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].

A REALIZATION OF THE ENVELOPING SUPERALGEBRA

2021 ◽  
pp. 1-36
Author(s):  
JIE DU ◽  
QIANG FU ◽  
YANAN LIN
Keyword(s):  
Symmetric Groups ◽  
Loop Algebra ◽  
Natural Representation ◽  
Schur Algebras ◽  
Tensor Spaces ◽  
Weyl Duality ◽  
Geometric Setting

Abstract In [2], Beilinson–Lusztig–MacPherson (BLM) gave a beautiful realization for quantum $\mathfrak {gl}_n$ via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ of the loop algebra $\widehat {\mathfrak {gl}}_{m|n}$ of ${\mathfrak {gl}}_{m|n}$ with those of affine symmetric groups ${\widehat {{\mathfrak S}}_{r}}$ . Then, we give a BLM type realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ via affine Schur superalgebras. The first application of the realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ is to determine the action of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ on tensor spaces of the natural representation of $\widehat {\mathfrak {gl}}_{m|n}$ . These results in epimorphisms from $\;{\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ to affine Schur superalgebras so that the bridging relation between representations of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ and ${\widehat {{\mathfrak S}}_{r}}$ is established. As a second application, we construct a Kostant type $\mathbb Z$ -form for ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur–Weyl duality in arbitrary characteristics.

Multiplication Formulas and Canonical Bases for Quantum Affine gln

2018 ◽  
Vol 70 (4) ◽  
pp. 773-803 ◽  
Author(s):  
Jie Du ◽  
Zhonghua Zhao
Keyword(s):  
Quantum Group ◽  
Canonical Basis ◽  
Recursive Formula ◽  
Loop Algebra ◽  
Hall Algebra ◽  
Monomial Basis ◽  
Multiplication Formula ◽  
Canonical Bases ◽  
Theoretic Proof ◽  
Cyclic Quiver

AbstractWe will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra of a cyclic quiver Δ(n). As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established by J. Y. Guo and C. M. Ringel, and derive a recursive formula to compute them. We will further use the formula and the construction of a monomial basis for given by Deng, Du, and Xiao together with the double Ringel-Hall algebra realisation of the quantum loop algebra given by Deng, Du, and Fu to develop some algorithms and to compute the canonical basis for . As examples, we will show explicitly the part of the canonical basis associated with modules of Lowey length at most 2 for the quantum group .

scholarly journals Canonical bases arising from quantum symmetric pairs of Kac–Moody type

2021 ◽  
Vol 157 (7) ◽  
pp. 1507-1537
Author(s):  
Huanchen Bao ◽  
Weiqiang Wang
Keyword(s):  
Quantum Group ◽  
Finite Type ◽  
Canonical Basis ◽  
Integral Form ◽  
New Approach ◽  
Canonical Bases ◽  
Highest Weight ◽  
Technical Assumption ◽  
Rank One ◽  
Divided Powers

For quantum symmetric pairs $(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct $\imath$ -canonical bases for the highest weight integrable $\textbf U$ -modules and their tensor products regarded as $\textbf {U}^\imath$ -modules, as well as an $\imath$ -canonical basis for the modified form of the $\imath$ -quantum group $\textbf {U}^\imath$ . A key new ingredient is a family of explicit elements called $\imath$ -divided powers, which are shown to generate the integral form of $\dot {\textbf {U}}^\imath$ . We prove a conjecture of Balagovic–Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi- $K$ -matrix and the constructions of $\imath$ -canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.

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.

scholarly journals Dominant and global dimension of blocks of quantised Schur algebras

2021 ◽  
Author(s):  
Ming Fang ◽  
Wei Hu ◽  
Steffen Koenig
Keyword(s):  
Hecke Algebras ◽  
Symmetric Groups ◽  
Global Dimension ◽  
Group Algebras ◽  
Dominant Dimension ◽  
Schur Algebras ◽  
Homological Dimensions ◽  
Weyl Duality

AbstractGroup algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and $$S_q(n,r)$$ S q ( n , r ) with $$n \geqslant r$$ n ⩾ r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59–85, 2021).

scholarly journals Schur–Weyl duality for infinitesimal q-Schur algebras sq(2,r)1

2008 ◽  
Vol 320 (3) ◽  
pp. 1099-1114 ◽  
Author(s):  
Karin Erdmann ◽  
Qiang Fu
Keyword(s):  
Schur Algebras ◽  
Weyl Duality

scholarly journals Canonical bases and higher representation theory

2014 ◽  
Vol 151 (1) ◽  
pp. 121-166 ◽  
Author(s):  
Ben Webster
Keyword(s):  
General Theory ◽  
Representation Theory ◽  
Finite Type ◽  
Canonical Basis ◽  
Canonical Bases ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Natural Categories ◽  
Grothendieck Groups ◽  
Quantized Universal Enveloping Algebra

AbstractThis paper develops a general theory of canonical bases and how they arise naturally in the context of categorification. As an application, we show that Lusztig’s canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is of finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest- and highest-weight integrable representations. This generalizes past work of the author’s in the highest-weight case.

Canonical bases of modified quantum algebras for type A2

2018 ◽  
Vol 17 (06) ◽  
pp. 1850113
Author(s):  
Weideng Cui
Keyword(s):  
Canonical Basis ◽  
Quantum Algebra ◽  
Explicit Description ◽  
Quantum Algebras ◽  
Canonical Bases ◽  
Type Formula

The modified quantum algebra [Formula: see text] associated to a quantum algebra [Formula: see text] was introduced by Lusztig. [Formula: see text] has a remarkable basis, which was defined by Lusztig, called the canonical basis. In this paper, we give an explicit description of all elements of the canonical basis of [Formula: see text] for type [Formula: see text].

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