On bases of quantum affine algebras

Let $\textbf{U}^{+}$ be the positive part of the quantum group U associated with a generalized Cartan matrix $A$.

In the case of finite type, Lusztig constructed the canonical basis B of $\textbf{U}^{+}$ via two approaches ([8]). The first one is an elementary algebraic construction via Ringel-Hall algebra realization of $\textbf{U}^{+}$. The isomorphism classes of representations of the corresponding Dynkin quiver form a PBW-type basis of $\textbf{U}^{+}$. By a lemma (Lemma 3.1) of Lusztig, one can construct a bar-invariant basis, which is the canonical basis B. A remarkable characteristic in his construction is that Lusztig reveals the triangular relations among three kind of bases: PBW-basis, monomial basis and canonical basis.

The second one is a geometric construction. In [8] and [9], Lusztig gave a geometric realization of $\mathbf{U}^{+}$ via the category of some semisimple complexes on the variety $E_{\nu}$ consisting of representations with dimension vector $\nu\in\mathbb{N}I$ of the corresponding quiver $Q$. The set of the isomorphism classes of simple perverse sheaves gives the canonical basis $\mathbf{B}$ of $\mathbf{U}^{+}$.

The geometric construction of canonical basis was generalized to the cases of all types in [9] (see also [11]). Furthermore, Lusztig in [10] gave the construction of affine canonical bases by the perverse sheaves associated with tame quivers, an important feature is that those perverse sheaves are indexed by the classes of aperiodic modules of tame quivers and irreducible modules of symmetric groups.

The generalization of this elementary algebraic construction to affine type is an important problem.

In [1] and [2], Beck-Chari-Pressley and Beck-Nakajima defined a PBW-type basis and gave an algebraic construction of canonical basis. However, the meaning of this construction is not clear in terms of representation theory of quivers, in particular, it is not clear how to parametrize the canonical basis in terms of aperiodic modules of tame quivers and irreducible modules of symmetric groups.

In the case of Kronecker quiver, Chen ([3]) and Zhang ([17]) defined a PBW-type basis by using Ringel-Hall algebra realization of $\textbf{U}^{+}$, McGerty ([12]) gave the interpretation of the PBW-type basis of Beck-Nakajima in terms of representation theory of the Kronecker quiver. In the case of cyclic quiver, Deng-Du-Xiao ([4]) defined a PBW-type basis and gave a concrete algebraic construction of the canonical basis by using the triangular relations between the monomial basis and the PBW-basis.

Lin-Xiao-Zhang in [7] provided a process to construct a PBW-type basis of $\textbf{U}^{+}$ and the canonical basis B by using Ringel-Hall algebra approach. Recently Xiao-Xu-Zhao ([16]) provided a direct method to construct a PBW-type basis of $\textbf{U}^{+}$ and the canonical basis B. Compared with the construction Lin-Xiao-Zhang in [7], the PBW-type basis of Xiao-Xu-Zhao is a $\mathbb{Z}[v,v^{-1}]$-basis. Particularly the parametrization of the basis by aperiodic modules of tame quivers and irreducible characters of $S_{n}$ naturally arises in the construction.

In this paper, we shall review these constructions. Since we shall use representations of quivers, we mainly consider the quantum group corresponding to a symmetric generalized Cartan matrix.

Let $I$ be a finite index set, $A=(a_{ij})_{i,j\in I}$ be a symmetric generalized Cartan matrix. Denote by $\mathbf{U}$ the quantum group associated with the Cartan matrix $A$ and $\mathbf{U}^{+}$ the positive part of $\mathbf{U}$ ([11]).

Fix an indeterminate $v$. For any $n\in\mathbb{Z}$, set

Let $[0]_{v}!=1$ and $[n]_{v}!=[n]_{v}[n-1]_{v}\cdots[1]_{v}$ for any $n\in\mathbb{Z}_{>0}$.

Note that $\mathbf{U}^{+}$ is an associate algebra over $\mathbb{Q}(v)$ generated by the elements $E_{i}$ for various $i\in I$ subject to the quantum Serre relations

for all $i\neq j\in I$, where $E_{i}^{(n)}=E_{i}^{n}/[n]_{v}!$.

Let $\,\,\bar{}:\mathbf{U}^{+}\rightarrow\mathbf{U}^{+}$ be the unique $\mathbb{Q}$-algebra involution such that

Let $Q=(I,H,s,t)$ be a quiver without loops, where $I$ is the set of vertices, $H$ is the set of arrows and $s,t:H\rightarrow I$ are two maps sending an arrow $h\in H$ to the source $s(h)$ and target $t(h)$ respectively. Let

Then $A=(a_{ij})_{i,j\in I}$ is a symmetric generalized Cartan matrix and called the generalized Cartan matrix associated to the quiver $Q$. The Euler form on $\mathbb{Z}I$ is defined as

and the symmetric Euler form is defined as $(\nu,\nu^{\prime})=\langle\nu,\nu^{\prime}\rangle+\langle\nu^{\prime},\nu\rangle$ for any $\nu=\sum_{i\in I}\nu_{i}i$ and $\nu^{\prime}=\sum_{i\in I}\nu^{\prime}_{i}i$.

Let $k=\mathbb{F}_{q}$ be a finite field with $q$ elements and $kQ$ be the path algebra. Denote by $\textrm{rep}_{k}Q$ the abelian category of finite dimensional representations of $Q$ over $k$. There is an isomorphism between $\textrm{rep}_{k}Q$ and the category $\textrm{mod-}kQ$ of finite dimensional $kQ$-modules.

Let $\mathcal{P}_{k}$ be the set of isomorphism classes of objects in $\textrm{rep}_{k}Q$. For any $\alpha\in\mathcal{P}_{k}$, choose an object $M_{\alpha}\in\textrm{rep}_{k}Q$ such that the isomorphism classes $[M_{\alpha}]=\alpha$. Define the dimension vector of $\alpha\in\mathcal{P}_{k}$ by $\underline{\dim}\,\alpha=\underline{\dim}M_{\alpha}$.

Given three elements $\alpha_{1},\alpha_{2}$ and $\alpha$ in $\mathcal{P}_{k}$, denote by $g^{\alpha}_{\alpha_{1}\alpha_{2}}$ the number of subrepresentations $N$ of $M_{\alpha}$ such that $N\simeq M_{\alpha_{2}}$ and $M_{\alpha}/N\simeq M_{\alpha_{1}}$ in $\textrm{rep}_{k}Q$. Let $v_{q}=\sqrt{q}\in\mathbb{C}$. The twisted Ringel-Hall algebra $\mathcal{H}^{\ast}(kQ)$ is the $\mathbb{Q}(v_{q})$-space with basis

whose multiplication is given by

For any $\alpha\in\mathcal{P}_{k}$, let

The set $\{\langle{M_{\alpha}}\rangle\,\,|\,\,\alpha\in\mathcal{P}_{k}\}$ is also a $\mathbb{Q}(v_{q})$-basis of $\mathcal{H}^{\ast}(kQ)$. There is a bilinear form $(-,-)$ on $\mathcal{H}^{\ast}(kQ)$ defined in [5].

Denote by $\mathcal{C}^{\ast}(kQ)$ the composition subalgebra of $\mathcal{H}^{\ast}(kQ)$ generated by $u_{i}=u_{[S_{i}]}$ for all $i\in I$, where $S_{i}$ is the simple $kQ$-module corresponding to $i\in I$.

Then we shall recall the generic form of $\mathcal{C}^{\ast}(kQ)$. Let $\mathcal{K}$ be a set of some finite fields $k$ such that the set $\{q_{k}=|k|\,\,|\,\,k\in\mathcal{K}\}$ is an infinite set. Consider the direct product

and the elements $v=(v_{q_{k}})_{k\in\mathcal{K}}$, $v^{-1}=(v_{q_{k}}^{-1})_{k\in\mathcal{K}}$ and $u_{i}=(u_{i}(k))_{k\in\mathcal{K}}$. By $\mathcal{C}^{\ast}(Q)_{\mathbb{Q}[v,v^{-1}]}$ we denote the subalgebra of $\mathcal{H}^{\ast}(Q)$ generated by $v$, $v^{-1}$ and $u_{i}$ over $\mathbb{Q}$. We may regard it as a $\mathbb{Q}[v,v^{-1}]$-algebra generated by $u_{i}$, where $v$ is viewed as an indeterminate. Finally, define $\mathcal{C}^{\ast}(Q)=\mathbb{Q}(v)\otimes_{\mathbb{Q}[v,v^{-1}]}\mathcal{C}^{\ast}(Q)_{\mathbb{Q}[v,v^{-1}]}$, which is called the generic twisted composition algebra of $Q$.

Under this isomorphism, the bar involution on $\mathbf{U}^{+}$ induces a bar involution $\bar{}:\mathcal{C}^{\ast}(Q)\rightarrow\mathcal{C}^{\ast}(Q)$ such that $\overline{v^{n}}=v^{-n}$ and $\overline{u_{i}}=u_{i}.$

In [8, 9], Lusztig gave a geometric realization of $\mathbf{U}^{+}$. Let $Q$ be a quiver without loops with associated generalized Cartan matrix $A$. Lusztig considered the variety $E_{\nu}$ consisting of representations with dimension vector $\nu\in\mathbb{N}I$ of the quiver $Q$ over algebraically closed field $\bar{k}$ and the category $\mathcal{Q}_{\nu}$ of some semisimple complexes of constructible sheaves on $E_{\nu}$. Let $K(\mathcal{Q}_{\nu})$ be the Grothendieck group of $\mathcal{Q}_{\nu}$. Considering all dimension vectors, let

Lusztig define induction functors on $\mathcal{Q}_{\nu}$ and get a $\mathbb{Z}[v,v^{-1}]$-algebra structure on $K(\mathcal{Q})$. This algebra is isomorphic to the integral form of $\mathbf{U}^{+}$. The set $\mathbf{B}$ of the isomorphism classes of simple perverse sheaves gives a basis of $\mathbf{U}^{+}$, which is called the canonical basis.

Assume that $Q$ is a Dynkin quiver. In this case, the algebraic construction of canonical basis was introduced by Lusztig in [8] (see also [15]).

Let $A=(a_{ij})_{i,j\in I}$ be the generalized Cartan matrix associated to the quiver $Q$ and denote by $\Delta^{+}$ the set of positive roots of the Lie algebra $\mathfrak{g}(A)$ with $i$ corresponding to simple roots. The dimension vector $\underline{\dim}$ induces a bijection between the set of isomorphism classes of indecomposable objects ind-$\mathcal{P}$ and the set $\Delta^{+}$ by Gabriel theorem. Given a positive root $\alpha$, choose an indecomposable representation $M_{\alpha}$ of $Q$ such that $[M_{\alpha}]=\alpha$.

Denote by $\mathbb{N}^{\Delta^{+}}$ the set of all functions $\phi:\Delta^{+}\rightarrow\mathbb{N}$. For each $\phi:\Delta^{+}\rightarrow\mathbb{N}$, define a representation

Then $\mathcal{P}=\{[M_{\phi}]\,\,|\,\,\phi\in\mathbb{N}^{\Delta^{+}}\}$.

Now $\mathcal{H}^{\ast}(kQ)$ is spanned by the set

as $\mathbb{Q}(v)$-vector space, which is called a PBW-type basis.

Since $Q$ is representation-directed, we can define a total order on the set $\Delta^{+}$ such that

for any $\alpha,\beta\in\Delta^{+}$. This total order induces an order on $\mathbb{N}^{\Delta^{+}}$. For any $\phi,\psi:\Delta^{+}\rightarrow\mathbb{N}$, define $\phi<\psi$ if and only if there exists $\alpha\in\Delta^{+}$ such that $\phi(\alpha)>\psi(\alpha)$ and $\phi(\beta)=\psi(\beta)$ for all $\alpha>\beta\in\Delta^{+}$.

For each $\phi:\Delta^{+}\rightarrow\mathbb{N}$, there exists a monomial $m_{\phi}$ on the divided powers of Chevalley generators $u_{i}$ satisfying

with $a^{\phi^{\prime}}_{\phi}\in\mathbb{Z}[v,v^{-1}]$. Since $\overline{m_{\phi}}=m_{\phi}$, we have

with $b^{\phi^{\prime}}_{\phi}\in\mathbb{Z}[v,v^{-1}]$ such that

1. (1)

   $b^{\phi}_{\phi}=1$ for all $\phi$ in $\mathbb{N}^{\Delta^{+}}$;

2. (2)

   for all $\phi^{\prime}\leq\phi$ in $\mathbb{N}^{\Delta^{+}}$,

   $$\sum_{\phi^{\prime\prime},\phi^{\prime}\leq\phi^{\prime\prime}\leq\phi}\overline{b^{\phi^{\prime}}_{\phi^{\prime\prime}}}b^{\phi^{\prime\prime}}_{\phi}=\delta_{\phi,\phi^{\prime}}.$$

Here we need a lemma by Lusztig, which can be obtained by an elementary linear algebra method.

By Lemma 3.1, there exists a unique family of elements $c^{\phi^{\prime}}_{\phi}\in\mathbb{Z}[v^{-1}]$ defined for all $\phi^{\prime}\leq\phi$ in $\mathbb{N}^{\Delta^{+}}$ such that

1. (1)

   $c^{\phi}_{\phi}=1$ for all $\phi$ in $\mathbb{N}^{\Delta^{+}}$;

2. (2)

   $c^{\phi^{\prime}}_{\phi}\in v^{-1}\mathbb{Z}[v^{-1}]$ for all $\phi^{\prime}\leq\phi$ in $\mathbb{N}^{\Delta^{+}}$;

3. (3)

   for all $\phi^{\prime}\leq\phi$ in $\mathbb{N}^{\Delta^{+}}$,

   $$c^{\phi^{\prime}}_{\phi}=\sum_{\phi^{\prime\prime},\phi^{\prime}\leq\phi^{\prime\prime}\leq\phi}\overline{c^{\phi^{\prime}}_{\phi^{\prime\prime}}}b^{\phi^{\prime\prime}}_{\phi}.$$

For any $\phi\in\mathbb{N}^{\Delta^{+}}$, let

These formulas hold for every finite fields and may be viewed as formulas in $\mathcal{H}^{\ast}(Q)=\mathcal{C}^{\ast}(Q)$. Then we have the following theorem.

Under the isomorphism between $\mathcal{H}^{\ast}(Q)$ and $\mathbf{U}^{+}$, the set

induces a basis of $\mathbf{U}^{+}$. This basis is just the canonical basis $\mathbf{B}$ of $\mathbf{U}^{+}$, by Theorem 3.2 and the uniqueness of canonical basis of $\mathbf{U}^{+}$.

In this section, we shall recall the construction of canonical basis given by Beck-Nakajima ([2][13]).

Let $A=(a_{ij})_{i,j\in I}$ be a generalized Cartan matrix of affine type, where $I=\{0,1,\ldots,n\}$, $0\in I$ is the exceptional point and $I_{0}=I\backslash\{0\}$. Let

be a diagonal matrix such that $DA$ is symmetric. Let $\Delta^{+}$ be the set of positive roots and $R$ the set of all positive real roots. Let $\{\alpha_{i}\,\,|\,\,i\in I\}$ be the set of simple roots. Let $v_{i}=v^{d_{i}}$.

We follow the notations of [2]. Denote by $\hat{W}$ the affine Weyl group generated by simple reflections $s_{i}$ for $i\in I$. Let $\tilde{W}$ be the extended affine Weyl group. Then $\tilde{W}=J\ltimes\hat{W}$, where $J$ is a subgroup of the group of Dynkin diagram automorphism and $\tau s_{i}=s_{\tau(i)}\tau\in\tilde{W}$ for $\tau\in J,s_{i}\in\hat{W}$. For any $i\in I_{0}$, there exists $t_{\tilde{w}_{i}}\in\tilde{W}$ such that

where the minimal imaginary positive root $\delta=\sum a_{i}\alpha_{i}$ and $a_{0}=1$.

Let $s_{i_{1}}s_{i_{2}}\cdots s_{i_{N}}\tau$ be a reduced expression of $t_{\tilde{w}_{n}}t_{\tilde{w}_{n-1}}\cdots t_{\tilde{w}_{1}}$. Define an infinite sequence

in $I$ such that $i_{k+N}=\tau(i_{k})$ for any $k\in\mathbb{Z}$. Let

and

It is well-known that

For all $j\in I$, denote by $T_{j}$ the Lusztig’s symmetries $T^{\prime\prime}_{j,1}$ in [11]. For any $k\in\mathbb{Z}_{>0}$, let

For any $k\in\mathbb{Z}_{\leq 0}$, let

Then $E_{\beta_{k}}$ are the root vectors for the real roots $\beta_{k}\in R$.

Then we shall define imaginary root vectors. For $k>0$ and $i\in I_{0}$, let

$\tilde{P}_{i,0}=1$ and

Let $\mathscr{P}$ be the set of all partitions and $c_{0}:I_{0}\rightarrow\mathscr{P}$ be a map. For $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots)\in\mathscr{P}$, define

where $t$ is the length of $\lambda$. Denote

Let $\mathcal{E}$ be the set of all such $\bar{c}=(c,c_{0})$, where $c_{0}:I_{0}\rightarrow\mathscr{P}$ is a map and $c:\mathbb{Z}\rightarrow\mathbb{N}$ is a function with finite support. For any $\bar{c}\in\mathcal{E}$ and $p\in\mathbb{Z}$, let

where $i_{p}$ are from the sequence $h$.

For any $p\in\mathbb{Z}$, Beck-Nakajima defined a partial ordering $<_{p}$ on $\mathcal{E}$ such that the following Theorem holds.

The set $\{L(\bar{c},p)\,\,|\,\,\bar{c}\in\mathcal{E},p\in\mathbb{Z}\}$ is called a PBW-type basis of $\mathbf{U}^{+}$.

Beck-Nakajima also proved the following Theorem.

Let $Q$ be the Kronecker quiver with $I=\{0,1\}$ and $H=\{\rho_{1},\rho_{2}\}$:

Let $kQ$ be the path algebra of $Q$ over finite field $k$.

The set of dimension vectors of indecomposable $kQ$-modules is

The dimension vectors $(l+1,l)$ and $(n,n+1)$ correspond to preprojective and preinjective indecomposable $kQ$-modules respectively. The subset consisting of $(l+1,l)$ (resp. $(n,n+1)$) is denoted by $Prep$ (resp. $Prei$).

For any $n\in\mathbb{N}$, let $M(n+1,n)$ and $M(n,n+1)$ be the indecomposable $kQ$-module of dimension vectors $(n+1,n)$ and $(n,n+1)$ respectively. For real root vectors, define

and

Then we shall define imaginary root vectors. Let $\delta=(1,1)$. As a special case of the definition in Section 4, let

$\tilde{P}_{0}=1$ and

for $k>0$.

For any partition $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{t})$, let

and

The relation between $\tilde{P}_{\lambda}$ and $S_{\lambda}$ is

where $K_{\lambda\mu}$ is the Kostka number associated to the partitions $\lambda$ and $\mu$.

Let $\mathcal{G}$ be the set of $(c=(c_{-},c_{+}),\lambda)$, where $c_{-}:Prep\rightarrow\mathbb{N}$, $c_{+}:Prei\rightarrow\mathbb{N}$ are functions with finite support and $\lambda$ is a partition.

For any $(c,\lambda)\in\mathcal{G}$, consider

and

where

and

The sets $\{N(c,\lambda)\,\,|\,\,(c,\lambda)\in\mathcal{G}\}$ and $\{N^{\prime}(c,\lambda)\,\,|\,\,(c,\lambda)\in\mathcal{G}\}$ are called PBW-type bases of $\mathcal{C}^{\ast}(Q)$.