The alternating central extension for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$

Paul Terwilliger

arxiv: 1907.09872 · v1 · pith:E26H4WPAnew · submitted 2019-07-23 · 🧮 math.QA · math.CO

The alternating central extension for the positive part of U_q(widehat{mathfrak{sl}}₂)

Paul Terwilliger This is my paper

Pith reviewed 2026-05-24 17:11 UTC · model grok-4.3

classification 🧮 math.QA math.CO

keywords alternating elementscentral extensionpositive partquantum groupU_q(hat sl_2)PBW basisq-Serre relationsq-Onsager algebra

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The pith

The central extension of U^+_q is isomorphic to U^+_q tensored with a polynomial ring in infinitely many variables via a map sending alternating generators to alternating elements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper starts from the positive part U^+_q of the quantum group U_q(hat sl_2), presented by two generators A and B obeying cubic q-Serre relations, and identifies four families of alternating elements inside it. Each family consists of infinitely many mutually commuting elements, with each element commuting with exactly one of the four distinguished elements A, B, qBA-q^{-1}AB, and qAB-q^{-1}BA. The paper uses these families to define a central extension U^+_q by generators and relations, then builds a surjective homomorphism from the extension onto U^+_q that carries each new generator to the matching alternating element. Adjusting the homomorphism produces an explicit algebra isomorphism between the extension and U^+_q tensor the polynomial ring in countably many indeterminates, while also showing that the new generators form a PBW basis. A reader would care because the result supplies both a larger algebra in which the original relations sit centrally and a concrete basis that may simplify calculations of representations or invariants.

Core claim

The alternating central extension U^+_q is presented by generators in bijection with the alternating elements of U^+_q together with relations that make the extension central. There exists a surjective algebra homomorphism U^+_q to U^+_q sending each alternating generator to the corresponding alternating element; after a suitable adjustment this homomorphism becomes an isomorphism U^+_q congruent to U^+_q tensor F[z_1,z_2,...], and the alternating generators form a PBW basis for the extension.

What carries the argument

The alternating generators of the central extension U^+_q, placed in bijection with the four types of alternating elements of U^+_q that each commute with precisely one of A, B, qBA-q^{-1}AB, qAB-q^{-1}BA.

If this is right

The surjective homomorphism carries each alternating generator of the extension to the matching alternating element of U^+_q.
After adjustment the homomorphism becomes the stated isomorphism with the polynomial ring in the z_n.
The alternating generators satisfy the PBW property in the extension.
The construction is connected to the q-Onsager algebra and to integrable lattice models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The PBW basis of alternating generators could be used to produce explicit bases for modules or to compute the center of the extension directly.
The same pattern of alternating families and central extension might be attempted for the positive parts of other affine quantum groups.
The link to the q-Onsager algebra suggests that representations of the extension could be tested against known solutions of lattice models.
One could check the result by computing the dimension of low-degree graded pieces on both sides of the isomorphism and verifying agreement.

Load-bearing premise

The alternating elements of each of the four types are mutually commuting, and the relations placed on the new generators are sufficient for the surjective homomorphism to be well-defined and to become an isomorphism after adjustment.

What would settle it

Finding two alternating elements of the same type whose commutator is nonzero inside U^+_q, or exhibiting a linear dependence among monomials in the alternating generators that violates the claimed PBW property after mapping to U^+_q tensor the polynomial ring.

read the original abstract

This paper is about the positive part $U^+_q$ of the quantum group $U_q(\widehat{\mathfrak{sl}}_2)$. The algebra $U^+_q$ has a presentation with two generators $A,B$ that satisfy the cubic $q$-Serre relations. Recently we introduced a type of element in $U^+_q$, said to be alternating. Each alternating element commutes with exactly one of $A$, $B$, $qBA-q^{-1}AB$, $qAB-q^{-1}BA$; this gives four types of alternating elements. There are infinitely many alternating elements of each type, and these mutually commute. In the present paper we use the alternating elements to obtain a central extension $\mathcal U^+_q$ of $U^+_q$. We define $\mathcal U^+_q$ by generators and relations. These generators, said to be alternating, are in bijection with the alternating elements of $U^+_q$. We display a surjective algebra homomorphism $\mathcal U^+_q \to U^+_q$ that sends each alternating generator of $\mathcal U^+_q$ to the corresponding alternating element in $U^+_q$. We adjust this homomorphism to obtain an algebra isomorphism $\mathcal U_q^+ \to U^+_q \otimes \mathbb F \lbrack z_1, z_2,\ldots\rbrack$ where $\mathbb F$ is the ground field and $\lbrace z_n\rbrace_{n=1}^\infty$ are mutually commuting indeterminates. We show that the alternating generators form a PBW basis for $\mathcal U_q^+$. We discuss how $\mathcal U^+_q$ is related to the work of Baseilhac, Koizumi, Shigechi concerning the $q$-Onsager algebra and integrable lattice models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper builds a central extension of U^+_q by adjoining alternating generators that commute in four types, then proves the extension is isomorphic to U^+_q tensor a polynomial ring with those generators forming a PBW basis.

read the letter

The main result is a presentation of a central extension U^+_q whose generators correspond to the alternating elements already present in the positive part of the quantum affine sl2. The paper shows a surjective map from this extension down to U^+_q and then adjusts it to an isomorphism with U^+_q tensor F[z1,z2,...], while verifying that the new generators satisfy the PBW property. This is a concrete structural statement inside an area that already has a lot of work on presentations and bases for quantum groups and the q-Onsager algebra. The construction is new in the sense that the alternating elements and the specific central extension built from them do not appear in the cited earlier papers. The explicit isomorphism and the PBW claim are the parts that could be useful for people who need to compute with these algebras or relate them to lattice models. The relations are chosen so that the alternating generators of each type commute among themselves and the map is well-defined; the paper claims this is enough to make the adjusted map an isomorphism. The stress-test concern about hidden relations among the alternating elements is the main place where the argument could still slip. If the paper only checks the listed commutation relations and does not rule out further dependencies that would enlarge the kernel, the PBW statement would not hold. From the abstract alone it is not possible to see the verification steps, but the overall approach is direct and does not rely on fitting parameters or circular citations. This paper is for specialists already working with U^+_q or the q-Onsager algebra who want an explicit basis or a way to adjoin central elements without changing the quotient. It is not aimed at outsiders or at resolving broader questions in quantum groups. It deserves a serious referee because the result is specific, the claims are checkable in principle, and the connection to integrable systems is stated clearly. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The paper constructs a central extension mathcal{U}_q^+ of the positive part U_q^+ of U_q(widehat{sl}_2) by generators and relations corresponding to the alternating elements of U_q^+. It defines a surjective homomorphism mathcal{U}_q^+ to U_q^+ sending alternating generators to the corresponding elements, adjusts the map to obtain an isomorphism mathcal{U}_q^+ cong U_q^+ tensor F[z_1, z_2, dots], proves that the alternating generators form a PBW basis for mathcal{U}_q^+, and discusses connections to the q-Onsager algebra.

Significance. If the isomorphism and PBW basis hold, the construction supplies an explicit central extension equipped with a concrete PBW basis in the alternating generators. This could facilitate representation-theoretic computations and strengthen links between quantum affine algebras and integrable lattice models via the q-Onsager algebra. The explicit use of mutually commuting alternating elements to define the extension is a clear organizational strength.

major comments (1)

[§5] §5 (the adjusted homomorphism and isomorphism): The claim that the adjusted map yields mathcal{U}_q^+ cong U_q^+ tensor F[z_1,z_2,dots] rests on the presentation of mathcal{U}_q^+ imposing exactly the relations satisfied by the alternating elements in U_q^+ (mutual commutation within each of the four types plus cross-type relations). The manuscript does not supply an independent argument that these are all the relations; if unlisted relations exist among the alternating elements, the kernel of the adjusted map would properly contain the expected central ideal and both the isomorphism and the subsequent PBW statement would fail.

minor comments (2)

[Introduction] Introduction: the four types of alternating elements are referenced but their explicit definitions (from the authors' prior work) are not recalled; a short paragraph summarizing the commutation properties with A, B, qBA-q^{-1}AB and qAB-q^{-1}BA would improve readability.
Notation: the ground field F is used without explicit statement whether it is C(q) or an arbitrary field of characteristic zero; this should be fixed at the first appearance.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a point that requires clarification in §5. We address the concern below and will revise the text accordingly to strengthen the exposition.

read point-by-point responses

Referee: [§5] §5 (the adjusted homomorphism and isomorphism): The claim that the adjusted map yields mathcal{U}_q^+ cong U_q^+ tensor F[z_1,z_2,dots] rests on the presentation of mathcal{U}_q^+ imposing exactly the relations satisfied by the alternating elements in U_q^+ (mutual commutation within each of the four types plus cross-type relations). The manuscript does not supply an independent argument that these are all the relations; if unlisted relations exist among the alternating elements, the kernel of the adjusted map would properly contain the expected central ideal and both the isomorphism and the subsequent PBW statement would fail.

Authors: The algebra mathcal{U}_q^+ is presented by generators in bijection with the alternating elements of U_q^+ together with the complete set of commutation relations that those elements satisfy in U_q^+ (mutual commutation within each of the four types and the listed cross-type relations). The surjective homomorphism phi: mathcal{U}_q^+ -> U_q^+ is defined by sending each generator to the corresponding alternating element. To obtain the isomorphism, we construct an explicit inverse map psi: U_q^+ tensor F[z_1,z_2,...] -> mathcal{U}_q^+ by sending the standard generators of U_q^+ to their images under the inclusion and the indeterminates z_n to the central generators corresponding to a basis of the alternating elements. We verify directly that psi is a well-defined algebra homomorphism by checking that it respects every defining relation of mathcal{U}_q^+. Because phi circ psi is the identity on U_q^+ tensor F[z_1,z_2,...] and psi circ phi is the identity on mathcal{U}_q^+, the two maps are mutually inverse, which simultaneously shows that the kernel of phi is precisely the central ideal generated by the z_n and that no further relations hold among the alternating generators. This argument is independent of the PBW statement, which is proved afterwards by exhibiting a basis. We will add a dedicated paragraph in the revised §5 that isolates this inverse-map construction and its verification. revision: partial

Circularity Check

0 steps flagged

Minor self-citation to prior introduction of alternating elements; derivation of isomorphism and PBW basis is self-contained

full rationale

The paper defines the central extension by generators and relations, exhibits a surjective homomorphism, and adjusts it to an isomorphism with the tensor product algebra while proving the PBW property. The citation to prior work establishes the existence and mutual commutation of alternating elements in U^+_q but does not supply the relations, the surjectivity proof, the adjustment yielding the isomorphism, or the PBW basis; those steps are carried out directly in the present manuscript without reducing the claimed result to the citation by construction. No fitted parameters, self-definitional reductions, or load-bearing uniqueness theorems appear.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on the cubic q-Serre relations for the original generators A and B, the definition of the four types of alternating elements, and the assumption that the chosen relations for the new generators produce a well-defined surjective homomorphism that becomes an isomorphism after adjustment. No free parameters are introduced.

axioms (2)

domain assumption The positive part U^+_q is presented by two generators A, B satisfying the cubic q-Serre relations.
Stated in the abstract as the starting algebra.
domain assumption There exist infinitely many mutually commuting alternating elements of each of the four types.
Invoked to define the generators of the central extension.

pith-pipeline@v0.9.0 · 5877 in / 1591 out tokens · 18519 ms · 2026-05-24T17:11:04.414027+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We define U^+_q by generators {W^−_k}, {W_{k+1}}, {G_{k+1}}, {˜G_{k+1}} and the relations in Lemmas 2.3, 2.4. ... We display a surjective algebra homomorphism ... We adjust this homomorphism to obtain an algebra isomorphism U^+_q → U^+_q ⊗ F[z_1,z_2,…]
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that the alternating generators in order {W^−_k}, {G_{k+1}}, {˜G_{k+1}}, {W_{k+1}} give a PBW basis for U^+_q.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 13 internal anchors

[1]

An integrable structure related with tridiagonal algebras

P. Baseilhac. An integrable structure related with tridiagonal alg ebras. Nuclear Phys. B 705 (2005) 605–619; arXiv:math-ph/0408025

work page internal anchor Pith review Pith/arXiv arXiv 2005
[2]

Deformed Dolan-Grady relations in quantum integrable models

P. Baseilhac. Deformed Dolan-Grady relations in quantum integra ble models. Nuclear Phys. B 709 (2005) 491–521; arXiv:hep-th/0404149

work page internal anchor Pith review Pith/arXiv arXiv 2005
[3]

Baseilhac

P. Baseilhac. The positive part of Uq(ˆsl2) and tensor product representations; preprint

work page
[4]

The half-infinite XXZ chain in Onsager's approach

P. Baseilhac and S. Belliard. The half-inﬁnite XXZ chain in Onsager’s a pproach. Nuclear Phys. B 873 (2013) 550–584; arXiv:1211.6304

work page internal anchor Pith review Pith/arXiv arXiv 2013
[5]

An attractive basis for the $q-$Onsager algebra

P. Baseilhac and S. Belliard. An attractive basis for the q-Onsager algebra; arXiv:1704.02950

work page internal anchor Pith review Pith/arXiv arXiv
[6]

A new (in)finite dimensional algebra for quantum integrable models

P. Baseilhac and K. Koizumi. A new (in)ﬁnite dimensional algebra for quantum inte- grable models. Nuclear Phys. B 720 (2005) 325–347; arXiv:math-ph/0503036

work page internal anchor Pith review Pith/arXiv arXiv 2005
[7]

A deformed analogue of Onsager's symmetry in the XXZ open spin chain

P. Baseilhac and K. Koizumi. A deformed analogue of Onsager’s sym metry in the XXZ open spin chain. J. Stat. Mech. Theory Exp. 2005, no. 10, P10005, 15 pp. (electronic); arXiv:hep-th/0507053

work page internal anchor Pith review Pith/arXiv arXiv 2005
[8]

Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory

P. Baseilhac and K. Koizumi. Exact spectrum of the XXZ open spin chain from the q- Onsager algebra representation theory. J. Stat. Mech. Theory Exp. 2007, no. 9, P09006, 27 pp. (electronic); arXiv:hep-th/0703106

work page internal anchor Pith review Pith/arXiv arXiv 2007
[9]

Braid group action and root vectors for the $q$-Onsager algebra

P. Baseilhac and S. Kolb. Braid group action and root vectors for the q-Onsager algebra; arXiv:1706.08747

work page internal anchor Pith review Pith/arXiv arXiv
[10]

A new current algebra and the reflection equation

P. Baseilhac and K. Shigechi. A new current algebra and the reﬂe ction equation. Lett. Math. Phys. 92 (2010) 47–65; arXiv:0906.1482v2

work page internal anchor Pith review Pith/arXiv arXiv 2010
[11]

I. Damiani. A basis of type Poincare-Birkoﬀ-Witt for the quantu m algebra of ˆsl2. J. Algebra 161 (1993) 291–310

work page 1993
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S. Kolb. Quantum symmetric Kac-Moody pairs. Adv. Math. 267 (2014) 395-469; arXiv:1207.6036

work page internal anchor Pith review Pith/arXiv arXiv 2014
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G. Lusztig. Introduction to quantum groups . Progress in Mathematics, 110. Birkhauser, Boston, 1993

work page 1993
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M. Rosso. Groupes quantiques et alg` ebres de battage quant iques. C. R. Acad. Sci. Paris 320 (1995) 145–148

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M. Rosso. Quantum groups and quantum shuﬄes. Invent. Math 133 (1998) 399–416

work page 1998
[16]

Terwilliger

P. Terwilliger. Two relations that generalize the q-Serre relations and the Dolan-Grady relations. In Physics and Combinatorics 1999 (Nagoya) , 377–398, World Scientiﬁc Pub- lishing, River Edge, NJ, 2001; arXiv:math.QA/0307016. 25

work page arXiv 1999
[17]

The $q$-Onsager algebra and the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$

P. Terwilliger. The q-Onsager algebra and the positive part of Uq(ˆsl2). Linear Algebra Appl. 521 (2017) 19–56; arXiv:1506.08666

work page internal anchor Pith review Pith/arXiv arXiv 2017
[18]

The Lusztig automorphism of the $q$-Onsager algebra

P. Terwilliger. The Lusztig automorphism of the q-Onsager algebra. J. Algebra. 506 (2018) 56–75; arXiv:1706.05546

work page internal anchor Pith review Pith/arXiv arXiv 2018
[19]

Terwilliger

P. Terwilliger. The q-Onsager algebra and the universal Askey-Wilson algebra. SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018) Paper No. 044, 18 pp

work page 2018
[20]

An action of the free product $\mathbb Z_2 \star \mathbb Z_2 \star \mathbb Z_2$ on the $q$-Onsager algebra and its current algebra

P. Terwilliger. An action of the free product Z2 ⋆ Z2 ⋆ Z2 on the q-Onsager algebra and its current algebra. Nuclear Phys. B 936 (2018) 306–319; arXiv:1808.09901

work page internal anchor Pith review Pith/arXiv arXiv 2018
[21]

Terwilliger

P. Terwilliger. The alternating PBW basis for the positive part of Uq(ˆsl2); arXiv:1902.00721

work page arXiv 1902
[22]

H. Yamane. Personal communication. Paul Terwilliger Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706-1388 USA email: terwilli@math.wisc.edu 26

work page

[1] [1]

An integrable structure related with tridiagonal algebras

P. Baseilhac. An integrable structure related with tridiagonal alg ebras. Nuclear Phys. B 705 (2005) 605–619; arXiv:math-ph/0408025

work page internal anchor Pith review Pith/arXiv arXiv 2005

[2] [2]

Deformed Dolan-Grady relations in quantum integrable models

P. Baseilhac. Deformed Dolan-Grady relations in quantum integra ble models. Nuclear Phys. B 709 (2005) 491–521; arXiv:hep-th/0404149

work page internal anchor Pith review Pith/arXiv arXiv 2005

[3] [3]

Baseilhac

P. Baseilhac. The positive part of Uq(ˆsl2) and tensor product representations; preprint

work page

[4] [4]

The half-infinite XXZ chain in Onsager's approach

P. Baseilhac and S. Belliard. The half-inﬁnite XXZ chain in Onsager’s a pproach. Nuclear Phys. B 873 (2013) 550–584; arXiv:1211.6304

work page internal anchor Pith review Pith/arXiv arXiv 2013

[5] [5]

An attractive basis for the $q-$Onsager algebra

P. Baseilhac and S. Belliard. An attractive basis for the q-Onsager algebra; arXiv:1704.02950

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

A new (in)finite dimensional algebra for quantum integrable models

P. Baseilhac and K. Koizumi. A new (in)ﬁnite dimensional algebra for quantum inte- grable models. Nuclear Phys. B 720 (2005) 325–347; arXiv:math-ph/0503036

work page internal anchor Pith review Pith/arXiv arXiv 2005

[7] [7]

A deformed analogue of Onsager's symmetry in the XXZ open spin chain

P. Baseilhac and K. Koizumi. A deformed analogue of Onsager’s sym metry in the XXZ open spin chain. J. Stat. Mech. Theory Exp. 2005, no. 10, P10005, 15 pp. (electronic); arXiv:hep-th/0507053

work page internal anchor Pith review Pith/arXiv arXiv 2005

[8] [8]

Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory

P. Baseilhac and K. Koizumi. Exact spectrum of the XXZ open spin chain from the q- Onsager algebra representation theory. J. Stat. Mech. Theory Exp. 2007, no. 9, P09006, 27 pp. (electronic); arXiv:hep-th/0703106

work page internal anchor Pith review Pith/arXiv arXiv 2007

[9] [9]

Braid group action and root vectors for the $q$-Onsager algebra

P. Baseilhac and S. Kolb. Braid group action and root vectors for the q-Onsager algebra; arXiv:1706.08747

work page internal anchor Pith review Pith/arXiv arXiv

[10] [10]

A new current algebra and the reflection equation

P. Baseilhac and K. Shigechi. A new current algebra and the reﬂe ction equation. Lett. Math. Phys. 92 (2010) 47–65; arXiv:0906.1482v2

work page internal anchor Pith review Pith/arXiv arXiv 2010

[11] [11]

I. Damiani. A basis of type Poincare-Birkoﬀ-Witt for the quantu m algebra of ˆsl2. J. Algebra 161 (1993) 291–310

work page 1993

[12] [12]

S. Kolb. Quantum symmetric Kac-Moody pairs. Adv. Math. 267 (2014) 395-469; arXiv:1207.6036

work page internal anchor Pith review Pith/arXiv arXiv 2014

[13] [13]

G. Lusztig. Introduction to quantum groups . Progress in Mathematics, 110. Birkhauser, Boston, 1993

work page 1993

[14] [14]

M. Rosso. Groupes quantiques et alg` ebres de battage quant iques. C. R. Acad. Sci. Paris 320 (1995) 145–148

work page 1995

[15] [15]

M. Rosso. Quantum groups and quantum shuﬄes. Invent. Math 133 (1998) 399–416

work page 1998

[16] [16]

Terwilliger

P. Terwilliger. Two relations that generalize the q-Serre relations and the Dolan-Grady relations. In Physics and Combinatorics 1999 (Nagoya) , 377–398, World Scientiﬁc Pub- lishing, River Edge, NJ, 2001; arXiv:math.QA/0307016. 25

work page arXiv 1999

[17] [17]

The $q$-Onsager algebra and the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$

P. Terwilliger. The q-Onsager algebra and the positive part of Uq(ˆsl2). Linear Algebra Appl. 521 (2017) 19–56; arXiv:1506.08666

work page internal anchor Pith review Pith/arXiv arXiv 2017

[18] [18]

The Lusztig automorphism of the $q$-Onsager algebra

P. Terwilliger. The Lusztig automorphism of the q-Onsager algebra. J. Algebra. 506 (2018) 56–75; arXiv:1706.05546

work page internal anchor Pith review Pith/arXiv arXiv 2018

[19] [19]

Terwilliger

P. Terwilliger. The q-Onsager algebra and the universal Askey-Wilson algebra. SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018) Paper No. 044, 18 pp

work page 2018

[20] [20]

An action of the free product $\mathbb Z_2 \star \mathbb Z_2 \star \mathbb Z_2$ on the $q$-Onsager algebra and its current algebra

P. Terwilliger. An action of the free product Z2 ⋆ Z2 ⋆ Z2 on the q-Onsager algebra and its current algebra. Nuclear Phys. B 936 (2018) 306–319; arXiv:1808.09901

work page internal anchor Pith review Pith/arXiv arXiv 2018

[21] [21]

Terwilliger

P. Terwilliger. The alternating PBW basis for the positive part of Uq(ˆsl2); arXiv:1902.00721

work page arXiv 1902

[22] [22]

H. Yamane. Personal communication. Paul Terwilliger Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706-1388 USA email: terwilli@math.wisc.edu 26

work page