Quantum Symmetric Pairs Research Articles

$$\imath $$Hall algebras of weighted projective lines and quantum symmetric pairs II: injectivity

$$\imath $$Hall algebras of weighted projective lines and quantum symmetric pairs II: injectivity

𝚤Hall algebras of weighted projective lines and quantum symmetric pairs

The ı \imath Hall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of 1 1 -periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the ı \imath quantum loop algebra, which is a generalization of the ı \imath quantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The ı \imath Hall algebra of the ı \imath quiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two ı \imath Hall algebras, explaining the isomorphism of the ı \imath quantum group of split affine type A under the two presentations.

Combinatorial Fock spaces and quantum symmetric pairs

A way to construct the natural representation of the quantized affine algebra U v ( s l ̂ l ) is via the deformed Fock space by Misra and Miwa. This relates the classes of Weyl modules for U q ( s l N ) were q is a root of unity to the action of U v ( s l ̂ l ) as N tends toward infinity. In this paper we investigate the situation outside of type A. In classical types, we construct embeddings of the Grothendieck group of finite dimensional U q ( g ) -modules into Fock spaces of different charges and define an action of an affine quantum symmetric pair that plays the role of the quantized affine algebra. We describe how the action is related to the linkage principal for quantum groups at a root of unity and tensor product multiplicities.

Duplex Hecke algebras of type B

As a sequel to [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.], in this article we first introduce a so-called duplex Hecke algebras of type [Formula: see text] which is a [Formula: see text]-algebra associated with the Weyl group [Formula: see text] of type [Formula: see text], and symmetric groups [Formula: see text] for [Formula: see text], satisfying some Hecke relations (see Definition 3.1). This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur–Weyl duality of Levi-type (see [B. Shu and Y. Yao, On enhanced reductive groups (I): Enhanced Schur algebras and the dualities related to degenerate duplex Hecke algebras, with an appendix by B. Liu, submitted (2023)]), extending the duplex Hecke algebra of type [Formula: see text] arising from the related [Formula: see text]-Schur–Weyl duality of Levi-type (see [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.]). A duplex Hecke algebra of type [Formula: see text] admits natural representations on certain tensor spaces. We then establish a Levi-type [Formula: see text]-Schur–Weyl duality of type [Formula: see text], which reveals the double centralizer property between such duplex Hecke algebras and [Formula: see text]quantum groups studied by Bao and Wang in [H. Bao and W. Wang, A new approach to Kazhdan–Lusztig theory of type B via quantum symmetric pairs, Astérisque 402 (2018)].

Weyl algebras for quantum homogeneous spaces

We present a new family of quantum Weyl algebras where the polynomial part is the quantum analog of functions on homogeneous spaces corresponding to symmetric matrices, skew symmetric matrices, and the entire space of matrices of a given size. The construction uses twisted tensor products and their deformations combined with invariance properties derived from quantum symmetric pairs. These quantum Weyl algebras admit Uq(glN)-module algebra structures compatible with standard ones on the polynomial part, have relations that are expressed nicely via matrices, and are closely related to an algebra arising in the theory of quantum bounded symmetric domains.

Whittaker categories of quasi-reductive lie superalgebras and quantum symmetric pairs

Abstract We show that, for an arbitrary finite-dimensional quasi-reductive Lie superalgebra over ${\mathbb {C}}$ with a triangular decomposition and a character $\zeta $ of the nilpotent radical, the associated Backelin functor $\Gamma _\zeta $ sends Verma modules to standard Whittaker modules provided the latter exist. As a consequence, this gives a complete solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category ${\mathcal {O}}$ . In the case of the ortho-symplectic Lie superalgebras, we show that the Backelin functor $\Gamma _\zeta $ and its target category, respectively, categorify a q-symmetrizing map and the corresponding q-symmetrized Fock space associated with a quasi-split quantum symmetric pair of type $AIII$ .

Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras

Let g be a complex simple Lie algebra and UqLg the corresponding quantum affine algebra. We construct a functor Fθ between finite-dimensional modules over a quantum symmetric pair subalgebra of affine type Uqk⊂UqLg and an orientifold KLR algebra arising from a framed quiver with a contravariant involution, providing a boundary analogue of the Kang-Kashiwara-Kim-Oh generalized Schur-Weyl duality. With respect to their construction, our combinatorial model is further enriched with the poles of a trigonometric K-matrix intertwining the action of Uqk on finite-dimensional UqLg-modules. By construction, Fθ is naturally compatible with the Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of monoidal categories, Fθ is a functor of module categories. Relying on a suitable isomorphism à la Brundan-Kleshchev-Rouquier, we prove that Fθ recovers the Schur-Weyl dualities due to Fan-Lai-Li-Luo-Wang-Watanabe in quasi-split type AIII.

Braid group action and quasi-split affine 𝚤quantum groups I

This is the first of our papers on quasi-split affine quantum symmetric pairs ( U ~ ( g ^ ) , U ~ ı ) \big (\widetilde {\mathbf U}(\widehat {\mathfrak g}), \widetilde {{\mathbf U}}^\imath \big ) , focusing on the real rank one case, i.e., g = s l 3 \mathfrak g = \mathfrak {sl}_3 equipped with a diagram involution. We construct explicitly a relative braid group action of type A 2 ( 2 ) A_2^{(2)} on the affine ı \imath quantum group U ~ ı \widetilde {{\mathbf U}}^\imath . Real and imaginary root vectors for U ~ ı \widetilde {{\mathbf U}}^\imath are constructed, and a Drinfeld type presentation of U ~ ı \widetilde {{\mathbf U}}^\imath is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine ı \imath quantum groups in the sequels.

An intrinsic approach to relative braid group symmetries on ı$\imath$quantum groups

AbstractWe initiate a general approach to the relative braid group symmetries on (universal) quantum groups, arising from quantum symmetric pairs of arbitrary finite types, and their modules. Our approach is built on new intertwining properties of quasi ‐matrices which we develop and braid group symmetries on (Drinfeld double) quantum groups. Explicit formulas for these new symmetries on quantum groups are obtained. We establish a number of fundamental properties for these symmetries on quantum groups, strikingly parallel to their well‐known quantum group counterparts. We apply these symmetries to fully establish rank 1 factorizations of quasi ‐matrices, and this factorization property, in turn, helps to show that the new symmetries satisfy relative braid relations. As a consequence, conjectures of Kolb–Pellegrini and Dobson–Kolb are settled affirmatively. Finally, the above approach allows us to construct compatible relative braid group actions on modules over quantum groups for the first time.

Hall algebras and quantum symmetric pairs of Kac-Moody type

We extend our ıHall algebra construction from acyclic to arbitrary ıquivers, where the ıquiver algebras are infinite-dimensional 1-Gorenstein in general. Then we establish an injective homomorphism from the universal ıquantum group of Kac-Moody type arising from quantum symmetric pairs to the ıHall algebra associated to a virtually acyclic ıquiver.

Hall Algebras and Quantum Symmetric Pairs of Kac–Moody Type II

Hall Algebras and Quantum Symmetric Pairs of Kac–Moody Type II

Representations of orientifold Khovanov–Lauda–Rouquier algebras and the Enomoto–Kashiwara algebra

We consider an "orientifold" generalization of Khovanov-Lauda-Rouquier algebras, depending on a quiver with an involution and a framing. Their representation theory is related, via a Schur-Weyl duality type functor, to Kac-Moody quantum symmetric pairs, and, via a categorification theorem, to highest weight modules over an algebra introduced by Enomoto and Kashiwara. Our first main result is a new shuffle realization of these highest weight modules and a combinatorial construction of their PBW and canonical bases in terms of Lyndon words. Our second main result is a classification of irreducible representations of orientifold KLR algebras and a computation of their global dimension in the case when the framing is trivial.

Affine Hecke algebras and quantum symmetric pairs

We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and canonical bases for the affine Schur algebra. The multiplication formula allows us to establish a stabilization property of the family of affine Schur algebras that leads to the modified version of an algebra K n c {\mathbf K}^{\mathfrak c}_n . We show that K n c {\mathbf K}^{\mathfrak c}_n is a coideal subalgebra of quantum affine algebra U ( g l ^ n ) {\mathbf {U}}(\widehat {\mathfrak {gl}}_n) , and ( U ( g l ^ n ) , K n c ) \big ({\mathbf {U}}(\widehat { \mathfrak {gl}}_n), {\mathbf K}^{\mathfrak c}_n) forms a quantum symmetric pair. The modified coideal subalgebra is shown to admit monomial and stably canonical bases. We also formulate several variants of the affine Schur algebra and the (modified) coideal subalgebra above, as well as their monomial and canonical bases. This work provides a new and algebraic approach which complements and sheds new light on our previous geometric approach on the subject. In the appendix by four of the authors, new length formulas for the Weyl groups of affine classical types are obtained in a symmetrized fashion.

Stokes Phenomenon and Reflection Equations

In this paper, we study the Stokes phenomenon of the cyclotomic Knizhnik–Zamolodchikov equation associated to the symmetric pair \((\mathfrak {gl}_n,\mathfrak {so}_n)\), and prove that its two types of Stokes matrices satisfy the Yang–Baxter and reflection equations respectively. We briefly discuss its isomonodromy deformation, and its relations with cyclotomic associators, twists and quantum symmetric pairs.

Serre-Lusztig relations for ıquantum groups III

Let U˜ı be a quasi-split universal ıquantum group associated to a quantum symmetric pair (U˜,U˜ı) of Kac-Moody type with a diagram involution τ. We establish the Serre-Lusztig relations for U˜ı associated to a simple root i such that i≠τi, complementary to the Serre-Lusztig relations associated to i=τi which we obtained earlier. A conjecture on braid group symmetries on U˜ı associated to i disjoint from τi is formulated.

A Drinfeld-type presentation of affine $$\imath $$quantum groups II: split BCFG type

Let $$\widetilde{{{\mathbf {U}}}}^\imath $$ be the universal $$\imath $$ quantum group arising from quantum symmetric pairs. Recently, Lu and Wang formulated a Drinfeld-type presentation for $$\widetilde{{{\mathbf {U}}}}^\imath $$ of split affine ADE type. In this paper, we generalize their results by establishing a Drinfeld-type presentation for $$\widetilde{{{\mathbf {U}}}}^\imath $$ of arbitrary split affine type.

ıQuantum groups of split type via derived Hall algebras

A quantum symmetric pair consists of a quantum group U and its coideal subalgebra Uςı (called an ıquantum group) with parameters ς. In this note, we use the derived Hall algebras of 1-periodic complexes to realize the ıquantum groups Uςı of split type.

Universal K-matrices for quantum Kac-Moody algebras

We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra H H endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups on tensor products of its representations. We prove that new examples of such universal K-matrices arise from quantum symmetric pairs of Kac-Moody type and depend upon the choice of a pair of generalized Satake diagrams. In finite type, this yields a refinement of a result obtained by Balagović and Kolb, producing a family of non-equivalent solutions interpolating between the quasi-K-matrix originally due to Bao and Wang and the full universal K-matrix. Finally, we prove that this construction yields formal solutions of the generalized reflection equation with a spectral parameter in the case of finite-dimensional representations over the quantum affine algebra U q L s l 2 U_qL\mathfrak {sl}_{2} .

POLYNOMIAL FUNCTORS AND TWO-PARAMETER QUANTUM SYMMETRIC PAIRS

We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups GLn, the two-parameter polynomial functors give a new interpretation of (polynomial) representations of the quantum symmetric pair ( {U}_{Q,q}^B ( mathfrak{gl} n), Uq( mathfrak{gl} n)) which specializes to type AIII/AIV quantum symmetric pairs. The coideal subalgebra {U}_{Q,q}^B ( mathfrak{gl} n) appears in a Schur–Weyl duality with the type B Hecke algebra {mathcal{H}}_{Q,q}^B (d). We endow two-parameter polynomial functors with a cylinder braided structure which we use to construct the two-parameter Schur functors. Our polynomial functors can be precomposed with the quantum polynomial functors of type A producing new examples of action pairs.

Serre–Lusztig relations for $$\imath $$quantum groups II

The $\imath$Serre relations and the corresponding Serre-Lusztig relations are formulated and established for arbitrary $\imath$quantum groups arising from quantum symmetric pairs of Kac-Moody type.