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).
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Acknowledgments
We are grateful to Christof Geiss, Jacob Greenstein, and Andrei Zelevinsky (who sadly passed away on April 10, 2013) for stimulating discussions. An important part of this work was done during the authors’ visit to the MSRI in the framework of the “Cluster algebras” program and they thank the Institute and the organizers for their hospitality and support. The first author benefited from the hospitality of Institut des Hautes Études Scientiques and Max-Planck-Institut für Mathematik, which he gratefully acknowledges. The authors are immensely grateful to Gleb Koshevoy for stimulating discussions on the final stage of work on this paper.
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To the memory of Andrei Zelevinsky.
A. Berenstein was supported in part by the NSF grant DMS-1101507.
Appendix: Twisted bialgebras in braided monoidal categories
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
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
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
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
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
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
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
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
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
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:
where
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.,
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
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):
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:
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
Proposition 9.4 is proved. \(\square \)
Finally, we define the pairing \({\mathcal {A}}\times \hat{\mathcal {U}}_{\mathcal {P}}\rightarrow {\mathcal {P}}\) by:
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
for any \(\tilde{u}_1,\tilde{u}_2\in \hat{\mathcal {U}}_{\mathcal {P}}\). Thus, we have
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:
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:
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