Spin weight spaces and rank-one Hall realization

Assume the Axiom of Choice as inherited through the cited Hall supplier. Let $V={\mathbb{C}}^2$ with ordered orthonormal basis $\mathrm{\mid }\uparrow ⟩,\mathrm{\mid }\downarrow ⟩$. On $V$ define $E=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),F=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),H=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\mathrm{.}$ They satisfy $[H,E]=2E$, $[H,F]=-2F$, and $[E,F]=H$, and therefore define the standard two-dimensional representation of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$. Write $S^{+}=E$, $S^{-}=F$, and $S^z=H\mathrm{/}2$.

For each integer $N\ge 1$, the spin space is ${\mathcal{H}}_N=V^{\otimes N}$ with its tensor-product inner product. If $A$ is an endomorphism of $V$, let $A_j$ act as $A$ on factor $j$ and as the identity on the other factors. Define the diagonal generators $S_N^{+}={\sum }_{j=1}^N{E}_j,S_N^{-}={\sum }_{j=1}^N{F}_j,S_N^z=\frac{1}{2}{\sum }_{j=1}^N{H}_j\mathrm{.}$ For $m\in \{-N\mathrm{/}2,-N\mathrm{/}2+1,\dots ,N\mathrm{/}2\}$, its weight space is ${\mathcal{H}}_{N,m}=\ker (S_N^z-mI)$. These are dimensionless matrices in the algebraic spin convention.

For each integer $r\ge 0$, define $V(r)$ to be the vector space $\mathbb{C}[x,y{]}_r$ of homogeneous polynomials of degree $r$, with ${\mathfrak{s}\mathfrak{l}}_2$ action $E=x{\mathrm{\partial }}_y,F=y{\mathrm{\partial }}_x,H=x{\mathrm{\partial }}_x-y{\mathrm{\partial }}_y\mathrm{.}$ The ordered monomials $x^{r-k}{y}^k$ for $0\le k\le r$ form its basis. Under $x\mapsto \mathrm{\mid }\uparrow ⟩$ and $y\mapsto \mathrm{\mid }\downarrow ⟩$, $V(1)$ is the representation on $V$ above. A tensor power $V(1{)}^{\otimes N}$ uses the diagonal action defined by the sums $S_N^{+},S_N^{-},2S_N^z$.

Let $q_H>1$ be a prime power, fix a finite field ${\mathbb{F}}_{q_H}$, and put $\nu =\sqrt{q_H}>1$ and $K=\mathbb{Q}(\nu )$. The representation category $\mathcal{A}={\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{{\mathbb{F}}_{q_H}}^{\mathrm{f}\mathrm{d}}$ of the one-vertex, no-arrow quiver has one simple isomorphism class, represented by $S={\mathbb{F}}_{q_H}$. For objects $M,N$, write ${\chi }_0(M,N)={\dim }_{{\mathbb{F}}_{q_H}}\mathrm{Hom}(M,N)-{\dim }_{{\mathbb{F}}_{q_H}}{\mathrm{YExt}}^1(M,N)$. In the Hall convention of The Euler-twisted Hall algebra, the Hall space is $H_{\mathcal{A},\nu }={⨁}_{[M]}K[M]$, with basis $[M]$ indexed by isomorphism classes and product $[M]\star [N]={\nu }^{{\chi }_0(M,N)}{\sum }_{[L]}{g}_{M,N}^L[L],$ where $g_{M,N}^L$ counts subspaces $N^{\mathrm{\prime }}\subseteq L$ with $N^{\mathrm{\prime }}\cong N$ and $L\mathrm{/}{N}^{\mathrm{\prime }}\cong M$, or equivalently short exact sequences $0\to N\stackrel{i}{\to }L\stackrel{p}{\to }M\to 0$ modulo changes of the endpoint labels: $(i,p)\sim (i\circ a,b\circ p)$ for $a\in \mathrm{Aut}(N)$ and $b\in \mathrm{Aut}(M)$, with $L$ fixed pointwise. Each orbit is determined by $\mathrm{im}i=N^{\mathrm{\prime }}$; without this quotient, each such subspace contributes $\mathrm{\mid }\mathrm{Aut}(N)\mathrm{\mid }\mathrm{\mid }\mathrm{Aut}(M)\mathrm{\mid }$ labelled sequences. Thus the twist is part of the multiplication. For $M\cong {\mathbb{F}}_{q_H}^m$ and $N\cong {\mathbb{F}}_{q_H}^n$, $\dim \mathrm{Hom}(M,N)=mn$ and every short exact sequence splits, so ${\mathrm{YExt}}^1(M,N)=0$ and ${\chi }_0(M,N)=mn$.

By The extended Hall algebra and composition subalgebra, the composition subalgebra is the subalgebra of this Hall algebra generated by the classes of simple objects; its torus extension is the distinct extended composition subalgebra. Denote the former by $\mathcal{C}$ here, so $\mathcal{C}=K⟨[S]⟩\subseteq H_{\mathcal{A},\nu }$. The rank-one positive quantum half is $U_{\nu }^{+}({\mathfrak{s}\mathfrak{l}}_2)=K[E]$, the polynomial algebra on one generator. The Ringel map $E\mapsto [S]$ embeds this positive half with image $\mathcal{C}$; since the one-vertex graph is finite Dynkin type $A_1$, the finite-Dynkin criterion identifies $\mathcal{C}$ with the full Hall algebra in this case. The simple object is unique up to isomorphism, not as a chosen representative. The Axiom of Choice is inherited through the cited Ringel supplier.

The Axiom of Choice

The Euler-twisted Hall algebra

The extended Hall algebra and composition subalgebra

The Ringel map embeds the extended positive Borel and is full Hall exactly in finite Dynkin type

Schiffmann, Theorem 3.16

Ringel, §7

The formal Hall parameter $q_H$ and its square root $\nu$ are mathematical parameters; $q_H$ is not a physical Potts-state parameter. The field-valued Ringel identification is stated for $\nu >1$; a specialization to $\nu =1$ requires an integral form. The positive half contains no negative generator or compact star structure. No categorical action on $V(1{)}^{\otimes N}$ is part of this definition.