Exercise 15, § 14

Keep the notation of Exercise 14; denote the ring End$_D(V)$ by A. Let G be a group of automorphisms of A and $G_0$ the subgroup of elements of G that are inner automorphisms of A. Denote by U(G) the (left and right) D-linear subspace of Ω generated by the semilinear bijections $s$ of V such that the automorphism $a\mapsto sas^{-1}$ belongs to G, and denote by $U_0(G)$ the Z-linear subspace of A generated by the automorphisms $u$ of V such that the automorphism $a\mapsto uau^{-1}$ belongs to $G_0$. These are Z-subalgebras of Ω.

a) Let $(s_{\alpha})$ be a basis of $U_0(G)$ over Z consisting of elements of $S_0$(G), and let $(g_{\beta})_{\beta\in G/G_0}$ be a system of representatives of the classes (mod $G_0$) in G. For each $\beta \in G/G_0$, choose an element $u_{\beta}$ of S(G) corresponding to $g_{\beta}$. Prove that the elements $s_{\alpha}u_{\beta}$ form a (left and right) basis of U(G) over D (use Exercise 14, a)). The degree [U(G) : D] is finite if and only if $[U_0(G) : Z]$ and $(G : G_0)$ are finite; if this is the case, then we have $[U(G) : D] = [U_0(G) : Z] (G : G_0)$.

b) Prove that $U_0(G)$ is equal to $U(G)\cap A$ and that $U(G_0) = Du_0(G)$ is canonically isomorphic to $U_0(G)\otimes_ZD$ (use Exercise 14 as well as Exercise 14 of VIII, p. 227). c) An element of U(G) is a semilinear mapping from V to itself if and only if it is of the form $\lambda su_{\beta}$ with $\lambda \in D,s\in U_0$(G), and $\beta \in G/G_0$ (use a), b), and Exercise 14, a)).

d) Denote the commutant of U(G) in Ω by $A^G$; it is the subring of A consisting of the elements invariant under G. Set $Z^G= A^G\cap Z$; it is the subfield of elements of Z invariant under G. Prove that Z and $A^G$ are linearly disjoint over $Z^G$ (verify that a basis of $A^G$ over $Z^G$ is a free subset of A over Z).

e) Prove that if $U_0(G)$ is a simple ring and one of the Z-algebras $U_0(G)$ and D has finite degree, then the ring U(G) is quasi-simple (deduce from c) that $U(G_0)$ is a simple ring, and apply Exercise 14, c) to a two-sided ideal of U(G) using c)).

f ) Suppose that V and U(G) are finite-dimensional over D and that the ring $U_0(G)$ is simple. Prove that the ring A[G] is simple and that we have $[A : A^G]_s=$ $[U_0(G) : Z] (G : G_0)$ (apply Proposition 12 of VIII, p. 205). Prove that the commutant of $A^G$ in Ω (resp. in A) is U(G) (resp. $U_0(G)$) (use Proposition 5 of VIII, p. 139).

g) Deduce that the group of automorphisms of A that fix $A^G$ consists of the automorphisms $a\mapsto vav^{-1}$, where $v$ runs through the set of semilinear bijections in U(G). Prove that it is generated by G and the inner automorphisms associated with the invertible elements of $U_0(G)$ (use c)).

h) If $G = G_0$, then the ring $A^G$ contains Z; prove the converse when D is finite-dimensional over Z (use Theorem 4 of VIII, p. 257).