Remark. Three different commutants for the left Clifford action [fcap-003Q]

✍️sourceAGENTDRAFTED

Keep the hypotheses and notation of Theorem [fcap-003P], and let \(L_x(c)=\delta (x)c\). The internal centralizer \(J\subseteq A\), the right-multiplication operators on \(A\), and the full algebra of \(\mathfrak g\)-module endomorphisms of \((A,L)\) have distinct roles.

Left multiplication by \(j\in J\) commutes with every \(L_x\). Right multiplication by any \(a\in A\) also commutes with every \(L_x\), purely by associativity. Thus there are injective algebra maps \[J\longrightarrow \operatorname {End}_{\mathfrak g}(A,L), \qquad j\longmapsto (c\longmapsto jc),\] \[A^{\mathrm {op}}\longrightarrow \operatorname {End}_{\mathfrak g}(A,L), \qquad a\longmapsto (c\longmapsto ca).\] Evaluation at \(1\) proves injectivity. The opposite algebra in the second map records the reversed order of right compositions.

By Theorem [fcap-003G], \((A,L)\simeq V_\rho \otimes \mathbb C^m\) with \(m=2^{\ell +r}\). Schur's lemma and the same block calculation as in Theorem [fcap-003P] give \[\operatorname {End}_{\mathfrak g}(A,L) \simeq M_m(\mathbb C).\] The dimensions of the three algebras are therefore \[\dim J=2^\ell ,\qquad \dim A^{\mathrm {op}}=2^{\ell +2r},\qquad \dim \operatorname {End}_{\mathfrak g}(A,L)=2^{2\ell +2r}.\] For \(\mathfrak g=\mathfrak {sl}_2\), these are \(2\), \(8\), and \(16\), respectively: \(J\simeq \mathbb C\oplus \mathbb C\), while \(\operatorname {End}_{\mathfrak g}(A,L)\simeq M_4(\mathbb C)\). The four-copy left-module calculation is explicit in Example [fcap-003F]. For every nonzero complex semisimple \(\mathfrak g\), the dimensions show that right multiplications form a proper subalgebra of the full module commutant. They do not by themselves identify that commutant.