Let ( be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories and of finite-dimensional representations over the quantum loop algebras of and , respectively. As a consequence, we solve long-standing problems: the positivity of the analogs of Kazhdan–Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced . In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur–Weyl dualities, we prove the analog of Kazhdan–Lusztig conjecture (formulated in [D. Hernandez, Algebraic approach to -characters, Adv. Math. 187 2004, 1, 1–52]) for simple modules in remarkable monoidal subcategories of for any non-simply-laced , and for any simple finite-dimensional modules in for of type . In the course of the proof we obtain and combine several new ingredients. In particular, we establish a quantum analog of T-systems, and also we generalize the isomorphisms of [D. Hernandez and B. Leclerc, Quantum Grothendieck rings and derived Hall algebras, J. reine angew. Math. 701 2015, 77–126, D. Hernandez and H. Oya, Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan–Lusztig algorithm, Adv. Math. 347 2019, 192–272] to all in a unified way, that is, isomorphisms between subalgebras of the quantum group of and subalgebras of the quantum Grothendieck ring of .
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: (A) JP17H01086
Award Identifier / Grant number: (B) JP19H01782
Award Identifier / Grant number: 19K14515
Funding source: European Research Council
Award Identifier / Grant number: 647353 Qaffine
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: NRF-2019R1A2C4069647
Funding statement: Ryo Fujita was supported in part by JSPS Grant-in-Aid for Scientific Research (A) JP17H01086, (B) JP19H01782 and by JSPS Overseas Research Fellowships. David Hernandez was supported by the European Research Council under the European Union’s Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine. Se-jin Oh was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2019R1A2C4069647). Hironori Oya is supported by JSPS Grant-in-Aid for Young Scientists No. 19K14515.
A Quantum unipotent minors
In this appendix section, we briefly review the definitions of the quantum coordinate algebra and the normalized quantum unipotent minors associated with a simply-laced Lie algebra . We will use the notation in Section 4.2.
A.1 Quantum coordinate algebra
Let v be an indeterminate with a formal square root . The quantized enveloping algebra of (over ) is the unital associative -algebra presented by the set of generators
and the following relations:
and for ,
For each generator, we set
which endows with a structure of -graded -algebra.
We denote by the -subalgebra of generated by . This is regarded as a quantized enveloping algebra of the nilpotent Lie subalgebra corresponding to the negative roots. As a -algebra, is presented by the set of generators and the quantum Serre relations (7.1).
For , we define the -linear maps and by
for homogeneous elements . There exists a unique symmetric -bilinear form on such that
Then is non-degenerate.
Let be the -subalgebra of generated by the divided powers . Then the quantized coordinate algebra is defined as
A.2 Quantum unipotent minors
Let V be a -module. For , we set
This is called the weight space of V of weight μ. A -module with weight space decomposition is said to be integrable if and act locally nilpotently on V for all .
For each , we denote by the (finite-dimensional) irreducible highest weight -module generated by a highest weight vector . The module is integrable. There exists a unique -bilinear form
for and , where φ is the -algebra anti-involution on defined by
for . The form is non-degenerate and symmetric.
For , we define the element by
for , where . It is known that this element does not depend on the choice of and (depends only on ). See, for example, [43, Proposition 39.3.7]. Then .
For and , we define an element by the property
for . By the nondegeneracy of the bilinear form , this element is uniquely determined. An element of this form is called a quantum unipotent minor. Moreover, we set
where . This element is called a normalized quantum unipotent minor.
B Quantum cluster algebras
In this appendix section, we briefly review the definition of a quantum cluster algebra following .
B.1 Quantum seed
Let v be an indeterminate with its formal square root . Let J be a finite set. For a -valued -skew-symmetric matrix , we define the quantum torus as the -algebra presented by the set of generators and the relations:
For , we write
where we fixed an arbitrary total ordering of the set J. Note that the resulting element is independent from the choice of the total ordering . Since is an Ore domain, it is embedded in its skew field of fractions .
Let be a subset and set . Let be a -valued -matrix whose principal part is skew-symmetric. Such a matrix is called an exchange matrix.
We say that a pair is compatible if we have
for some positive integer d.
If is a compatible pair, the datum is called a quantum seed. Then the set is called the quantum cluster of Σ and each element is called a quantum cluster variable.
Given a quantum seed and an element , we can associate a new quantum seed as follows.
Define the -matrix and the -matrix by
In addition, define and by
Then the new datum is given by
One can show that the datum actually defines a quantum seed, which is called the mutation of Σ in direction k. This operation is involutive, i.e., we have .
Let be a compatible pair and the associated quantum seed. The quantum cluster algebra is the -subalgebra of the skew field generated by all the quantum cluster variables in the quantum seeds obtained from Σ by any sequence of mutations.
Theorem B.2 (Quantum Laurent phenomenon [2, Corollary 5.2]).
The quantum cluster algebra is contained in the quantum torus .
The authors are grateful to Hiraku Nakajima for helpful discussions, comments and correspondence. The authors would like also to thank Bernard Leclerc and Alex Weekes for their comments. The authors are also thankful to the anonymous referee whose suggestions improve our present paper. A part of this work was done when the first named author enjoyed short-term postdoctoral positions in Kyoto University and in Kavli IPMU during the spring-summer semester in 2020. He thanks both institutions and colleagues for their supports in the complicated situation.
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