Symmetrizable Cartan data for quantum groups

A symmetrizable Cartan datum for a quantum group consists of a finite nonempty index set $I=\{1,\dots ,n\}$, a symmetrizable generalized Cartan matrix $A=(a_{ij}{)}_{i,j\in I}$ (Generalized cartan matrix, Symmetrizable generalized cartan matrix) and a chosen diagonal symmetrizer $D=\mathrm{diag}(d_1,\dots ,d_n)$ with $d_i\in {\mathbb{Z}}_{>0}$ and $d_i{a}_{ij}=d_j{a}_{ji}$ for all $i,j$; free abelian groups $P^{\vee }$ and $P$ (Free abelian group on a set) with an integer-valued bilinear pairing $⟨\cdot ,\cdot ⟩:P\times P^{\vee }\to \mathbb{Z}$; simple coroots $h_i\in P^{\vee }$ that are linearly independent in $P^{\vee }{\otimes }_{\mathbb{Z}}\mathbb{Q}$; and simple roots ${\alpha }_i\in P$ that freely generate the root lattice $Q={⨁}_{i\in I}\mathbb{Z}{\alpha }_i\subseteq P$ (Kac Moody root lattice height and positive cone) and satisfy $⟨{\alpha }_j,h_i⟩=a_{ij}$ for all $i,j$ (Realization of a generalized cartan matrix, Minimal realizations exist and are unique up to isomorphism).

The index set is finite and $A$ need not be nonsingular. We do not require the simple coroots to span $P^{\vee }{\otimes }_{\mathbb{Z}}\mathbb{Q}$, and the pairing is not required to be perfect; choosing $P$ and $P^{\vee }$ is part of the datum, not something determined by the matrix alone.

We fix an indeterminate $q$ over $\mathbb{Q}$, work over $\mathbb{Q}(q)$, and set $q_i=q^{d_i}$. For $h\in P^{\vee }$, $⟨{\alpha }_i,h⟩\in \mathbb{Z}$ is the exponent in $q^{⟨{\alpha }_i,h⟩}$. We use the row convention $⟨{\alpha }_j,h_i⟩=a_{ij}$ throughout. No fundamental weights with prescribed values on all $h_i$ are part of this definition.