of $k$-vector spaces such that $f^{\otimes p}\otimes (f^{\ast-1})^{\otimes q}: V^{\otimes p}\otimes_k (V^\ast)^{\otimes q}\to W^{\otimes p}\otimes_k (W^\ast)^{\otimes q}$ maps $\Phi$ to $\Psi$.

Let $K/k$ be a finite Galois extension. Given a $k$-automorphism $\sigma: K\to K$, tensor by $V$ to give a $K$-automorphism $V_K \to V_K$ which we again denote by $\sigma$. Each $K$-linear map $f: V_K\to W_K$ induces a map $\sigma(f): V_K\to W_K$ defined by $\sigma(f)=\sigma\circ f\circ \sigma^{-1}$.

How do I show that if $f$ is a $K$-isomorphism from $(V_K,\Phi_K)$ to $(W_K, \Psi_K)$ then $\sigma(f)$ is also?

Many thanks in advance!

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  • $\begingroup$ When you write $\sigma \circ f \circ \sigma^{-1}$, your two $\sigma$ maps are not really the same (one on $V_K$, one on $W_K$). Why not think of your $\sigma(f)$ as a composite of three isomorphisms between appropriate spaces? $\endgroup$
    – KCd
    Aug 3 '14 at 18:22
  • $\begingroup$ Yes you are right, I should have made the distinction clearer. My main query is why $(\sigma f \sigma^{-1})^{\otimes p}\otimes ((\sigma f \sigma^{-1})^{*-1})^{\otimes q}$ maps $\Phi_K$ to $\Psi_K$. $\endgroup$
    – user149343
    Aug 3 '14 at 20:03
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Of course, no. All non-degenerate symmentic bilinear forms over $\mathbb{C}$ are the same. But not over $\mathbb{R}$.

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  • $\begingroup$ Sorry I don't quite follow you. $\endgroup$
    – user149343
    Aug 3 '14 at 17:47
  • $\begingroup$ en.wikipedia.org/wiki/Metric_signature $\endgroup$
    – quinque
    Aug 3 '14 at 17:57
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    $\begingroup$ Let as take tensors of tipe p=0, q=2. Let $k$ be $\mathbb{R}$ and $K$ be $\mathbb{C}$. Moreover we will consider only symmetric non degenerate tensors. For fixed vector space $V$ isomorphism classes are indexed by signature. However over $\mathbb{C}$ there is only one such class. $\endgroup$
    – quinque
    Aug 3 '14 at 18:04

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