Proposition 2

Let A be an $\mathbf{R}$-algebra (associative and unital), and W a finite dimensional simple $A_{(\mathbf{C})}$-module over $\mathbf{C}$. Then, we must be in one of the following three situations:

a) There exists an $A_{(\mathbf{C})}$-isomorphism $\theta$ from W to $\overline{W}$ with $\theta \circ \theta = 1_W$. Then the set V of fixed points of $\theta$ is an $\mathbf{R}$-structure on W, and a simple A-module with commutant $\mathbf{R}.1_V$. Moreover, $W_{[\mathbf{R}]}$ is the direct sum of two isomorphic simple A-modules.

b) The $A_{(\mathbf{C})}$-modules W and $\overline{W}$ are not isomorphic; then $W_{[\mathbf{R}]}$ is a simple A-module with commutant $\mathbf{C}.1_W$.

c) There exists an $A_{(\mathbf{C})}$-isomorphism $\theta$ from W to $\overline{W}$ with $\theta \circ \theta =-1_W$. Then the A-module $W_{[\mathbf{R}]}$ is simple, and its commutant is the field $\mathbf{C}.1_W\oplus \mathbf{C}.\theta$, which is isomorphic to $\mathbf{H}$.

The complex vector space Hom$_{A_{(\mathbf{C})}}(W,W)$ is of dimension $\leq 1 ($Algebra, Chap. VIII, §3, no. 2); if $\theta \in$ Hom$_{A_{(\mathbf{C})}}(W,W)$, the endomorphism $\theta \circ \theta$ of W is a homothety, with ratio $\alpha \in \mathbf{C}$. For all $w\in W$, we have $\alpha \theta (w) =\theta \circ \theta \circ \theta (w) =$ $\theta (\alpha w) = \overline{\alpha \theta}(w)$, so $\alpha$ is real. If $\theta '=\lambda \theta$, with $\lambda \in \mathbf{C}$, then $\theta '\circ \theta '=|\lambda |^2\theta \circ \theta$; thus, exactly one of the following three possibilities is realised:

a) There exists $\theta \in$ Hom$_{A_{(\mathbf{C})}}(W,W)$ with $\theta \circ \theta = 1_W$;

b) Hom$_{A_{(\mathbf{C})}}(W,W) =\{0\}$;

c) There exists $\theta \in$ Hom$_{A_{(\mathbf{C})}}(W,W)$ with $\theta \circ \theta =-1_W$.

In case a), the set V of fixed points of $\theta$ is an $\mathbf{R}$-structure on W (Algebra, Chap. V, p. 61, Prop. 7); since $V_{(\mathbf{C})}$ is isomorphic to W, the A-module V is simple with commutant $\mathbf{R}.1_V$ (Prop. 1), and $W_{[\mathbf{R}]}$ is not simple.

Conversely, if $W_{[\mathbf{R}]}$ is not simple, let V be a simple A-submodule of $W_{[\mathbf{R}]}$; since the $A_{(\mathbf{C})}$-module W is simple, $V +iV = W$ and $V\cap iV =\{0\}$, that is, $W = V\oplus iV$. Thus, V is an $\mathbf{R}$-structure on W, and the isomorphism $\theta$ from W to $\overline{W}$ such that $\theta (v+iv') =v-iv'$ for $v$ and $v'$ in V satisfies $\theta \circ \theta = 1_W$.

Consequently, in cases b) and c), the A-module $W_{[\mathbf{R}]}$ is simple; by Prop. 1, its commutant E is isomorphic to $\mathbf{C}$ in case b), and to $\mathbf{H}$ in case c). Moreover, it is clear that E contains $\mathbf{C}.1_W$, and $\mathbf{C}.\theta$ in case c), hence the proposition.

With the assumptions in the proposition, the $A_{(\mathbf{C})}$-module W is said to be of real, complex or quaternionic type (relative to A) in case $a),b)$ or c), respectively.

For $K =\mathbf{R}$ or $\mathbf{C}$, denote by $\mathfrak{S}_K(A)$ the set of classes of finite dimensional simple $A_{(K)}$-modules over K. The group $\Gamma =$ Gal($\mathbf{C}/\mathbf{R}$) operates on $\mathfrak{S}_{\mathbf{C}}(A)$; the two preceding propositions establish a bijective correspondence between $\mathfrak{S}_{\mathbf{R}}(A)$ and the quotient set $\mathfrak{S}_{\mathbf{C}}(A)/\Gamma$.