Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter February 15, 2022

Isomorphisms among quantum Grothendieck rings and propagation of positivity

Ryo Fujita ORCID logo, David Hernandez ORCID logo, Se-jin Oh ORCID logo and Hironori Oya ORCID logo

Abstract

Let (𝔤,𝗀) be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with 𝗀 being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories 𝒞𝔤 and 𝒞𝗀 of finite-dimensional representations over the quantum loop algebras of 𝔤 and 𝗀, 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 𝔤. 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 q,t-characters, Adv. Math. 187 2004, 1, 1–52]) for simple modules in remarkable monoidal subcategories of 𝒞𝔤 for any non-simply-laced 𝔤, and for any simple finite-dimensional modules in 𝒞𝔤 for 𝔤 of type Bn. 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 𝔤 in a unified way, that is, isomorphisms between subalgebras of the quantum group of 𝗀 and subalgebras of the quantum Grothendieck ring of 𝒞𝔤.

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 𝒜v[N-] and the normalized quantum unipotent minors associated with a simply-laced Lie algebra 𝗀. We will use the notation in Section 4.2.

A.1 Quantum coordinate algebra 𝒜v[N-]

Let v be an indeterminate with a formal square root v1/2. The quantized enveloping algebra 𝒰v(𝗀) of 𝗀 (over (v1/2)) is the unital associative (v1/2)-algebra presented by the set of generators

{eı,fı,Kı±1ıΔ0}

and the following relations:

  1. KıKı-1=Kı-1Kı=1 and KıKȷ=KȷKı for ı,ȷΔ0,

  2. Kıeȷ=v(αı,αȷ)eȷKı,Kıfȷ=v-(αı,αȷ)fȷKı for ı,ȷΔ0,

  3. (v-v-1)[eı,fȷ]=δı,ȷ(Kı-Kı-1) for ı,ȷΔ0,

  4. eı2eȷ-(v+v-1)eıeȷeı-eȷeı2=fı2fȷ-(v+v-1)fıfȷfı-fȷfı2=0 if ıȷ,

  5. [eı,eȷ]=[fı,fȷ]=0 if ıȷ.

For each generator, we set

wt(eı)=αı,wt(fı)=-αı,wt(Kı±1)=0,

which endows 𝒰v(𝗀) with a structure of 𝖰-graded (v1/2)-algebra.

We denote by 𝒰v(𝗇-) the (v1/2)-subalgebra of 𝒰v(𝗀) generated by {fı}ıΔ0. This is regarded as a quantized enveloping algebra of the nilpotent Lie subalgebra 𝗇-𝗀 corresponding to the negative roots. As a (v1/2)-algebra, 𝒰v(𝗇-) is presented by the set of generators {fı}ıΔ0 and the quantum Serre relations (7.1).

For ıΔ0, we define the (v1/2)-linear maps eı and eı:𝒰v(𝗇-)𝒰v(𝗇-) by

eı(xy)=eı(x)y+v(αı,wt(x))xeı(y),eı(fȷ)=δı,ȷ,eı(xy)=v(αı,wt(y))eı(x)y+xeı(y),eı(fȷ)=δı,ȷ

for homogeneous elements x,y𝒰v(𝗇-). There exists a unique symmetric (v1/2)-bilinear form (,)L on 𝒰v(𝗇-) such that

(1,1)L=1,(fıx,y)L=11-v2(x,eı(y))L,(xfı,y)L=11-v2(x,eı(y))L.

Then (,)L is non-degenerate.

Let 𝒰v(𝗇-) be the [v±1/2]-subalgebra of 𝒰v(𝗇-) generated by the divided powers {fı(m)fım/[m]v!ıΔ0,m0}. Then the quantized coordinate algebra𝒜v[N-] is defined as

𝒜v[N-]{x𝒰v(𝗇-)(x,𝒰v(𝗇-))L[v±1/2]}.

A.2 Quantum unipotent minors

Let V be a 𝒰v(𝗀)-module. For μ𝖯, we set

Vμ{uVKıu=v(αı,μ)u for all ıΔ0}.

This is called the weight space of V of weight μ. A 𝒰v(𝗀)-module with weight space decomposition V=μ𝖯Vμ is said to be integrable if eı and fı act locally nilpotently on V for all ıΔ0.

For each λ𝖯+ıΔ00ϖı, we denote by V(λ) the (finite-dimensional) irreducible highest weight 𝒰v(𝗀)-module generated by a highest weight vector uλV(λ)λ. The module V(λ) is integrable. There exists a unique (v1/2)-bilinear form

(,)λφ:V(λ)×V(λ)(v1/2)

such that

(uλ,uλ)λφ=1,(xu1,u2)λφ=(u1,φ(x)u2)λφ

for u1,u2V(λ) and x𝒰v(𝗀), where φ is the (v1/2)-algebra anti-involution on 𝒰v(𝗀) defined by

φ(eı)=fı,φ(fı)=eı,φ(Kı)=Kı

for ıΔ0. The form (,)λφ is non-degenerate and symmetric.

For w𝖶, we define the element uwλV(λ)wλ by

uwλ=fı1(m1)fıl-1(ml-1)fıl(ml)uλ

for (ı1,,ıl)(w), where mk(αık,sık+1sıl-1sılλ)0. It is known that this element does not depend on the choice of (ı1,,ıl)(w) and w𝖶 (depends only on wλ). See, for example, [43, Proposition 39.3.7]. Then (uwλ,uwλ)λφ=1.

For λ𝖯+ and w,w𝖶, we define an element Dwλ,wλ𝒰v(𝗇-) by the property

(Dwλ,wλ,x)L=(uwλ,xuwλ)λφ

for x𝒰v(𝗇-). By the nondegeneracy of the bilinear form (,)L, this element is uniquely determined. An element of this form is called a quantum unipotent minor. Moreover, we set

D~wλ,wλv-(wλ-wλ,wλ-wλ)/4+(wλ-wλ,ρ)/2Dwλ,wλ,

where ρıΔ0ϖı𝖯+. 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 [2].

B.1 Quantum seed

Let v be an indeterminate with its formal square root v1/2. Let J be a finite set. For a -valued J×J-skew-symmetric matrix Λ=(Λij)i,jJ, we define the quantum torus𝒯(Λ) as the [v±1/2]-algebra presented by the set of generators {Xj±1jJ} and the relations:

  1. XjXj-1=Xj-1Xj=1 for jJ,

  2. XiXj=vΛijXjXi for i,jJ.

For 𝒂=(aj)jJJ, we write

X𝒂v-12i<jaiajΛijjJXjaj,

where we fixed an arbitrary total ordering < of the set J. Note that the resulting element X𝒂𝒯(Λ) is independent from the choice of the total ordering <. Since 𝒯(Λ) is an Ore domain, it is embedded in its skew field of fractions 𝔽(𝒯(Λ)).

Let JfJ be a subset and set JeJJe. Let B~=(bij)iJ,jJe be a -valued J×Je-matrix whose principal part(bij)i,jJe is skew-symmetric. Such a matrix B~ is called an exchange matrix.

We say that a pair (Λ,B~) is compatible if we have

kJbkiΛkj=dδi,j(iJe,jJ)

for some positive integer d.

If (Λ,B~) is a compatible pair, the datum Σ=((Xj)jJ,Λ,B~) is called a quantum seed. Then the set {Xj}jJ𝔽(𝒯(Λ)) is called the quantum cluster of Σ and each element Xj is called a quantum cluster variable.

B.2 Mutations

Given a quantum seed Σ=((Xj)jJ,Λ,B~) and an element kJe, we can associate a new quantum seed μk(Σ) as follows.

Define the J×J-matrix E=(eij)i,jJ and the Je×Je-matrix F=(fij)i,jJe by

eij{δi,jif jk,-1if i=j=k,max(0,-bik)if ij=k,fij{δi,jif ik,-1if i=j=k,max(0,-bik)if i=kj.

In addition, define 𝒂=(aj)jJ and 𝒂=(aj′′)jJ by

aj{-1if i=k,max(0,bik)if ik,aj′′{-1if i=k,max(0,-bik)if ik,

Then the new datum μk(Σ)=((Xj)jJ,Λ,B~) is given by

ΛETΛE,B~EB~F,Xj{X𝒂+X𝒂′′if j=k,Xjif jk.

One can show that the datum μk(Σ) actually defines a quantum seed, which is called the mutation of Σ in direction k. This operation is involutive, i.e., we have μk(μk(Σ))=Σ.

Definition B.1.

Let (Λ,B~) be a compatible pair and Σ=((Xj)jJ,Λ,B~) the associated quantum seed. The quantum cluster algebra𝒜v(Λ,B~) is the [v±1/2]-subalgebra of the skew field 𝔽(𝒯(Λ)) 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 Av(Λ,B~) is contained in the quantum torus T(Λ).

Acknowledgements

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.

References

[1] T. Akasaka and M. Kashiwara, Finite-dimensional representations of quantum affine algebras, Publ. Res. Inst. Math. Sci. 33 (1997), no. 5, 839–867. 10.2977/prims/1195145020Search in Google Scholar

[2] A. Berenstein and A. Zelevinsky, Quantum cluster algebras, Adv. Math. 195 (2005), no. 2, 405–455. 10.1016/j.aim.2004.08.003Search in Google Scholar

[3] L. Bittmann, A quantum cluster algebra approach to representations of simply laced quantum affine algebras, Math. Z. 298 (2021), no. 3–4, 1449–1485. 10.1007/s00209-020-02664-9Search in Google Scholar

[4] V. Chari, Minimal affinizations of representations of quantum groups: The rank 2 case, Publ. Res. Inst. Math. Sci. 31 (1995), no. 5, 873–911. 10.2977/prims/1195163722Search in Google Scholar

[5] V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge 1994. Search in Google Scholar

[6] V. Chari and A. Pressley, Quantum affine algebras and their representations, Representations of groups (Banff 1994), CMS Conf. Proc. 16, American Mathematical Society, Providence (1995), 59–78. Search in Google Scholar

[7] V. Chari and A. Pressley, Minimal affinizations of representations of quantum groups: The simply laced case, J. Algebra 184 (1996), no. 1, 1–30. 10.1006/jabr.1996.0247Search in Google Scholar

[8] E. Date and M. Okado, Calculation of excitation spectra of the spin model related with the vector representation of the quantized affine algebra of type An(1), Internat. J. Modern Phys. A 9 (1994), no. 3, 399–417. 10.1142/S0217751X94000194Search in Google Scholar

[9] E. Frenkel and E. Mukhin, Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, Comm. Math. Phys. 216 (2001), no. 1, 23–57. 10.1007/s002200000323Search in Google Scholar

[10] E. Frenkel and N. Reshetikhin, Deformations of 𝒲-algebras associated to simple Lie algebras, Comm. Math. Phys. 197 (1998), no. 1, 1–32. Search in Google Scholar

[11] R. Fujita, Geometric realization of Dynkin quiver type quantum affine Schur-Weyl duality, Int. Math. Res. Not. IMRN 2020 (2020), no. 22, 8353–8386. 10.1093/imrn/rny226Search in Google Scholar

[12] R. Fujita, Graded quiver varieties and singularities of normalized R-matrices for fundamental modules, Selecta Math. (N.S.) 28 (2022), no. 1, Paper No. 2. 10.1007/s00029-021-00715-5Search in Google Scholar

[13] R. Fujita and S.-j. Oh, Q-data and representation theory of untwisted quantum affine algebras, Comm. Math. Phys. 384 (2021), no. 2, 1351–1407. 10.1007/s00220-021-04028-8Search in Google Scholar

[14] C. Geiß, B. Leclerc and J. Schröer, Cluster structures on quantum coordinate rings, Selecta Math. (N.S.) 19 (2013), no. 2, 337–397. 10.1007/s00029-012-0099-xSearch in Google Scholar

[15] C. Geiss, B. Leclerc and J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017), no. 1, 61–158. 10.1007/s00222-016-0705-1Search in Google Scholar

[16] D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339–389. 10.1007/BF02564452Search in Google Scholar

[17] D. Hernandez, Algebraic approach to q,t-characters, Adv. Math. 187 (2004), no. 1, 1–52. 10.1016/j.aim.2003.07.016Search in Google Scholar

[18] D. Hernandez, Représentations des algèbres affinisées quantiques : q,t-caractères et produit de fusion, PhD thesis, Université Paris 6, 2004. Search in Google Scholar

[19] D. Hernandez, Monomials of q and q,t-characters for non simply-laced quantum affinizations, Math. Z. 250 (2005), no. 2, 443–473. 10.1007/s00209-005-0762-4Search in Google Scholar

[20] D. Hernandez, The Kirillov–Reshetikhin conjecture and solutions of T-systems, J. reine angew. Math. 596 (2006), 63–87. 10.1515/CRELLE.2006.052Search in Google Scholar

[21] D. Hernandez, On minimal affinizations of representations of quantum groups, Comm. Math. Phys. 276 (2007), no. 1, 221–259. 10.1007/s00220-007-0332-1Search in Google Scholar

[22] D. Hernandez and M. Jimbo, Asymptotic representations and Drinfeld rational fractions, Compos. Math. 148 (2012), no. 5, 1593–1623. 10.1112/S0010437X12000267Search in Google Scholar

[23] D. Hernandez and B. Leclerc, Cluster algebras and quantum affine algebras, Duke Math. J. 154 (2010), no. 2, 265–341. 10.1007/978-3-030-63849-8_2Search in Google Scholar

[24] D. Hernandez and B. Leclerc, Quantum Grothendieck rings and derived Hall algebras, J. reine angew. Math. 701 (2015), 77–126. 10.1515/crelle-2013-0020Search in Google Scholar

[25] D. Hernandez and B. Leclerc, A cluster algebra approach to q-characters of Kirillov–Reshetikhin modules, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 5, 1113–1159. 10.4171/JEMS/609Search in Google Scholar

[26] D. Hernandez and H. Oya, Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan–Lusztig algorithm, Adv. Math. 347 (2019), 192–272. 10.1016/j.aim.2019.02.024Search in Google Scholar

[27] S.-J. Kang, M. Kashiwara and M. Kim, Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras, II, Duke Math. J. 164 (2015), no. 8, 1549–1602. 10.1215/00127094-3119632Search in Google Scholar

[28] S.-J. Kang, M. Kashiwara and M. Kim, Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras, Invent. Math. 211 (2018), no. 2, 591–685. 10.1007/s00222-017-0754-0Search in Google Scholar

[29] S.-J. Kang, M. Kashiwara, M. Kim and S.-j. Oh, Simplicity of heads and socles of tensor products, Compos. Math. 151 (2015), no. 2, 377–396. 10.1112/S0010437X14007799Search in Google Scholar

[30] S.-J. Kang, M. Kashiwara, M. Kim and S.-j. Oh, Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras IV, Selecta Math. (N.S.) 22 (2016), no. 4, 1987–2015. 10.1007/s00029-016-0267-5Search in Google Scholar

[31] S.-J. Kang, M. Kashiwara, M. Kim and S.-j. Oh, Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), no. 2, 349–426. 10.1090/jams/895Search in Google Scholar

[32] M. Kashiwara, On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. 10.1215/S0012-7094-91-06321-0Search in Google Scholar

[33] M. Kashiwara, On level-zero representations of quantized affine algebras, Duke Math. J. 112 (2002), no. 1, 117–175. 10.1215/S0012-9074-02-11214-9Search in Google Scholar

[34] M. Kashiwara, M. Kim and S.-j. Oh, Monoidal categories of modules over quantum affine algebras of type A and B, Proc. Lond. Math. Soc. (3) 118 (2019), no. 1, 43–77. 10.1112/plms.12160Search in Google Scholar

[35] M. Kashiwara, M. Kim, S.-j. Oh and E. Park, Monoidal categories associated with strata of flag manifolds, Adv. Math. 328 (2018), 959–1009. 10.1016/j.aim.2018.02.013Search in Google Scholar

[36] M. Kashiwara, M. Kim, S.-j. Oh and E. Park, Monoidal categorification and quantum affine algebras, Compos. Math. 156 (2020), no. 5, 1039–1077. 10.1112/S0010437X20007137Search in Google Scholar

[37] M. Kashiwara, M. Kim, S.-j. Oh and E. Park, PBW theory for quantum affine algebras, preprint (2020), https://arxiv.org/abs/2011.14253; to appear in J. Eur. Math. Soc. Search in Google Scholar

[38] M. Kashiwara and S.-j. Oh, Categorical relations between Langlands dual quantum affine algebras: Doubly laced types, J. Algebraic Combin. 49 (2019), no. 4, 401–435. 10.1007/s10801-018-0829-zSearch in Google Scholar

[39] Y. Kimura, Quantum unipotent subgroup and dual canonical basis, Kyoto J. Math. 52 (2012), no. 2, 277–331. 10.1215/21562261-1550976Search in Google Scholar

[40] A. Kuniba, T. Nakanishi and J. Suzuki, T-systems and Y-systems in integrable systems, J. Phys. A 44 (2011), no. 10, 103001. Search in Google Scholar

[41] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. 10.1090/S0894-0347-1990-1035415-6Search in Google Scholar

[42] G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365–421. 10.1090/S0894-0347-1991-1088333-2Search in Google Scholar

[43] G. Lusztig, Introduction to quantum groups, Reprint of the 1994 edition, Modern Birkhäuser Class., Birkhäuser/Springer, New York 2010. 10.1007/978-0-8176-4717-9Search in Google Scholar

[44] H. Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), no. 1, 145–238. 10.1090/S0894-0347-00-00353-2Search in Google Scholar

[45] H. Nakajima, t-analogs of q-characters of Kirillov–Reshetikhin modules of quantum affine algebras, Represent. Theory 7 (2003), 259–274. 10.1090/S1088-4165-03-00164-XSearch in Google Scholar

[46] H. Nakajima, Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. of Math. (2) 160 (2004), no. 3, 1057–1097. 10.4007/annals.2004.160.1057Search in Google Scholar

[47] H. Nakajima, Modules of quantized Coulomb branches, in preparation. Search in Google Scholar

[48] H. Nakajima and A. Weekes, Coulomb branches of quiver gauge theories with symmetrizers, preprint (2019), https://arxiv.org/abs/1907.06552; to appear in J. Eur. Math. Soc. 10.4171/JEMS/1176Search in Google Scholar

[49] K. Naoi, Equivalence between module categories over quiver Hecke algebras and Hernandez–Leclerc’s categories in general types, Adv. Math. 389 (2021), Paper No. 107916. 10.1016/j.aim.2021.107916Search in Google Scholar

[50] S.-j. Oh, The denominators of normalized R-matrices of types A2n-1(2), A2n(2), Bn(1) and Dn+1(2), Publ. Res. Inst. Math. Sci. 51 (2015), no. 4, 709–744. 10.4171/PRIMS/170Search in Google Scholar

[51] S.-j. Oh and T. Scrimshaw, Categorical relations between Langlands dual quantum affine algebras: Exceptional cases, Comm. Math. Phys. 368 (2019), no. 1, 295–367. 10.1007/s00220-019-03287-wSearch in Google Scholar

[52] S.-j. Oh and U. R. Suh, Combinatorial Auslander–Reiten quivers and reduced expressions, J. Korean Math. Soc. 56 (2019), no. 2, 353–385. Search in Google Scholar

[53] F. Qin, Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. J. 166 (2017), no. 12, 2337–2442. 10.1215/00127094-2017-0006Search in Google Scholar

[54] U. R. Suh and S.-j. Oh, Twisted and folded Auslander–Reiten quivers and applications to the representation theory of quantum affine algebras, J. Algebra 535 (2019), 53–132. 10.1016/j.jalgebra.2019.06.013Search in Google Scholar

[55] M. Varagnolo and E. Vasserot, Perverse sheaves and quantum Grothendieck rings, Studies in memory of Issai Schur (Chevaleret/Rehovot 2000), Progr. Math. 210, Birkhäuser Boston, Boston (2003), 345–365. 10.1007/978-1-4612-0045-1_13Search in Google Scholar

Received: 2021-02-07
Revised: 2021-10-09
Published Online: 2022-02-15
Published in Print: 2022-04-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Scroll Up Arrow