The cyclic affine type A Cartan datum and its residues

$A$ is a generalized Cartan matrix. By construction $a_{ii}=2$ and $a_{ij}\le 0$ for $i\mathrm{\ne }j$; off-diagonal entries vanish except for the neighbouring entries $-1$ and, at $e=2$, the double edge $-2$, and $A$ is symmetric, so $a_{ij}=0⟺a_{ji}=0$. Hence [F1] applies, $D=\mathrm{diag}(1,\dots ,1)$ is a symmetrizer, and $A$ is symmetrizable.

The pairing is perfect on the displayed bases. The families $\{{\mathrm{\Lambda }}_i{\}}_{i\in I}\cup \{\delta \}$ and $\{h_i{\}}_{i\in I}\cup \{d\}$ are bases of the free abelian groups $P$ and $P^{\vee }$, and the four displayed rules give $⟨\cdot ,\cdot ⟩$ on all basis pairs as the identity matrix. Hence every homomorphism $P\to \mathbb{Z}$ is $⟨\cdot ,\xi ⟩$ for a unique $\xi \in P^{\vee }$, and in particular the simple coroots are linearly independent in $P^{\vee }{\otimes }_{\mathbb{Z}}\mathbb{Q}$.

The row and column sums of $A$ vanish: ${\sum }_{j\in I}{a}_{ij}=0={\sum }_{j\in I}{a}_{ji}$ for every $i\in I$, by [F7] and symmetry of $A$.

The simple roots are linearly independent. Let ${\sum }_i{x}_i{\alpha }_i=0$ with $x_i\in \mathbb{Z}$. Comparing coefficients of $\delta$ in ${\alpha }_i={\sum }_j{a}_{ji}{\mathrm{\Lambda }}_j+{\delta }_{i0}\delta$ gives $x_0=0$, and comparing coefficients of ${\mathrm{\Lambda }}_k$ gives ${\sum }_i{x}_i{a}_{ki}=0$ for every $k$. For $e\ge 3$ this reads $2x_k-x_{k+1}-x_{k-1}=0$, so all differences $x_{k+1}-x_k$ are equal; their sum over the cycle $I$ is $0$, so each difference is $0$ and $x$ is constant, and $x_0=0$ forces $x=0$. For $e=2$ the two equations $2x_0-2x_1=0$ and $2x_1-2x_0=0$ together with $x_0=0$ give $x_1=0$. Hence the ${\alpha }_i$ are independent and freely generate $Q$.

The simple-root identities of the datum hold: $⟨{\alpha }_j,h_i⟩={\sum }_k{a}_{kj}⟨{\mathrm{\Lambda }}_k,h_i⟩+{\delta }_{j0}⟨\delta ,h_i⟩=a_{ij}$; $⟨{\alpha }_i,d⟩={\sum }_k{a}_{ki}⟨{\mathrm{\Lambda }}_k,d⟩+{\delta }_{i0}⟨\delta ,d⟩={\delta }_{i0}$; and ${\sum }_{i\in I}{\alpha }_i={\sum }_k\left({\sum }_i{a}_{ki}\right){\mathrm{\Lambda }}_k+\delta =\delta$, because every column sum of $A$ vanishes by step 1.3. In particular ${\alpha }_i\in P$ and its coordinates in the basis of $P$ are the displayed integers.

The level functional and the central element are as claimed. By step 1.2 the rules $⟨c,{\mathrm{\Lambda }}_i⟩=1$ and $⟨c,\delta ⟩=0$ define a homomorphism $P\to \mathbb{Z}$, the level; and $⟨c,{\alpha }_j⟩={\sum }_i⟨h_i,{\alpha }_j⟩={\sum }_i{a}_{ij}=0$ for every $j$ by step 1.3, so all simple roots, hence all of $Q$, have level $0$. Every $\lambda \in P$ has integral labels $⟨\lambda ,h_i⟩$ on the coroots by the displayed rules, so $P$ meets the integral-weight conditions of [F5] on $P^{\vee }$.

The datum $(I,A,D,P,P^{\vee })$ is a symmetrizable Cartan datum for a quantum group. The index set is finite and nonempty; $A$ is a symmetrizable GCM with symmetrizer $D$ by step 1.1; $P$ and $P^{\vee }$ are free abelian groups with an integer pairing, perfect on the displayed bases, by step 1.2; the simple coroots are independent by step 1.2; and the simple roots are independent, generate the root lattice freely, and satisfy $⟨{\alpha }_j,h_i⟩=a_{ij}$ by steps 2.1 and 2.2. Hence [F2] holds and $q_i=q$ for all $i$. For [F3], take $\mathfrak{h}=P^{\vee }{\otimes }_{\mathbb{Z}}\mathbb{C}$ and extend the perfect pairing to identify $P{\otimes }_{\mathbb{Z}}\mathbb{C}$ with ${\mathfrak{h}}^{\ast }$. The kernel calculation in step 2.1 works over $\mathbb{C}$ and gives $\ker A=\mathbb{C}(1,\dots ,1)$, so $\mathrm{rank}(A)=e-1$. The same calculation with $x_0=0$ proves independence of the roots over $\mathbb{C}$; the coroots are part of the displayed basis. Thus $\dim \mathfrak{h}=e+1=2e-\mathrm{rank}(A)$, and the pairings of step 2.2 establish a minimal realization.

The diagram reflection $\tau$ is a well-defined involution preserving the pairing and the root data. It permutes the basis $\{{\mathrm{\Lambda }}_i{\}}_{i\in I}\cup \{\delta \}$ and the basis $\{h_i{\}}_{i\in I}\cup \{d\}$, so it is an automorphism of $P$ and of $P^{\vee }$ with ${\tau }^2=\mathrm{id}$. On basis pairs, $⟨\tau {\mathrm{\Lambda }}_j,\tau h_i⟩=⟨{\mathrm{\Lambda }}_{-j},h_{-i}⟩={\delta }_{-j,-i}={\delta }_{ij}=⟨{\mathrm{\Lambda }}_j,h_i⟩$, while $⟨\tau \delta ,\tau h_i⟩=0$, $⟨\tau {\mathrm{\Lambda }}_j,\tau d⟩=0$ and $⟨\tau \delta ,\tau d⟩=1$, so the pairing is preserved. Finally $\tau ({\alpha }_i)={\sum }_j{a}_{ji}{\mathrm{\Lambda }}_{-j}+{\delta }_{i0}\delta ={\sum }_k{a}_{-k,i}{\mathrm{\Lambda }}_k+{\delta }_{i0}\delta ={\alpha }_{-i}$, because $a_{-k,i}=a_{k,-i}$ holds by the symmetry of $A$ under index negation; hence $\tau$ permutes the simple roots, preserves $Q$ and $Q^{+}$, and fixes ${\mathrm{\Lambda }}_0$, $\delta$ and $d$.

The normalized form exists and pairs weights with roots by the Cartan labels. By step 2.1, $Q_{\mathbb{Q}}$ has dimension $e$, and by step 3.1 it has level $0$, whereas ${\mathrm{\Lambda }}_0$ has level $1$. Since $P_{\mathbb{Q}}$ has dimension $e+1$, $P_{\mathbb{Q}}=Q_{\mathbb{Q}}\oplus \mathbb{Q}{\mathrm{\Lambda }}_0$. The displayed formula therefore defines a unique symmetric rational bilinear form with the specified values, and its restriction to $Q$ is integral. For $\lambda ={\sum }_j{x}_j{\alpha }_j+t{\mathrm{\Lambda }}_0$, one has $(\lambda ,{\alpha }_i)={\sum }_j{x}_j{a}_{ji}+t{\delta }_{i0}={\sum }_j{x}_j⟨{\alpha }_j,h_i⟩+t⟨{\mathrm{\Lambda }}_0,h_i⟩=⟨\lambda ,h_i⟩$, using symmetry of $A$. Summing over $i$ and using $\delta ={\sum }_i{\alpha }_i$ yields $(\lambda ,\delta )=⟨\lambda ,c⟩$; in particular $({\mathrm{\Lambda }}_0,\delta )=1$, while $(\delta ,Q)=0$ and $(\delta ,\delta )=0$.

The normalized form is invariant under the diagram reflection and the Weyl group. By step 3.3, $\tau$ permutes ${\alpha }_i$ by $i\mapsto -i$ and fixes ${\mathrm{\Lambda }}_0$. The matrix is invariant under this permutation, and ${\delta }_{-i,0}={\delta }_{i0}$, so all defining form values are preserved and $\tau$ is an isometry. For $s_i$, write $b=(\lambda ,{\alpha }_i)$ and $b^{\mathrm{\prime }}=(\mu ,{\alpha }_i)$; then $(s_i\lambda ,s_i\mu )=(\lambda -b{\alpha }_i,\mu -b^{\mathrm{\prime }}{\alpha }_i)=(\lambda ,\mu )-b{b}^{\mathrm{\prime }}-b{b}^{\mathrm{\prime }}+2b{b}^{\mathrm{\prime }}=(\lambda ,\mu )$, since $({\alpha }_i,{\alpha }_i)=2$ and step 4.1 identifies $b,b^{\mathrm{\prime }}$ with the Cartan labels. Also $s_i^2=1$ by the same identity and $s_i$ preserves $P$ because its labels are integers. Every product in $W$ is consequently an isometry. These computations apply to the double-edge matrix when $e=2$ as well.

Conclusion. The cyclic affine type $A$ datum is a legitimate symmetrizable Cartan datum of rank $e+1$ with $q_i=q$: the matrix, symmetrizer and lattices satisfy every axiom by step 3.2; the simple roots satisfy $⟨{\alpha }_j,h_i⟩=a_{ij}$, $⟨{\alpha }_i,d⟩={\delta }_{i0}$ and ${\sum }_i{\alpha }_i=\delta$ by step 2.2; the central element $c={\sum }_i{h}_i$ has $⟨c,{\alpha }_j⟩=0$ and pairs with every ${\mathrm{\Lambda }}_i$ to $1$, so level is a well-defined integral functional with the ${\mathrm{\Lambda }}_i$ of level $1$ and $\delta$ of level $0$ by step 3.1; the normalized affine weight form extends the root form, is distinct from the lattice pairing, satisfies $(\lambda ,{\alpha }_i)=⟨\lambda ,h_i⟩$, and is invariant under $W$ and $\tau$ by steps 4.1 and 5.1; and the diagram reflection $\tau$ is an involutive automorphism of the datum fixing ${\mathrm{\Lambda }}_0,\delta ,d$ and negating the remaining indices by step 3.3.