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user9072

You are correct: there is a connection to the Galois theory of C/R$\mathbb{C}/\mathbb{R}$ here.

To give a Hodge structure on a real vector space V$V$ -- i.e., a direct sum decomposition of its complexification into (p,q)$(p,q)$ subspaces such that H^{q,p}$H^{q,p}$ is the complex conjugate of H^{p,q}$H^{p,q}$ -- is equivalent to giving an action of G = Res{C/R} C^{\times}$G = \operatorname{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{C}^{\times}$ on V$V$. Here Res{C/R} C^{\times} $\operatorname{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{C}^{\times}$ means the "restriction of scalars" from C/R$\mathbb{C}/\mathbb{R}$ of the complex multiplicative group C^{\times}$\mathbb{C}^{\times}$. In plainer terms, it means that we view C^{\times}$\mathbb{C}^{\times}$ not as a one-dimensional complex algebraic group, but as a 2-dimensional real algebraic group, a "nonsplit torus". Then the fact that we have a homomorphism of real groups

G -> GL(V)$$G \to \operatorname{GL}(V)$$

implies an extra condition on the complexified representation

C^{\times} -> GL(V \otimes C) $\mathbb{C}^{\times} \to \operatorname{GL}(V \otimes \mathbb{C})$: namely that the space V^{p,q}$V^{p,q}$ on which z$z$ in C^{\times}$\mathbb{C}^{\times}$ acts as z^{p} \overline{z}^q$z^{p} \overline{z}^q$ is the complex conjugate of the space V^{q,p}$V^{q,p}$.

A brief (but accurate!) discussion of this can be found at

http://en.wikipedia.org/wiki/Hodge_structure#Hodge_structures

You are correct: there is a connection to the Galois theory of C/R here.

To give a Hodge structure on a real vector space V -- i.e., a direct sum decomposition of its complexification into (p,q) subspaces such that H^{q,p} is the complex conjugate of H^{p,q} -- is equivalent to giving an action of G = Res{C/R} C^{\times} on V. Here Res{C/R} C^{\times} means the "restriction of scalars" from C/R of the complex multiplicative group C^{\times}. In plainer terms, it means that we view C^{\times} not as a one-dimensional complex algebraic group, but as a 2-dimensional real algebraic group, a "nonsplit torus". Then the fact that we have a homomorphism of real groups

G -> GL(V)

implies an extra condition on the complexified representation

C^{\times} -> GL(V \otimes C): namely that the space V^{p,q} on which z in C^{\times} acts as z^{p} \overline{z}^q is the complex conjugate of the space V^{q,p}.

A brief (but accurate!) discussion of this can be found at

http://en.wikipedia.org/wiki/Hodge_structure#Hodge_structures

You are correct: there is a connection to the Galois theory of $\mathbb{C}/\mathbb{R}$ here.

To give a Hodge structure on a real vector space $V$ -- i.e., a direct sum decomposition of its complexification into $(p,q)$ subspaces such that $H^{q,p}$ is the complex conjugate of $H^{p,q}$ -- is equivalent to giving an action of $G = \operatorname{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{C}^{\times}$ on $V$. Here $\operatorname{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{C}^{\times}$ means the "restriction of scalars" from $\mathbb{C}/\mathbb{R}$ of the complex multiplicative group $\mathbb{C}^{\times}$. In plainer terms, it means that we view $\mathbb{C}^{\times}$ not as a one-dimensional complex algebraic group, but as a 2-dimensional real algebraic group, a "nonsplit torus". Then the fact that we have a homomorphism of real groups

$$G \to \operatorname{GL}(V)$$

implies an extra condition on the complexified representation $\mathbb{C}^{\times} \to \operatorname{GL}(V \otimes \mathbb{C})$: namely that the space $V^{p,q}$ on which $z$ in $\mathbb{C}^{\times}$ acts as $z^{p} \overline{z}^q$ is the complex conjugate of the space $V^{q,p}$.

A brief (but accurate!) discussion of this can be found at

http://en.wikipedia.org/wiki/Hodge_structure#Hodge_structures

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Pete L. Clark
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You are correct: there is a connection to the Galois theory of C/R here.

To give a Hodge structure on a real vector space V -- i.e., a direct sum decomposition of its complexification into (p,q) subspaces such that H^{q,p} is the complex conjugate of H^{p,q} -- is equivalent to giving an action of G = Res{C/R} C^{\times} on V. Here Res{C/R} C^{\times} means the "restriction of scalars" from C/R of the complex multiplicative group C^{\times}. In plainer terms, it means that we view C^{\times} not as a one-dimensional complex algebraic group, but as a 2-dimensional real algebraic group, a "nonsplit torus". Then the fact that we have a homomorphism of real groups

G -> GL(V)

implies an extra condition on the complexified representation

C^{\times} -> GL(V \otimes C): namely that the space V^{p,q} on which z in C^{\times} acts as z^{p} \overline{z}^q is the complex conjugate of the space V^{q,p}.

A brief (but accurate!) discussion of this can be found at

http://en.wikipedia.org/wiki/Hodge_structure#Hodge_structures