Ungraded real and quaternionic spinor Morita evaluation

Write $A=M_k(D)$ and $S=D^k$ as a right $D$-space. Any real-linear endomorphism commuting with all matrix units has the same real-linear map $t:D\to D$ in every coordinate. Commuting also with left multiplication by every quaternion gives $t(a)=at(1)$, so it is right multiplication $R_q$. In the real case this is just a real scalar. Composition satisfies $R_q{R}_r=R_{rq}$, so the commutant is $D^{\mathrm{o}\mathrm{p}}$. For $D=\mathbb{H}$ the map $q\mapsto R_{\bar{q}}$ is an additive, real-linear ring isomorphism $\mathbb{H}\to D^{\mathrm{o}\mathrm{p}}$. Conjugation, rather than inversion on arbitrary nonzero quaternions, is essential here.

For a left $A$-module $L$, the matrix idempotents split $L={⨁}_i{e}_{ii}L$. The maps $e_{i1}$ identify the summands with $e_{11}L$, a left $D$-space, so choosing a $D$-basis identifies $L$ with $S^{\oplus t}$. For $D=\mathbb{H}$, the proposed Hom action is additive in both variables, has unit, and satisfies $((f\cdot q)\cdot r)(s)=f(s\bar{r}\bar{q})=f(s\overline{qr})$. The left action on $S$ satisfies $q\cdot (r\cdot s)=s\bar{r}\bar{q}=(qr)\cdot s$. Consequently evaluation is balanced, since $(f\cdot q)(s)=f(s\bar{q})=f(q\cdot s)$.

First take $L=S$. Its Hom is the commutant, and its right-$\mathbb{H}$ coordinate is $R_a\mapsto \bar{a}$, because $(R_a\cdot q)=R_{\bar{q}a}$ and $\overline{\stackrel{ˉ}{q}a}=\bar{a}q$. Under this coordinate evaluation is $\mathbb{H}{\otimes }_{\mathbb{H}}S\to S$, $h\otimes s\mapsto s\bar{h}$, the tensor-unit isomorphism for the left action just defined. For $L=S^{\oplus t}$ take its direct sum to obtain ${\mathrm{Hom}}_A(S,L){\otimes }_{\mathbb{H}}S\cong L$. For $D=\mathbb{R}$ the identical argument uses ordinary scalar multiplication. For a right-$\mathbb{H}$ space $E$, the inverse identification is $E\to {\mathrm{Hom}}_A(S,E{\otimes }_{\mathbb{H}}S)$, $e\mapsto (s\mapsto e\otimes s)$; it is right-linear by the balancing identity and bijective after taking any finite basis of $E$. No dimension-only surjectivity assertion is needed.

In a Spin frame, all these constructions have the fixed matrix form of step 1.1, step 2.1 and step 3.1. The right-$D$ action on the spinor commutes with every left Clifford multiplication, so it is preserved by the spinor transition functions in [F2]. Hom is a vector subbundle of the ordinary Hom bundle: locally it is the fixed kernel of finitely many matrix intertwining equations, of constant rank because the algebra and module types are locally constant. Changes of frame transport the Hom by conjugation and transport evaluation to evaluation. Thus the fiberwise isomorphisms are continuous bundle isomorphisms, glue globally, and commute with pullback and module morphisms.

The inverse equivalences preserve direct sums and isomorphism classes, so they identify the additive monoids of bundles and hence their Grothendieck groups without any cancellation assumption. If a stable presentation is desired, a right-$\mathbb{H}$ bundle embeds in a finite trivial quaternionic bundle: finite charts and a partition give the quaternionic-linear embedding, and its invariant Hermitian orthogonal complement is quaternionic. The real proof is the same. Tensoring this complement with $\mathrm{\Sigma }$ gives a module complement in a finite sum of $\mathrm{\Sigma }$. This proves the stable-complement compatibility as well. Arbitrary graded modules were never substituted for the ungraded category.