Lie and Hall algebra actions with controlled decategorification

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.

Let $N\ge 2$, let ${\mathcal{H}}_N=({\mathbb{C}}^2{)}^{\otimes N}$, and let the diagonal spin generators $S_N^{+},S_N^{-},S_N^z$ be the sums of the one-site matrices $E,F,H\mathrm{/}2$ defined in the spin convention. For any distinct sites $i,j$, let $P_{ij}$ exchange tensor factors $i$ and $j$. Then $[P_{ij},S_N^{+}]=[P_{ij},S_N^{-}]=[P_{ij},S_N^z]=0.$

For $N\ge 3$, take an auxiliary copy $V_a={\mathbb{C}}^2$ and define $P_{aj}$ as the flip of $V_a$ and site $j$. Fix the ordered orthonormal basis of $V_a$ inherited from $V$. For $X\in \mathrm{End}(V_a\otimes {\mathcal{H}}_N)$, write $X={\sum }_{r,s=1}^2{E}_{rs}\otimes X_{rs}$ in this basis, where $E_{rs}$ are its matrix units and $X_{rs}\in \mathrm{End}({\mathcal{H}}_N)$, and define the partial trace by ${\mathrm{tr}}_{a}X:=X_{11}+X_{22}$. With $R_{aj}(u)=uI+i{P}_{aj},T_a(u)=R_{aN}(u)\cdots R_{a1}(u),t(u)={\mathrm{tr}}_a{T}_a(u)(u\in \mathbb{C}),$ where products act rightmost first, every transfer matrix preserves the diagonal spin action: $[t(u),S_N^{+}]=[t(u),S_N^{-}]=[t(u),S_N^z]=0.$ Use the periodic convention $P_{N,N+1}=P_{N,1}$. For real $J,c$, the periodic XXX matrix $H_N=J{\sum }_{j=1}^N{P}_{j,j+1}+cI$ is Hermitian and commutes with all three generators. When $H_N$ is interpreted as a physical energy, $J$ and $c$ have joule units; $u$ is dimensionless.

For $m\in \{-N\mathrm{/}2,-N\mathrm{/}2+1,\dots ,N\mathrm{/}2\}$, set ${\mathcal{H}}_{N,m}=\ker (S_N^z-mI)$. For real $\beta$, define the finite matrix exponential by $\exp (A)={\sum }_{r=0}^{\mathrm{\infty }}{A}^r\mathrm{/}r!$, converging in operator norm, and let ${\mathrm{Tr}}_W(B)$ denote the trace of a finite-dimensional endomorphism $B:W\to W$, namely the matrix trace in any basis (independent of that basis). Put $Z_N(\beta )={\mathrm{Tr}}_{{\mathcal{H}}_N}(\exp (-\beta H_N))$. Then $Z_N(\beta )={\sum }_m{\mathrm{Tr}}_{{\mathcal{H}}_{N,m}}(\exp (-\beta H_N){\mid }_{{\mathcal{H}}_{N,m}})\mathrm{.}$ In a physical Gibbs interpretation, $\beta$ has inverse-joule units.

The Axiom of Choice

Spin weight spaces and rank-one Hall realization

Ordered auxiliary-space monodromy

Logarithmic derivative is periodic exchange

Finite-dimensional matrix exponential

The basis-independent trace of an endomorphism of a finite-dimensional vector space

Faddeev, §3

Classical and Quantum Integrable Systems, Lecture 14

Classical and Quantum Integrable Systems, Lecture 15, §15.2

The crossing represents the flip $P_{i,i+1}(v_i\otimes v_{i+1})=v_{i+1}\otimes v_i$. Strands are oriented from top to bottom; their horizontal positions label the tensor factors before and after the flip. The proof uses the corresponding operator identity directly and does not rely on the diagram.

Let $D$ be a KLR datum on a symmetrizable Cartan datum with no imaginary simple roots, over a field $k$, in the conventions of The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum. Let $R_D$ be the resulting graded KLR algebra, and let $K_0(R_D)$ be the Grothendieck group of its graded projectives. Put $\mathcal{A}=\mathbb{Z}[q_{\mathrm{g}\mathrm{r}},q_{\mathrm{g}\mathrm{r}}^{-1}]$, where $q_{\mathrm{g}\mathrm{r}}$ is the Laurent grading variable denoted by $q$ in the divided-power integral form, and let $\mathcal{A}{f}_D$ be the integral negative quantum half. The canonical map ${\gamma }_D:\mathcal{A}{f}_D\to K_0(R_D)$ is an isomorphism of $\mathcal{A}$-algebras and graded twisted bialgebras; $q_{\mathrm{g}\mathrm{r}}$ records the grading shift. Assume the Axiom of Choice inherited through this supplier.

For each integer $n\ge 1$, the irreducible highest-weight-$n$ module $V(n)$ defined by homogeneous polynomials of degree $n$ has $H$-weights $n,n-2,\dots ,-n$, each with multiplicity one. The diagonal ${\mathfrak{s}\mathfrak{l}}_2$-module $V(1{)}^{\otimes n}$ has $H$-weight $n-2k$ with multiplicity $\left(\genfrac{}{}{0px}{}{n}{k}\right)$ for $0\le k\le n$. Here $H=2S^z$.

For real $h$ and $\beta$, define the zero-exchange finite matrix $H_n^{(0)}=-h{S}_n^z$ on $V(1{)}^{\otimes n}$. Use $\exp (A)={\sum }_{r=0}^{\mathrm{\infty }}{A}^r\mathrm{/}r!$ for the finite-dimensional matrix exponential, convergent in operator norm, and let $\mathrm{Tr}$ denote the basis-independent trace of a finite-dimensional endomorphism. Define its partition trace by $Z_n(\beta )={\mathrm{Tr}}_{V(1{)}^{\otimes n}}\exp (-\beta H_n^{(0)})$. Then $Z_n(\beta )=(2\cosh (\beta h\mathrm{/}2){)}^n,\cosh x:=\frac{e^x+e^{-x}}{2}\mathrm{.}$ For a Gibbs interpretation, $h$ is an energy coefficient, $S_n^z$ is dimensionless, and $\beta$ has inverse-energy units. The KLR isomorphism supplies the negative-half categorification stated above; it supplies no identification with either finite-dimensional spin module, no tensor-product categorical realization, and no spin-chain energy spectrum.

The Axiom of Choice

The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum

Spin weight spaces and rank-one Hall realization

The KLR category categorifies the negative quantum half

Finite-dimensional matrix exponential

The basis-independent trace of an endomorphism of a finite-dimensional vector space

Khovanov–Lauda, Theorem 1.1 and §3.2

Khovanov–Lauda II, Theorem 8

Kleshchev, §9.1, Theorem 9.1

The KLR Laurent variable $q_{\mathrm{g}\mathrm{r}}$ records internal grading shifts. It is distinct from the finite-field cardinality $q_H$ and from the Hall parameter $\nu =\sqrt{q_H}$ in Spin weight spaces and rank-one Hall realization. The character and trace computations are finite-dimensional algebraic identities; their physical interpretation is conditional on the stated Hamiltonian convention.