Quantum cluster characters of Hall algebras

Abstract

The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object $V$ of a finitary Abelian category ${\mathcal {C}}$ over a finite field ${\mathbb {F}}_q$ and any sequence $\mathbf {i}$ of simple objects in ${\mathcal {C}}$ the element $X_{V,\mathbf {i}}$ of the corresponding algebra $P_{{\mathcal {C}},\mathbf {i}}$ of $q$-polynomials. We prove that if ${\mathcal {C}}$ was hereditary, then the assignments $V\mapsto X_{V,\mathbf {i}}$ define algebra homomorphisms from the (dual) Hall–Ringel algebra of ${\mathcal {C}}$ to the $P_{{\mathcal {C}},\mathbf {i}}$, which generalize the well-known Feigin homomorphisms from the upper half of a quantum group to $q$-polynomial algebras. If ${\mathcal {C}}$ is the representation category of an acyclic valued quiver $(Q,\mathbf {d})$ and $\mathbf {i}=(\mathbf {i}_0,\mathbf {i}_0)$, where $\mathbf {i}_0$ is a repetition-free source-adapted sequence, then we prove that the $\mathbf {i}$-character $X_{V,\mathbf {i}}$ equals the quantum cluster character $X_V$ introduced earlier by the second author in Rupel (Int Math Res Not 14:3207–3236, 2011; Quantum cluster characters of valued quivers, arXiv:1109.6694). Using this identification, we deduce a quantum cluster structure on the quantum unipotent cell corresponding to the square of a Coxeter element. As a corollary, we prove a conjecture from the joint paper Berenstein and Zelevinsky (Adv Math 195(2):405–455, 2005) of the first author with A. Zelevinsky for such quantum unipotent cells. As a by-product, we construct the quantum twist and prove that it preserves the triangular basis introduced by A. Zelevinsky and the first author in Berenstein and Zelevinsky (Int Math Res Not. doi:10.1093/imrn/rns268, 2014).

Appendix: Twisted bialgebras in braided monoidal categories

Let $\Bbbk$ be a field and $\varGamma$ an additive monoid. For any unitary bicharacter $\chi :\varGamma \times \varGamma \rightarrow \Bbbk ^\times$, let ${\mathcal {C}}_\chi$ be the tensor category of $\varGamma$-graded vector spaces $V=\bigoplus _{\gamma \in \varGamma } V(\gamma )$ such that each component $V(\gamma )$ is finite dimensional. Clearly, this category is braided via $\varPsi _{U,V}:U\otimes V\rightarrow V\otimes U$ given by

$$\begin{aligned} \varPsi _{U,V}\left( u_\gamma \otimes v_{\gamma '}\right) =\chi (\gamma ,\gamma ') \cdot v_{\gamma '}\otimes u_\gamma \end{aligned}$$

for any $u_\gamma \in U(\gamma ), v_{\gamma '}\in V(\gamma ')$.

Let ${\mathcal {U}}=\bigoplus _{\gamma \in \varGamma }{\mathcal {U}}(\gamma )$ be a bialgebra in ${\mathcal {C}}_\chi$. Denote by $\hat{\mathcal {U}}$ the completion of ${\mathcal {U}}$ with respect to the grading, that is, the space of all formal series $\tilde{u}=\sum _{\gamma \in \varGamma } u_\gamma$, where $u_\gamma \in {\mathcal {U}}(\gamma )$. For each such $\tilde{u}$ denote by $\mathrm{Supp}(\tilde{u})$ the submonoid of $\varGamma$ generated by $\{\gamma :u_\gamma \ne 0\}$.

From now on, we assume that for any $\gamma \in \varGamma$ the set

$$\begin{aligned} A_\gamma =\left\{ (\gamma ',\gamma ''):\gamma '+\gamma ''=\gamma \right\} \end{aligned}$$

of two-part partitions of $\gamma$ is finite. It is easy to see that, under this assumption, $\hat{\mathcal {U}}$ has a well-defined multiplication. The coproduct on ${\mathcal {U}}$ extends to $\hat{\varDelta }:\hat{\mathcal {U}}\rightarrow {\mathcal {U}}\hat{{\bigotimes }} {\mathcal {U}}$ so that $\hat{\mathcal {U}}$ becomes a complete bialgebra. The following fact is obvious.

Lemma 9.1

Let $E=\sum _{\gamma \in \varGamma } E^{(\gamma )}$ be a formal series, where each $E^{(\gamma )}\in {\mathcal {U}}(\gamma )$. Then, $E$ is grouplike in $\hat{{\mathcal {U}}}$ (i.e., $\hat{\varDelta }(E)=E\otimes E$) if and only if

$$\begin{aligned} \varDelta (E^{(\gamma )})=\sum _{(\gamma ',\gamma '')\in A_\gamma } E^{(\gamma ')}\otimes E^{(\gamma '')} \end{aligned}$$

for each $\gamma \in \varGamma$.

As a corollary of Lemma 9.1, we have the following well-known result.

Lemma 9.2

If $x\in {\mathcal {U}}$ is primitive, i.e., $\varDelta (x)=x\otimes 1+1\otimes x$ and $\varPsi _{{\mathcal {U}},{\mathcal {U}}}(x\otimes x)=qx\otimes x$ for some non-root of unity $q\in \Bbbk ^\times$, then the braided exponential

$$\begin{aligned} \mathrm{exp}_q(x)=\sum _{k=0}^\infty \frac{1}{(k)_q!} x^k \end{aligned}$$

of $x$ is grouplike in $\hat{{\mathcal {U}}}$, where $(k)_q!=(1)_q\ldots (k)_q$ and $(\ell )_q=\frac{q^\ell -1}{q-1}$.

However, the product of grouplike elements is not always grouplike. We can sometimes restore the grouplike property of a product by twisting the factors with elements of an appropriate non-commutative algebra ${\mathcal {P}}$ in ${\mathcal {C}}_\chi$. This, contained in Proposition 9.4, is the main idea behind the forthcoming theorem.

Now we define the restricted dual algebra ${\mathcal {A}}$ of ${\mathcal {U}}$. As a vector space, ${\mathcal {A}}$ is the set of all $\Bbbk$-linear forms $x:{\mathcal {U}}\rightarrow \Bbbk$ such that $x$ vanishes on ${\mathcal {U}}(\gamma )$ for all but finitely many $\gamma \in \varGamma$. In other words, ${\mathcal {A}}\cong \bigoplus _{\gamma \in \varGamma } {\mathcal {A}}(\gamma )$ where ${\mathcal {A}}(\gamma )={\mathrm{Hom }}_\Bbbk ({\mathcal {U}}(\gamma ),\Bbbk )$. Clearly, ${\mathcal {A}}$ is an algebra in ${\mathcal {C}}_\chi$ with the product (resp. unit) adjoint of the coproduct $\varDelta$ (resp. counit) on ${\mathcal {U}}$.

Let $\mathbf{E}=(E_1,\ldots ,E_m)$ be a family of grouplike elements

$$\begin{aligned} E_k=\sum _{\gamma \in \varGamma } E_k^{(\gamma )} \end{aligned}$$

in $\hat{\mathcal {U}}$. We say that $\mathbf{E}$ is ${\mathcal {P}}$-adapted if for each $k=1,\ldots ,m$ there exists a homomorphism $\tau _k$ from the monoid $\mathrm{Supp}(E_k)$ to the multiplicative monoid of ${\mathcal {P}}$ such that

$$\begin{aligned} \tau _\ell (\gamma _\ell )\tau _k(\gamma _k)=\chi (\gamma _k,\gamma _\ell )\cdot \tau _k(\gamma _k) \tau _\ell (\gamma _\ell ) \end{aligned}$$

(9.1)

for all $k<\ell$ and $\gamma _k\in \mathrm{Supp}(E_k)$. For every ${\mathcal {P}}$-adapted family $\mathbf{E}$, we define a map $\varPsi _\mathbf{E}:{\mathcal {A}} \rightarrow {\mathcal {P}}$ by the formula

$$\begin{aligned} \varPsi _\mathbf{E}(x)=\sum _{\gamma _1\in \mathrm{Supp}(E_1),\ldots ,\gamma _m\in \mathrm{Supp}(E_m)}x\big (E_1^{(\gamma _1)}\ldots E_m^{(\gamma _m)}\big )\tau _1(\gamma _1)\ldots \tau _m(\gamma _m), \end{aligned}$$

(9.2)

where we denote by $(x,u)\mapsto x(u)$ the natural non-degenerate evaluation pairing ${\mathcal {A}}\times {\mathcal {U}}\rightarrow \Bbbk$. Note that the sum in (9.2) is always finite because $x$ vanishes on all but finitely many homogeneous components of ${\mathcal {U}}$.

Theorem 9.3

For any ${\mathcal {P}}$-adapted family $\mathbf{E}$ of grouplike elements, the map $\varPsi _\mathbf{E}:{\mathcal {A}} \rightarrow {\mathcal {P}}$ defined by (9.2) is a homomorphism of $\varGamma$-graded algebras.

Proof

For any $\Bbbk$-algebra ${\mathcal {P}}$ denote ${\mathcal {U}}_{\mathcal {P}}:={\mathcal {U}} \bigotimes {\mathcal {P}}$ and view it as an algebra with the standard (NOT braided!) algebra structure. We will often abbreviate $u\cdot t:=u\otimes t$ for $u\in {\mathcal {U}}, t\in {\mathcal {P}}$.

Denote by $\hat{\mathcal {U}}_{\mathcal {P}}$ the completion of ${\mathcal {U}}_{\mathcal {P}}$, i.e., $\hat{\mathcal {U}}_{\mathcal {P}}=\hat{\mathcal {U}} \bigotimes {\mathcal {P}}$ is the space of formal series of the form $\sum _{\gamma \in \varGamma } u_\gamma \cdot t_\gamma$, where $u_\gamma \in {\mathcal {U}}(\gamma )$ and $t_\gamma \in {\mathcal {P}}$. Consider the tensor square ${\mathcal {V}}_{\mathcal {P}}={\mathcal {U}}_{\mathcal {P}}\bigotimes _{\mathcal {P}}{\mathcal {U}}_{\mathcal {P}}$ where the left factor is regarded as a right ${\mathcal {P}}$-module and the right factor as a left ${\mathcal {P}}$-module. Note that ${\mathcal {V}}_{\mathcal {P}}$ is a ${\mathcal {P}}$-bimodule in which we can write $t(u\otimes v)=(tu)\otimes v=u\otimes (tv)=(u\otimes v)t$ for any $u,v\in {\mathcal {U}}, t\in {\mathcal {P}}$. Under the standard identification

$$\begin{aligned} {\mathcal {V}}_{\mathcal {P}}\cong ({\mathcal {U}} \bigotimes {\mathcal {U}})\bigotimes {\mathcal {P}}, \end{aligned}$$

this bimodule ${\mathcal {V}}_{\mathcal {P}}$ becomes an algebra.

We will also need the completed tensor square $\hat{\mathcal {V}}_{\mathcal {P}}={\mathcal {U}}_{\mathcal {P}}{\hat{\bigotimes }_{\mathcal {P}}} {\mathcal {U}}_{\mathcal {P}}$. There is a natural morphism of ${\mathcal {P}}$-bimodules

$$\begin{aligned} \hat{\varDelta }_{\mathcal {P}}:\hat{\mathcal {U}}_{\mathcal {P}}\rightarrow \hat{\mathcal {V}}_{\mathcal {P}}\end{aligned}$$

which is the ${\mathcal {P}}$-linear extension of the coproduct $\hat{\varDelta }$ on $\hat{\mathcal {U}}$. Clearly, $\hat{\varDelta }_{\mathcal {P}}$ is an algebra homomorphism.

For each ${\mathcal {P}}$-adapted family $\mathbf{E}$ define an element $\tilde{E} \in \hat{\mathcal {U}}_{\mathcal {P}}$ as follows:

$$\begin{aligned} \tilde{E}=\tilde{E}_1\ldots \tilde{E}_m, \end{aligned}$$

where

$$\begin{aligned} \tilde{E}_k=\sum _{\gamma \in \mathrm{Supp}(E_k)} E_k^{(\gamma )} \cdot \tau _k(\gamma ). \end{aligned}$$

Proposition 9.4

For any ${\mathcal {P}}$-adapted family of grouplike elements $\mathbf{E}$, the element $\tilde{E}\in \hat{\mathcal {U}}_{\mathcal {P}}$ is grouplike, i.e.,

$$\begin{aligned} \varDelta _{\mathcal {P}}(\tilde{E})=\tilde{E}\otimes \tilde{E}. \end{aligned}$$

Proof

We need the following fact.

Lemma 9.5

In the assumptions of Proposition 9.4, one has:

• (a) each $\tilde{E}_k$ is a grouplike element in $\hat{\mathcal {U}}_{\mathcal {P}}$.

• (b) $(1\otimes \tilde{E}_k)(\tilde{E}_\ell \otimes 1)=(\tilde{E}_\ell \otimes 1) (1 \otimes \tilde{E|}_k)$ for any $1\le k<l\le m$.

Proof

To prove (a), note that by Lemma 9.1 we have

$$\begin{aligned} \hat{\varDelta }_{\mathcal {P}}(\tilde{E}_k)&= \sum _{\gamma \in \mathrm{Supp}(E_k)}\varDelta (E_k^{(\gamma )})\cdot \tau _k(\gamma )=\sum _{\gamma ',\gamma ''\in \mathrm{Supp}(E_k)}E_k^{(\gamma ')}\otimes E_k^{(\gamma '')}\tau _k(\gamma '+\gamma '') \\&= \sum _{\gamma ',\gamma ''\in \mathrm{Supp}(E_k)}E_k^{(\gamma ')}\cdot \tau _k(\gamma ')\otimes E_k^{(\gamma ')}\tau _k(\gamma '')=\tilde{E}_k\otimes \tilde{E}_k, \end{aligned}$$

where we have used the multiplicativity of $\tau _k$: $\tau _k(\gamma '+\gamma '')=\tau _k(\gamma ')\tau _k(\gamma '')$.

To prove (b), abbreviate $\tilde{E}_k^{(\gamma )}:=E_k^{(\gamma )}\cdot \tau _k(\gamma )$ for $k=1,\ldots ,m, \gamma \in \mathrm{Supp}(E_k)$. Then for $k<\ell$ and $\gamma _k\in \mathrm{Supp}(E_k), \gamma _\ell \in \mathrm{Supp}(E_\ell )$ we deduce the following commutation relation using (9.1):

$$\begin{aligned} \left( 1\otimes \tilde{E}_k^{\left( \gamma _k\right) }\right) \left( \tilde{E}_\ell ^{\left( \gamma _\ell \right) }\otimes 1\right)&= \left( 1\otimes E_k^{\left( \gamma _k\right) }\right) \left( E_\ell ^{\left( \gamma _\ell \right) }\otimes 1\right) \cdot \tau _k\left( \gamma _k\right) \tau _\ell \left( \gamma _\ell \right) \\&= \left( E_\ell ^{\left( \gamma _\ell \right) }\otimes 1\right) \left( 1\otimes E_k^{\left( \gamma _k\right) }\right) \cdot \chi \left( \gamma _k,\gamma _\ell \right) \tau _k\left( \gamma _k\right) \tau _\ell \left( \gamma _\ell \right) \\&= \left( E_\ell ^{\left( \gamma _\ell \right) }\otimes 1\right) \left( 1\otimes E_k^{\left( \gamma _k\right) }\right) \cdot \tau _\ell \left( \gamma _\ell \right) \tau _k\left( \gamma _k\right) \\&= \left( \tilde{E}_\ell ^{\left( \gamma _\ell \right) }\otimes 1\right) \left( 1\otimes \tilde{E}_k^{\left( \gamma _k\right) }\right) . \end{aligned}$$

Since $\tilde{E}_k=\sum _{\gamma _k\in \mathrm{Supp}(E_k)}\tilde{E}_k^{(\gamma _k)}$ and $\tilde{E}_\ell =\sum _{\gamma _\ell \in \mathrm{Supp}(E_\ell )} \tilde{E}_\ell ^{(\gamma _\ell )}$, we obtain the desired result:

$$\begin{aligned} \left( 1\otimes \tilde{E}_k\right) \left( \tilde{E}_\ell \otimes 1\right)&= \sum \limits _{\gamma _k\in \mathrm{Supp}\left( E_k\right) ,\gamma _\ell \in \mathrm{Supp}\left( E_\ell \right) }\left( 1\otimes \tilde{E}_k^{\left( \gamma _k\right) }\right) \left( \tilde{E}_\ell ^{\left( \gamma _\ell \right) }\otimes 1\right) \\&= \sum \limits _{\gamma _k\in \mathrm{Supp}\left( E_k\right) ,\gamma _\ell \in \mathrm{Supp}\left( E_\ell \right) }\left( \tilde{E}_\ell ^{\left( \gamma _\ell \right) }\otimes 1\right) \left( 1\otimes \tilde{E}_k^{\left( \gamma _k\right) }\right) \\&= \left( \tilde{E}_\ell \otimes 1\right) \left( 1 \otimes \tilde{E}_k\right) . \end{aligned}$$

Lemma 9.5 is proved. $\square$

Now we are ready to finish the proof of Proposition 9.4. Using Lemma 9.5 and the identities $\tilde{u}\otimes \tilde{v}=(\tilde{u}\otimes 1)(1\otimes \tilde{v}), (\tilde{u}\otimes 1)(\tilde{v}\otimes 1)=\tilde{u}\tilde{v}\otimes 1$ and $(1\otimes \tilde{u})(1\otimes \tilde{v})=1\otimes \tilde{u}\tilde{v}$, for any $\tilde{u},\tilde{v}\in \hat{\mathcal {U}}_{\mathcal {P}}$, we compute

$$\begin{aligned} \hat{\varDelta }_{\mathcal {P}}(\tilde{E})&= \hat{\varDelta }_{\mathcal {P}}(\tilde{E}_1\ldots \tilde{E}_m)=\hat{\varDelta }_{\mathcal {P}}(\tilde{E}_1)\ldots \hat{\varDelta }_{\mathcal {P}}(\tilde{E}_m)=(\tilde{E}_1\otimes \tilde{E}_1)\ldots (\tilde{E}_m\otimes \tilde{E}_m) \\&= (\tilde{E}_1\otimes 1)(1\otimes \tilde{E}_1)\ldots (\tilde{E}_m\otimes 1)(1\otimes \tilde{E}_m)\\&= \big ((\tilde{E}_1\otimes 1)\ldots (\tilde{E}_m\otimes 1)\big )\big ((1\otimes \tilde{E}_1)\ldots (1\otimes \tilde{E}_m)\big )\\&= (\tilde{E}\otimes 1)(1\otimes \tilde{E})=\tilde{E}\otimes \tilde{E}. \end{aligned}$$

Proposition 9.4 is proved. $\square$

Finally, we define the pairing ${\mathcal {A}}\times \hat{\mathcal {U}}_{\mathcal {P}}\rightarrow {\mathcal {P}}$ by:

$$\begin{aligned} x\left( \sum u_\gamma \cdot t_\gamma \right) =\sum x(u_\gamma )t_\gamma . \end{aligned}$$

Clearly, the pairing is well-defined because all but finitely many terms $x(u_\gamma )$ are $0$ for each $u\in {\mathcal {A}}$. For every ${\mathcal {P}}$-adapted family $\mathbf{E}$, we see that the map $\varPsi _\mathbf{E}:{\mathcal {A}} \rightarrow P$ defined in (9.2) is given by the formula $\varPsi _\mathbf{E}(x):=x(\tilde{E})$.

The definition of the multiplication in ${\mathcal {A}}$ implies that $(xy)(\tilde{u})=(x\otimes y)(\hat{\varDelta }_{\mathcal {P}}(\tilde{u}))$ for all $x,y\in {\mathcal {A}}$ and $\tilde{u}\in \hat{\mathcal {U}}_{\mathcal {P}}$, where

$$\begin{aligned} (x\otimes y)\left( \tilde{u}_1\otimes \tilde{u}_2\right) :=x(\tilde{u}_1)y(\tilde{u}_2) \end{aligned}$$

for any $\tilde{u}_1,\tilde{u}_2\in \hat{\mathcal {U}}_{\mathcal {P}}$. Thus, we have

$$\begin{aligned} \varPsi _\mathbf{E}(xy)&= (xy)(\tilde{E})=(x\otimes y)(\hat{\varDelta }_{\mathcal {P}}(\tilde{E}))=(x\otimes y)(\tilde{E}\otimes \tilde{E})\\&= x(\tilde{E})y(\tilde{E})=\varPsi _\mathbf{E}(x)\varPsi _\mathbf{E}(y), \end{aligned}$$

which finishes the proof of Theorem 9.3. $\square$

For each $u\in \mathcal {U}$ define the linear operators $x\mapsto u(x)$ and $x\mapsto u^{op}(x)$ on ${\mathcal {A}}$ by:

$$\begin{aligned} u(x)(u')=x(u'u),~ u^{op}(x)(u')=x(uu') \end{aligned}$$

for all $u'\in \mathcal {U}, x\in {\mathcal {A}}$.

Clearly, the operators $x\mapsto u(x)$ and $x\mapsto u^{op}(x)$ define, respectively, the left and the right $\mathcal {U}$-action on ${\mathcal {A}}$ and $u(x),u^{op}(x)\in {\mathcal {A}}_{\gamma '-\gamma }$ for each homogeneous $u\in \mathcal {U}_\gamma$ and $x\in {\mathcal {A}}_{\gamma '}$.

Using this in the form $x(u_1\ldots u_m)=(u_1\ldots u_m(x))(1)=u_m^{op}\ldots u_1^{op}(x)(1)$, we rewrite (9.2) for any homogeneous $x\in {\mathcal {A}}_\gamma$ as:

$$\begin{aligned}&\varPsi _\mathbf{E}(x)=\sum E_1^{(\gamma _1)}\ldots E_m^{(\gamma _m)}(x)\cdot \tau _1(\gamma _1)\ldots \tau _m(\gamma _m), \end{aligned}$$

(9.3)

$$\begin{aligned}&\varPsi _\mathbf{E}(x)=\sum {E_m^{(\gamma _m)}}^{op}\ldots {E_1^{(\gamma _1)}}^{op}(x)\cdot \tau _1(\gamma _1)\ldots \tau _m(\gamma _m), \end{aligned}$$

(9.4)

where the summation is over all $(\gamma _1,\ldots ,\gamma _m)\in \mathrm{Supp}(E_1)\times \cdots \times \mathrm{Supp}(E_m)$ such that $\gamma _1+\cdots +\gamma _m=\gamma$.

We finish with the following obvious, however, useful fact.

Lemma 9.6

Let $E\in \mathcal {U}_\alpha$ be any homogeneous primitive element. Then for any $x\in {\mathcal {A}}_\gamma$ and $y\in {\mathcal {A}}$, one has

$$\begin{aligned} E(yx)=\chi (\gamma ,\alpha )\cdot E(y)x+ yE(x),~E^{op}(xy)=E^{op}(x)y+ \chi (\alpha ,\gamma )\cdot xE^{op}(y). \end{aligned}$$