Dual canonical bases of the quantum general linear supergroup are constructed which are invariant under the multiplication of the quantum Berezinian. By setting the quantum Berezinian to identity, we obtain dual canonical bases of the quantum special linear supergroup ${\s O}_q(SL_{m\mid n})$. We apply the canonical bases to study invariant subalgebras of the quantum supergroups under left and right translations. In the case $n=1$, it is shown that each invariant subalgebra is spanned by a part of the dual canonical bases. This in turn leads to dual canonical bases for any Kac module constructed by using an analogue of Borel-Weil theorem.
Following Kashiwara's algebraic approach, we construct crystal bases and canonical bases for quantum supergroups with no isotropic odd roots and for their integrable modules.
We construct a canonical basis for a class of tensor product modules of a quantum covering group associated to a Kac-Moody Lie superalgebra of anisotropic type, and use these bases to construct a canonical basis for the modified form of a quantum covering group.
We give a combinatorial construction for the canonical bases of the $\pm$-parts of the quantum enveloping superalgebra $\bfU(\mathfrak{gl}_{m|n})$ and discuss their relationship with the Kazhdan-Lusztig bases for the quantum Schur superalgebras $\bsS(m|n,r)$ introduced in \cite{DR}. We will also extend this relationship to the induced bases for simple polynomial representations of $\bfU(\mathfrak{gl}_{m|n})$.
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 …
We initiate the study of several distinguished bases for the positive half of a quantum supergroup $U_q$ associated to a general super Cartan datum $(\mathrm{I}, (\cdot,\cdot))$ of basic type inside a quantum shuffle superalgebra. The combinatorics of words for an arbitrary total ordering on $\mathrm{I}$ is developed in connection with the root system associated to $\mathrm{I}$. The monomial, Lyndon, and PBW bases of $U_q$ are constructed, and moreover, a direct proof of the orthogonality of the PBW basis is provided within the framework of quantum shuffles. Consequently, the canonical basis i…
The universal R-matrix of two-parameter quantum general linear supergroups is computed explicitly based on the RTT realization of Faddeev--Reshetikhin--Takhtajan.
By using certain quantum differential operators, we construct a super representation for the quantum queer supergroup U_v(q_n). The underlying space of this representation is a deformed polynomial superalgebra in 2n^2 variables whose homogeneous components can be used as the underlying spaces of queer q-Schur superalgebras. We then extend the representation to its formal power series algebra which contains a (super) submodule isomorphic to the regular representation of U_v(q_n). A monomial basis M for U_v(q_n) plays a key role in proving the isomorphism. In this way, we may present the quantum…
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.
For $\imath$quantum covering groups $(\mathbf{U}, \mathbf{U}^\imath)$ of super Kac-Moody type, we construct $\imath$-canonical bases for the highest weight integrable $\mathbf{U}$-modules and their tensor products regarded as $\mathbf{U}^\imath$-modules, as well as a canonical basis for the modified form of the $\imath$quantum group $\mathbf{U}^\imath$, using the $\imath^π$-divided powers, rank one canonical basis for $\mathbf{U}^\imath$.
For the quantum symmetric pair $(\textbf{U}, \textbf{U}^{\imath})$ of type AIII/AIV, we show various positivity properties of the $\imath$-canonical bases on finite-dimensional simple $\textbf{U}$-modules, as well as their tensor product.
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.
In this paper, we show that quantum twist maps, introduced by Lenagan-Yakimov, induce bijections between dual canonical bases of quantum nilpotent subalgebras. As a corollary, we show the unitriangular property between dual canonical bases and Poincaré-Birkhoff-Witt type bases under the "reverse" lexicographic order. We also show that quantum twist maps induce bijections between certain unipotent quantum minors.
We give some closed formulas for certain vectors of the canonical bases of the Fock space representation of U_v(sl^_n). As a result, a combinatorial description of certain parabolic Kazhdan-Lusztig polynomials for affine type A is obtained.
We derive a formula for the entries of the (unitriangular) transition matrices between the standard monomial and dual canonical bases of the irreducible polynomial representations of U_q(gl_n) in terms of Kazhdan-Lusztig polynomials.
Denote by $N \subset SL_k$ the subgroup of unipotent upper triangular matrices. In this paper, we show that the dual canonical basis of $\mathbb{C}[N]$ (and base affine spaces) can be parameterized by semi-standard Young tableaux. Moreover, we give an explicit formula for every element in the the dual canonical basis using the data of the corresponding semistandard Young tableau. We apply our results to study cluster variables in $\mathbb{C}[N]$.
It has been known since their birth in Bao and Wang's work that the $\imath$canonical bases of $\imath$quantum groups are not stable in general. In the author's previous work, the stability of $\imath$canonical bases of certain quasi-split types turned out to be closely related to the theory of $\imath$crystals. In this paper, we prove the stability of $\imath$canonical bases of irreducible finite type of real rank $1$, for which the theory of $\imath$crystals has not been developed, by means of global and local crystal bases.
The modified quantized enveloping algebra $\dot{\mathbf{U}}$ has a remarkable canonical basis, which was introduced by Lusztig. In this paper, we give an explicit description of all elements of the canonical basis of $\dot{\mathbf{U}}$ for type $A_{2}$.
n^{th} root of a Lie algebra and its dual (that is fractional supergroup) based on the permutation group $S_n$ invariant forms are formulated in the Hopf algebra formalism. Detailed discussion of $S_3$-graided $sl(2)$ algebras is done.
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.
Expanding the classical work of Kazhdan-Lusztig, we construct a bar involution and canonical bases on the $q$-Brauer algebra introduced by Wenzl. We define explicit actions of the $q$-Brauer algebra on the tensor spaces, and formulate $\imath$Schur dualities between the $q$-Brauer algebra and the $\imath$quantum groups of type AI and AII respectively.
We show that canonical bases in $\dot{U}(\mathfrak{sl}_n)$ and the Schur algebra are compatible; in fact we extend this result to $p$-canonical bases. This follows immediately from a fullness result from a functor categorifying this map. In order to prove this result, we also explain the connections between categorifications of the Schur algebra which arise from parity sheaves on partial flag varieties, singular Soergel bimodules and Khovanov and Lauda's "flag category," which are of some independent interest.
We relate the canonical basis of the Fock space representation of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_{n})$, as defined by Leclerc and Thibon, to the canonical basis of its restriction to $U_q(\mathfrak{sl}_{n})$, regarded as a based module in the sense of Lusztig. More generally we consider the restriction to any parabolic subalgebra. We deduce results on decomposition numbers and branching coefficients of Schur algebras over fields of positive characteristic, generalising those of Kleshchev and of Tan and Teo.
We develop a general theory of canonical bases for quantum symmetric pairs $(\mathbf{U}, \mathbf{U}^\imath)$ with parameters of arbitrary finite type. We construct new canonical bases for the simple integrable $\mathbf{U}$-modules and their tensor products regarded as $\mathbf{U}^\imath$-modules. We also construct a canonical basis for the modified form of the $\imath$quantum group $\mathbf{U}^\imath$. To that end, we establish several new structural results on quantum symmetric pairs, such as bilinear forms, braid group actions, integral forms, Levi subalgebras (of real rank one), and integra…
We prove part of the conjectures in [Li10a] (arXiv:1007.5384). We also relate the construction of quantum modified algebras in [Li10a] with the functorial construction in [ZH08](arXiv:0803.3668).
We construct a crystal basis for the negative half of the quantum group U associated to the standard super Cartan datum of gl(m|1), which is compatible with known crystals on Kac modules and simple modules. We show that these crystals admit globalization functions which produce compatible canonical bases. We then define a braid group action on a family of quantum groups including U, and use this action to show that our canonical basis agrees with those constructed from PBW bases. Finally, we explicitly compute some small-rank examples.
The positivity property for canonical bases asserts that the structure constants of the multiplication for the canonical basis are in ${\mathbb N}[v,v^{-1}]$. Let $\mathbf U$ be the quantum group over ${\mathbb Q}(v)$ associated with a symmetric Cartan datum. The positivity property for the positive part ${\mathbf U}^+$ of ${\mathbf U}$ was proved by Lusztig. He conjectured that the positivity property holds for the modified form $\dot{\mathbf U}$ of ${\mathbf U}$. In this paper, we prove that the structure constants for the canonical basis of $\dot{\mathbf U}(\widehat{\frak{sl}}_n)$ coincide …
The dual basis of the canonical basis of the modified quantized enveloping algebra is studied, in particular for type $A$. The construction of a basis for the coordinate algebra of the $n\times n$ quantum matrices is appropriate for the study the multiplicative property. It is shown that this basis is invariant under multiplication by certain quantum minors including the quantum determinant. Then a basis of quantum SL(n) is obtained by setting the quantum determinant to one. This basis turns out to be equivalent to the dual canonical basis.
Some filtrations of the tensor product of a highest weight module and a lowest weight module over quantum group $U_q(\mathfrak g)$ are constructed in \cite{LZ:2009} and one can use them to define some ideals of the modified quantized enveloping algebra. It is shown that the quotient algebras inherit canonical bases from the modified quantized enveloping algebra and are dual to the quantum coordinate ring defined by Kashiwara for symmetrizable Kac-Moody algebra $\mathfrak g$.
It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis for quantum groups. For each reduced expression of a Weyl group element, Geiß, Leclerc and Schröer has recently given a parametrization of irreducible components of nilpotent varieties in studying cluster algebras. In this paper we show that their parametrization coincides with Lusztig's parametrization of the canonical basis.
A criterion of irreducibility for induction products of evaluation modules of type A affine Hecke algebras is given. It is derived from multiplicative properties of the canonical basis of a quantum deformation of the Bernstein-Zelevinsky ring.
For arbitrary reduced words we give formulas for the crystal structures on string and Lusztig data of type A as well as the defining inequalities of the corresponding polytopes revealing certain dualities between them.
We define canonical bases of the higher-level q-deformed Fock space modules of the affine Lie algebra sl(n)^. This generalizes the result of Leclerc and Thibon for the case of level 1. We express the transition matrices between the canonical bases and the natural bases of the Fock spaces in terms of affine Kazhdan-Lusztig polynomials.
We provide some necessary details to several arguments appearing in our previous paper ``Canonical bases for quantum generalized Kac-Moody algebras''. We also make the link with some other work on the same subject.
We give a fast algorithm for computing the canonical basis of an irreducible highest-weight module for $U_q(\hat{\mathfrak{sl}}_e)$, generalising the LLT algorithm.
We construct a canonical basis for quantum generalized Kac-Moody algebra via semisimple perverse sheaves on varieties of representations of quivers. We compare this basis with the one recently defined purely algebraically by Jeong, Kang and Kashiwara.
We study the combinatorics of fully commutative elements in Coxeter groups of type $H_n$ for any $n > 2$. Using the results, we construct certain canonical bases (i.e., IC bases) for non-simply-laced generalized Temperley--Lieb algebras and show how to relate them to morphisms in the category of decorated tangles.
We give closed formulas for all vectors of the canonical basis of a level 2 irreducible integrable representation of $U_v(sl_\infty)$. These formulas coincide at v=1 with Lusztig's formulas for the constructible characters of the Iwahori-Hecke algebras of type B and D.
We introduce a new class of bases for quantized universal enveloping algebras $U_q(\mathfrak g)$ and other doubles attached to semisimple and Kac-Moody Lie algebras. These bases contain dual canonical bases of upper and lower halves of $U_q(\mathfrak g)$ and are invariant under many symmetries including all Lusztig's symmetries if $\mathfrak g$ is semisimple. It also turns out that a part of a double canonical basis of $U_q(\mathfrak g)$ spans its center.
Let ${\mathbf U}^-_q$ be the negative part of the quantum enveloping algebra associated to a simply laced Kac-Moody Lie algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the algebra corresponding to the fixed point subalgebra of ${\mathfrak g}$ obtained from a diagram automorphism $σ$ on ${\mathfrak g}$. Let ${\mathbf B}^σ$ be the set of $σ$-fixed elements in the canonical basis of ${\mathbf U}_q^-$, and $\underline{\mathbf B}$ the canonical basis of $\underline{\mathbf U}_q^-$. Lusztig proved that there exists a canonical bijection ${\mathbf B}^σ \simeq \underline{\mathbf B}$ based on …
We consider tensor products of finite-dimensional representations of a coideal subalgebra in $U_{q}(\mathfrak{sl}_2)$. We present an explicit expression for the dual of the canonical bases through a diagrammatic presentation. We show that the decomposition of tensor products of dual canonical bases and the action of the coideal subalgebra have integral and positive properties. As an application, we consider the eigensystem of the generator of the coideal subalgebra on the dual canonical bases. We provide all the eigenvalues and obtain an explicit expression of the eigenfunction for the largest…
The aim of this paper is to generalize several aspects of the recent work of Leclerc-Thibon and Varagnolo-Vasserot on the canonical bases of the level 1 q-deformed Fock spaces due to Hayashi. Namely, we define canonical bases for the higher-level q-deformed Fock spaces of Jimbo-Misra-Miwa-Okado and establish a relation between these bases and (parabolic) Kazhdan-Lusztig polynomials for the affine Weyl group of type $A^{(1)}_{r-1}.$ As an application we derive an inversion formula for a sub-family of these polynomials. This article is an extended version of math.QA/9901032
We establish the existence of an IC basis for the generalized Temperley--Lieb algebra associated to a Coxeter system of arbitrary type. We determine this basis explicitly in the case where the Coxeter system is simply laced and the algebra is finite dimensional.
A string basis is constructed for each subalgebra of invariants of the function algebra on the quantum special linear group. By analyzing the string basis for a particular subalgebra of invariants, we obtain a ``canonical basis'' for every finite dimensional irreducible $U_q({\mathfrak{sl}}(n))$-module. It is also shown that the algebra of functions on any quantum homogeneous space is generated by quantum minors.
We establish the relation of the potential function constructed by Gross-Hacking-Keel-Kontsevich's and Berenstein-Kazhdan's decoration function on the open double Bruhat cell in the base affine space $G/\mathcal{N}$ of a simple, simply connected, simply laced algebraic group $G$. As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on $G/\mathcal{N}$ arising from the tropicalizations of the potential and decoration function with the classical string and Lusztig parametrizations.
We propose a new mwthod of constructing 4D-TQFTs. The method uses a new type of algebraic structure called a Hopf Category. We also outline the construction of a family of Hopf categories related to the quantum groups, using the canonical bases.
This paper is a sequel (and contains a better treatment) of math.QA/0404032; We construct some canonical bases for quantum toroidal algebras of types D_4, E_6, E_7, E_8 using perverse sheaves on quot schemes parametrizing equivariant coherent sheaves on elliptic curves. We also study some compatibility properties of these bases with some natural filtrations and modules.
Let U be the quantum group associated to a Lie algebra g of rank n. The negative part U^- of U has a canonical basis B with favourable properties, introduced by Kashiwara and Lusztig. The approaches of Kashiwara and Lusztig lead to a set of alternative parametrizations of the canonical basis, one for each reduced expression for the longest word in the Weyl group of g. We describe the authors' recent work establishing close relationships between the Lusztig cones, canonical basis elements and the regions of linearity of reparametrization functions arising from the above parametrizations in type…
We define a canonical basis of the $q$-deformed Fock space representation of the affine Lie algebra $\glchap_n$. We conjecture that the entries of the transition matrix between this basis and the natural basis of the Fock space are $q$-analogues of decomposition numbers of the $v$-Schur algebras for $v$ specialized to a $n$th root of unity.
We describe a natural $q$-deformation of Fock and Goncharov's canonical basis for the algebra of regular functions on a cluster variety associated to a quiver of type $A$. We then describe an extension of this construction involving a cluster variety called the symplectic double.
