Lie and Hall algebra actions with controlled decategorification — Examples

In the ordered tensor basis, define $v_{++}=\mathrm{\mid }\uparrow \uparrow ⟩$, $v_0=(\mathrm{\mid }\uparrow \downarrow ⟩+\mathrm{\mid }\downarrow \uparrow ⟩)\mathrm{/}\sqrt{2}$, $v_{--}=\mathrm{\mid }\downarrow \downarrow ⟩$, and $v_s=(\mathrm{\mid }\uparrow \downarrow ⟩-\mathrm{\mid }\downarrow \uparrow ⟩)\mathrm{/}\sqrt{2}$. These four vectors are pairwise orthonormal by the tensor-product inner product, and they span the four-dimensional space. The swap $P$ fixes the first three and sends $v_s$ to $-v_s$, so the same space decomposes into the symmetric eigenspace $W_{+}=\ker (P-I)$ of dimension three and the antisymmetric eigenspace $W_{-}=\ker (P+I)$ of dimension one. The swap is an involutive permutation of an orthonormal basis, so $P=P^{\ast }$. By [F2], it commutes with $\rho (X)$ for every $X\in {\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$; hence both eigenspaces are invariant under the action.

Let $A_0=0$ and $A_1=JP$. Both are self-adjoint because $J$ is real and $P=P^{\ast }$, and both belong to ${\mathcal{H}}_{\rho }$: the zero map commutes with every endomorphism and [F2] gives the required commutation for $P$. The spectrum of $A_0$ is $0$ with multiplicity four; that of $A_1$ is $J$ with multiplicity three and $-J$ with multiplicity one. Therefore, by [F4] and [F5], $Z_{A_0}(\beta )={\mathrm{Tr}}_W(\exp (-\beta A_0))=4,Z_{A_1}(\beta )={\mathrm{Tr}}_W(\exp (-\beta A_1))=3e^{-\beta J}+e^{\beta J}\mathrm{.}$

Since $\beta J=\ln 2$, the second trace is $3\mathrm{/}2+2=7\mathrm{/}2$, which differs from $4$. Hence the same spin representation decomposition admits distinct symmetry-preserving spectra and Gibbs traces.

The Hall data in [F1] and the KLR Grothendieck isomorphism in [F3] are fixed for both $A_0$ and $A_1$; neither specifies a selected endomorphism of $W$. Since both operators lie in the quantified class but have different spectra and different partition traces, they refute the stated uniqueness claim.