Searcharxiv⌕ Search

arXiv subjects

Yiqiang Li

Publications and source records attributed to Yiqiang Li.

At least 37 records · Page 2Linked to original sources
arXiv2017

Positivity vs negativity of canonical bases

We provide examples for negativity of structure constants of the stably canonical basis of modified quantum $\mathfrak{gl}_n$ and an analogous basis of modified quantum coideal algebra of $\mathfrak{gl}_n$. In contrast, we construct the canonical basis of the modified quantum coideal algebra of $\mathfrak{sl}_n$, establish the positivity of its structure constants, the positivity with respect to a geometric bilinear form as well as the positivity of its action on the tensor powers of the natural representation. The matrix coefficients of the transfer map on these Schur algebras with respect to the canonical bases are shown to be positive. Formulas for canonical basis of the iSchur algebra of rank one are obtained.

math.RT↗
arXiv2015

Positivity of Canonical Bases under Comultiplication

We show the positivity of the canonical basis for a modified quantum affine $\mathfrak{sl}_n$ under the comultiplication. Moreover, we establish the positivity of the i-canonical basis in [LW15] with respect to the coideal subalgebra structure.

math.QA↗
arXiv2014

Geometric Schur Duality of Classical Type, II

We establish algebraically and geometrically a duality between the Iwahori-Hecke algebra of type D and two new quantum algebras arising from the geometry of N-step isotropic flag varieties of type D. This duality is a type D counterpart of the Schur-Jimbo duality of type A and the Schur-like duality of type B/C discovered by Bao-Wang. The new algebras play a role in the type D duality similar to the modified quantum gl(N) in type A, and the modified coideal subalgebras of quantum gl(N) in type B/C. We construct canonical bases for these two algebras.

math.RT↗
arXiv2013

An identification of Lusztig's modified forms of quantum algebras and their analogues

We introduce a multi-parameter twisted Hopf algebra associated with a root datum and show that its modified form is isomorphic to Lusztig's modified quantum algebra under certain restrictions on the parameters. By taking various specializations of the parameters, we obtain similar results for two/multi-parameter quantum algebras and Kang-Kashiwara-Oh's multi-parameter quantum superalgebras. As a consequence, the categories of weight modules of these algebras are identical.

math.QA↗
arXiv2013

Quantum Supergroups III. Twistors

We establish direct connections at several levels between quantum groups and supergroups associated to bar-consistent super Cartan datum by constructing an automorphism (called twistor) in the setting of covering quantum groups. The canonical bases of the halves of quantum groups and supergroups are shown to match under the twistor up to powers of a square root of -1. We further show that the modified quantum group and supergroup are isomorphic over the rational function field adjoint with a square root of -1, by constructing a twistor on the modified covering quantum group. An equivalence of categories of weight modules for quantum groups and supergroups follows.

math.QA↗
arXiv2013

Two-parameter quantum algebras, canonical bases and categorifications

A theory of canonical basis for a two-parameter quantum algebra is developed in parallel with the one in one-parameter case. A geometric construction of the negative part of a two-parameter quantum algebra is given by using mixed perverse sheaves and Deligne's weight theory based on Lusztig's work. A categorification of the negative part of a two-parameter quantum algebra is provided. A two-parameter quantum algebra is shown to be a two-cocycle deformation, depending only on the second parameter, of its one-parameter analogue.

math.RT↗
arXiv2013

A geometric construction of generalized q-Schur algebras

We show that the algebras constructed in [Li10] and [Li12] are generalized q-Schur algebras as defined in [D03]. This provides a geometric construction of generalized q-Schur algebras in types A, D and E. We give a parameterization of Nakajima's Lagrangian quiver variety of type D associated to a certain highest weight.

math.RT↗
arXiv2013

A geometric setting for quantum osp(1|2)

A geometric categorification is given for arbitrary-large-finite-dimensional quotients of quantum osp(1|2) and the tensor product of its simple modules. The modified quantum osp(1|2) of Clark-Wang, a new version in this paper and the modified quantum sl(2) are isomorphic to each other over a field containing Q(v) and a root of -1.

math.RT↗
arXiv2012

A geometric realization of modified quantum algebras

A geometric construction of Lusztig's modified quantum algebra of symmetric type is presented by using certain localized equivariant derived categories of double framed representation varieties of quivers.

math.RT↗
arXiv2012

On quantum GL(n)

The quantum GL(n) of Faddeev-Reshetikhin-Takhtajan and Dipper-Donkin are realized geometrically by using double partial flag varieties. As a consequence, the difference of these two Hopf algebras is caused by a twist of a cocycle in the multiplication.

math.RT↗
arXiv2011

Tensor product varieties, perverse sheaves and stability conditions

We show that the space spanned by the class of simple perverse sheaves in [Zh08] without localization is isomorphic to the tensor product of a Verma module with a tensor product of irreducible integrable modules of the quantum enveloping algebra associated with a graph. Under the isomorphism, the simple perverse sheaves get identified with the canonical basis elements of the tensor product module. We also show that the two stability conditions coincide in the localization process in [Zh08], by using supports and singular supports of complexes of sheaves, respectively.

math.RT↗
arXiv2008

Canonical Bases of Borcherds-Cartan Type

We study the canonical basis for the negative part of the quantum generalized Kac-Moody algebra associated to a symmetric Borcherds-Cartan matrix. The algebras associated to two different matrices satisfying certain conditions may coincide. We show that the canonical bases coincide provided that the algebras coincide. We also answer partially a question by Lusztig.

math.RT↗
arXiv2007

Notes on affine canonical and monomial bases

We investigate the affine canonical basis and the monomial basis constructed in [LXZ] in Lusztig's geometric setting. We show that the transition matrix between the two bases is upper triangular with 1's in the diagonal and coefficients in the upper diagonal entries in N[v, v^{-1}]. As a consequence, we show that part of the monomial basis elements give rise to resolutions of support varieties of the affine canonical basis elements as simple perverse sheaves.

math.RT↗
arXiv2006

Generalized Kac-Moody Lie Algebras And Product Quivers

We construct the entire generalized Kac-Moody Lie algebra as a quotient of the positive part of another generalized Kac-Moody Lie algebra. The positive part of a generalized Kac-Moody Lie algebra can be constructed from representations of quivers using Ringel's Hall algebra construction. Thus we give a direct realization of the entire generalized Kac-Moody Lie algebra.

math.RT↗