The comultiplication of modified quantum affine 𝔰𝔩n

2019 ◽  
Vol 23 (01) ◽  
pp. 1950048
Author(s):  
Qiang Fu
Keyword(s):  
Quantum Groups ◽  
Canonical Basis ◽  
Positive Part ◽  
Schur Algebras ◽  
Positivity Property ◽  
Structure Constants

Let [Formula: see text] be the modified quantum affine [Formula: see text] and let [Formula: see text] be the positive part of quantum affine [Formula: see text]. Let [Formula: see text] be the canonical basis of [Formula: see text] and let [Formula: see text] be the canonical basis of [Formula: see text]. In this paper, we use the theory of affine quantum Schur algebras to prove that the structure constants for the comultiplication with respect to [Formula: see text] are determined by the structure constants for the comultiplication with respect to [Formula: see text] for [Formula: see text]. In particular, from the positivity property for the comultiplication of [Formula: see text], we obtain the positivity property for the comultiplication of [Formula: see text], which is conjectured by Lusztig [Introduction to Quantum Groups, Progress in Mathematics, Vol. 110 (Birkhäuser, Boston, 1993), 25.4.2].

scholarly journals Quantum groups via cyclic quiver varieties I

2015 ◽  
Vol 152 (2) ◽  
pp. 299-326 ◽  
Author(s):  
Fan Qin
Keyword(s):  
Quantum Groups ◽  
Bilinear Form ◽  
Canonical Basis ◽  
Natural Basis ◽  
Grothendieck Ring ◽  
Quiver Varieties ◽  
Enveloping Algebra ◽  
Dual Canonical Basis ◽  
Quantized Enveloping Algebra ◽  
Cyclic Quiver

We construct the quantized enveloping algebra of any simple Lie algebra of type $\mathbb{A}\mathbb{D}\mathbb{E}$ as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig’s bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland’s recent work for type $\mathbb{A}\mathbb{D}\mathbb{E}$.

On the combinatorial rank of the quantum groups of type G2

2020 ◽  
pp. 2150208
Author(s):  
Vanusa Dylewski ◽  
Barbara Pogorelsky ◽  
Carolina Renz
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Quantum Groups ◽  
Main Parameter ◽  
Positive Part ◽  
Simple Lie Algebra ◽  
Type Formula ◽  
Part Formula ◽  
Multiplicative Order

In this paper, we calculate the combinatorial rank of the positive part [Formula: see text] of the multiparameter version of the small Lusztig quantum group, where [Formula: see text] is a simple Lie algebra of type [Formula: see text]. Supposing that the main parameter of quantization [Formula: see text] has multiplicative order [Formula: see text], where [Formula: see text] is finite, [Formula: see text], we prove that the combinatorial rank equals 3.

Presenting Little q-Schur Algebras uk(2; r)

2017 ◽  
Vol 24 (02) ◽  
pp. 297-308
Author(s):  
Zhihao Bian ◽  
Mingqiang Liu
Keyword(s):  
Quantum Groups ◽  
Schur Algebras ◽  
Du Fu ◽  
Generators And Relations

Little q-Schur algebras were introduced as homomorphic images of the infinitesimal quantum groups by Du, Fu and Wang. In this paper, we obtain a presentation by generators and relations for little q-Schur algebras uk(2, r).

scholarly journals Schur Algebras and Quantum Symmetric Pairs With Unequal Parameters

2019 ◽  
Author(s):  
Chun-Ju Lai ◽  
Li Luo
Keyword(s):  
Weight Function ◽  
Canonical Basis ◽  
Hecke Algebras ◽  
Symmetric Pair ◽  
Type B ◽  
Schur Algebras ◽  
Algebraic Version ◽  
Type D ◽  
Invariant Basis

Abstract We study the (quantum) Schur algebras of type B/C corresponding to the Hecke algebras with unequal parameters. We prove that the Schur algebras afford a stabilization construction in the sense of Beilinson–Lusztig–MacPherson that constructs a multiparameter upgrade of the quantum symmetric pair coideal subalgebras of type AIII/AIV with no black nodes. We further obtain the canonical basis of the Schur/coideal subalgebras, at the specialization associated with any weight function. These bases are the counterparts of Lusztig’s bar-invariant basis for Hecke algebras with unequal parameters. In the appendix we provide an algebraic version of a type D Beilinson–Lusztig–MacPherson construction, which is first introduced by Fan–Li from a geometric viewpoint.

scholarly journals The Integral Quantum Loop Algebra of $\mathfrak {gl}_{n}$

2018 ◽  
Vol 2019 (20) ◽  
pp. 6179-6215 ◽  
Author(s):  
Jie Du ◽  
Qiang Fu
Keyword(s):  
Canonical Basis ◽  
Alternative Algebra ◽  
Integral Form ◽  
Loop Algebra ◽  
Algebra Structure ◽  
Schur Algebras ◽  
Canonical Bases ◽  
Weyl Duality ◽  
Integral Quantum

Abstract We will construct the Lusztig form for the quantum loop algebra of $\mathfrak {gl}_{n}$ by proving the conjecture [4, 3.8.6] and establish partially the Schur–Weyl duality at the integral level in this case. We will also investigate the integral form of the modified quantum affine $\mathfrak {gl}_{n}$ by introducing an affine stabilisation property and will lift the canonical bases from affine quantum Schur algebras to a canonical basis for this integral form. As an application of our theory, we will also discuss the integral form of the modified extended quantum affine $\mathfrak {sl}_{n}$ and construct its canonical basis to provide an alternative algebra structure related to a conjecture of Lusztig in [29, §9.3], which has been already proved in [34].

scholarly journals On the centre of two-parameter quantum groups

2006 ◽  
Vol 136 (3) ◽  
pp. 445-472 ◽  
Author(s):  
Georgia Benkart ◽  
Seok-Jin Kang ◽  
Kyu-Hwan Lee
Keyword(s):  
Quantum Groups ◽  
Polynomial Ring ◽  
Bilinear Form ◽  
Skew Polynomial Ring ◽  
Positive Part ◽  
Invariant Bilinear Form ◽  
Central Elements ◽  
Skew Polynomial ◽  
Two Parameter

We describe Poincaré–Birkhoff–Witt bases for the two-parameter quantum groups U = Ur,s(sln) following Kharchenko and show that the positive part of U has the structure of an iterated skew polynomial ring. We define an ad-invariant bilinear form on U, which plays an important role in the construction of central elements. We introduce an analogue of the Harish-Chandra homomorphism and use it to determine the centre of U.

THE -SCHUR ALGEBRAS AND -SCHUR DUALITIES OF FINITE TYPE

2020 ◽  
pp. 1-32 ◽  
Author(s):  
Li Luo ◽  
Weiqiang Wang
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
Finite Type ◽  
Canonical Basis ◽  
Hecke Algebra ◽  
Schur Algebras ◽  
Schur Algebra ◽  
Simple Lie Algebra ◽  
Finite Set

We formulate a $q$ -Schur algebra associated with an arbitrary $W$ -invariant finite set $X_{\text{f}}$ of integral weights for a complex simple Lie algebra with Weyl group $W$ . We establish a $q$ -Schur duality between the $q$ -Schur algebra and Hecke algebra associated with $W$ . We then realize geometrically the $q$ -Schur algebra and duality and construct a canonical basis for the $q$ -Schur algebra with positivity. With suitable choices of $X_{\text{f}}$ in classical types, we recover the $q$ -Schur algebras in the literature. Our $q$ -Schur algebras are closely related to the category ${\mathcal{O}}$ , where the type $G_{2}$ is studied in detail.

scholarly journals Non-perturbative geometries for planar $$ \mathcal{N} $$ = 4 SYM amplitudes

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Nima Arkani-Hamed ◽  
Thomas Lam ◽  
Marcus Spradlin
Keyword(s):  
Configuration Space ◽  
Cluster Algebra ◽  
Canonical Basis ◽  
Canonical Forms ◽  
Positive Part ◽  
Algebraic Functions ◽  
Square Roots ◽  
Open Questions ◽  
Natural Way

Abstract There is a remarkable well-known connection between the G(4, n) cluster algebra and n-particle amplitudes in $$ \mathcal{N} $$ N = 4 SYM theory. For n ≥ 8 two long-standing open questions have been to find a mathematically natural way to identify a finite list of amplitude symbol letters from among the infinitely many cluster variables, and to find an explanation for certain algebraic functions, such as the square roots of four-mass-box type, that are expected to appear in symbols but are not cluster variables. In this letter we use the notion of “stringy canonical forms” to construct polytopal realizations of certain compactifications of (the positive part of) the configuration space Confn(ℙk−1) ≅ G(k, n)/T that are manifestly finite for all k and n. Some facets of these polytopes are naturally associated to cluster variables, while others are naturally associated to algebraic functions constructed from Lusztig’s canonical basis. For (k, n) = (4, 8) the latter include precisely the expected square roots, revealing them to be related to certain “overpositive” functions of the kinematical invariants.

scholarly journals Remarks on PBW bases of Ringel–Hall algebras of cyclic quivers

2019 ◽  
Vol 19 (03) ◽  
pp. 2050054
Author(s):  
Zhonghua Zhao
Keyword(s):  
Canonical Basis ◽  
Recursive Formula ◽  
Schur Algebras ◽  
Canonical Bases ◽  
Hall Algebras ◽  
Pbw Basis ◽  
Loewy Length ◽  
Generic Extensions

In this paper, we give a recursive formula for the interesting PBW basis [Formula: see text] of composition subalgebras [Formula: see text] of Ringel–Hall algebras [Formula: see text] of cyclic quivers after [Generic extensions and canonical bases for cyclic quivers, Canad. J. Math. 59(6) (2007) 1260–1283], and another construction of canonical bases of [Formula: see text] from the monomial bases [Formula: see text] following [Multiplication formulas and canonical basis for quantum affine, [Formula: see text], Canad. J. Math. 70(4) (2018) 773–803]. As an application, we will determine all the canonical basis elements of [Formula: see text] associated with modules of Loewy length [Formula: see text]. Finally, we will discuss the canonical bases between Ringel–Hall algebras and affine quantum Schur algebras.

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