Quasi-Split ι-Quantum Groups

• Quasi-split ι-quantum groups are coideal subalgebras defined by symmetric pairs and diagram involutions, generalizing classical quantum groups with explicit Serre-type relations.
• They feature a uniform presentation including both Serre and Drinfeld current relations, enabling detailed construction of finite-dimensional standard and irreducible modules.
• Geometric and categorification approaches, via Steinberg varieties and Hall algebras, link their representation theory to intersection cohomology and invariant theory.

Quasi-split $\imath$quantum groups are coideal subalgebras of Drinfeld–Jimbo quantum groups defined by symmetric pairs $(\mathfrak g, \mathfrak k)$ whose Satake diagram has no black nodes, admitting a diagram involution $T$ that encodes symmetric pair data. They generalize quantum groups by incorporating diagram involutions and parameters into their algebraic and categorical structure, and are characterized by explicit Serre-type relations, a Drinfeld–type current presentation, and intricate relationships to Hall algebras, convolution algebras on equivariant K-theory of Steinberg varieties, and categorification frameworks. These structures yield rich families of finite-dimensional standard modules and irreducible modules, with their composition multiplicities computed by intersection cohomology of orbit closures, thereby connecting the representation theory of $\imath$quantum groups to geometric representation theory and categorical approaches.

1. Algebraic Foundations and Serre Presentations

A quasi-split $\imath$quantum group $U^\imath$ is defined via a uniform Serre presentation involving generators $B_i$ and toral elements $K_\mu$: $B_i = F_i + S_i E_{T(i)} K_i^{-1}$ where $T$ is the diagram involution, $S_i$ are parameters, and $K_i$ are rescalings of Cartan generators when $T(i) = i$ (Chen et al., 2018).

The relations satisfied by these generators are quantum analogues of classical Serre relations, known as $\imath$-Serre relations. For $T(i)=i$ and $j\neq i$, the primary Serre relation reads: $$\sum_{n=0}^{1-d_{ij}} (-1)^n B_{i,\overline{p}}^{(n)} B_j B_{i,\overline{p}+1}^{(1-d_{ij}-n)}=0$$ with $d_{ij}=1-a_{ij}$ for Cartan matrix entries $a_{ij}$, and the $B_{i,\overline{p}}^{(m)}$ are divided powers controlled by a natural parity parameter (Chen et al., 2018). The verification that these relations hold is reduced to vanishing of novel $q$-binomial combinations (e.g., $T(w,u,\ell)=0$), with these combinatorial identities forming the backbone for the algebra's definition.

In the "split" case (trivial diagram involution), the presentation simplifies: $B_i$ coincide with the Chevalley generators, and the toral part becomes trivial.

Additional structure arises under variation of parameters (as in covering groups with parameter $\pi$, interpolating between Lusztig's quantum groups and quantum supergroups), generalizing the Serre presentation and allowing for a uniform framework incorporating both classical and super analogues (Chung, 2019).

2. Drinfeld Current Presentations and Higher Rank Constructions

Beyond the Serre presentation, recent advances establish a Drinfeld type (current) presentation for quasi-split $\imath$quantum groups, notably in affine types (Su et al., 9 Jul 2024, Lu et al., 2023). The main generators are encoded as formal series: $$\begin{align*} {}_i(z) &= 1 + \sum_{k\geq1} (q-q^{-1}) \Theta_{i,k} z^k \ B_i(z) &= \sum_{\ell} B_{i,\ell} z^\ell \end{align*}$$ with central elements and grading operators included.

Defining relations include generalized quantum current relations for Cartan–B generators, for instance: $${}_i(z) B_j(w) = \frac{(1-q^{c_{ij}}z/w)(1-q^{-c_{\tau(i)j}}zwC)}{(1-q^{-c_{ij}}z/w)(1-q^{c_{\tau(i)j}}zwC)} B_j(w) {}_i(z)$$ and elaborate current Serre relations, such as: $$\operatorname{Sym}_{w_1,w_2}\left(B_i(w_1) B_i(w_2) B_j(z) - [2] B_i(w_1) B_j(z) B_i(w_2) + B_j(z) B_i(w_1) B_i(w_2)\right)=0$$ for pairs $(i,j)$ with $c_{ij}=-1$, and specific adjustments on fixed nodes (Su et al., 9 Jul 2024).

To construct the root data, one crucially employs a relative braid group action associated to the restricted Weyl group, yielding real and imaginary root vectors. The real root series $B_{i,k}$ are constructed inductively via powers of braid group automorphisms, while imaginary root vectors are constructed recursively involving commutators and Cartan elements (see (Lu et al., 2023), §4).

This Drinfeld current presentation provides a basis for explicit calculations of relations, for the construction of finite-dimensional representations, and for geometric and categorification approaches. The Serre-type and current relations reflect both the folded nature of the Satake diagram and the mixture of real and imaginary root data appropriate to the quasi-split context.

3. Geometric Realization and Standard Modules via Steingberg Varieties

A fundamental geometric approach realizes quasi-split affine $\imath$quantum groups as convolution algebras on the equivariant $K$-theory (or Borel–Moore homology) of Steinberg varieties. Specifically, for type $\mathrm{AIII}_{2n-1}^{(\tau)}$:

• The isotropic flag variety $\mathcal F$ for $G = \mathrm{Sp}(2d)$ is decomposed into partial flag varieties indexed by $$\mathbf v \in \Lambda_{\mathfrak{c},d} = \{(v_1, ..., v_n, ..., v_1) \in \mathbb N^{2n}\}$$, with $v_i = v_{2n+1-i}$ and $\sum v_i = d$.
• The Steinberg variety $\mathcal Z$ is constructed as the fiber product $\mathcal M \times_\mathcal N \mathcal M$, where $\mathcal M = T^* \mathcal F$ and $\mathcal N$ is the nilpotent cone in $\mathfrak{sp}(2d)$.

The equivariant $K$-group $K^{G \times \mathbb C^*}(\mathcal Z)$ admits a convolution product, and is shown to be generated by explicit classes corresponding to geometrically defined orbits. Algebraic generators of the $\imath$quantum group are matched to these geometric generators under an algebra homomorphism $\Psi$ [(Su et al., 9 Jul 2024), Thm 4.12].

• "Raising" and "lowering" currents correspond to explicit convolution operators acting on polynomial or Laurent representations (essentially as difference/difference operators on torus-fixed points).
• The Cartan currents act by multiplication by explicit rational functions in the polynomial variables.
• The central node receives special treatment matching the $\imath$-structure.

The Drinfeld relations and Serre relations are verified to hold on these polynomial representations via equivariant calculations involving the orbit structure and the geometry of isotropic flags.

4. Standard Modules, Irreducibles, and Composition Multiplicities

Given this realization, a rich family of finite-dimensional modules is constructed:

• Standard modules: For each pair $(\mathcal O_\phi, \chi_\phi)$ (orbit and local system) in the relevant fixed-point variety, define the Springer fiber $M_x = \pi^{-1}(x)$ for $x \in \mathcal O_\phi$, and the standard module as $H_*(M_x)_\phi$ (the $\chi_\phi$-isotypic component in the Borel–Moore homology).
• Irreducible modules: The semisimple summands $L_\psi$ in the decomposition of $\pi_* \mathbb C_{\mathcal M^a}$ correspond to irreducible modules.

There is then a precise formula for the composition multiplicities of the irreducibles in the standard modules [(Su et al., 9 Jul 2024), Prop 5.15]: $$[H_*(M_x)_\phi : L_\psi] = \sum_k \dim H^k(i_x^! IC_\psi)_\phi$$ where $IC_\psi$ is the intersection cohomology sheaf attached to $(\mathcal O_\psi, \chi_\psi)$ and $i_x$ is the inclusion of the point $x$. Thus, the composition multiplicities are determined by the stalk cohomology of the IC-sheaves. This is directly analogous to the multiplicity formulae in affine Hecke algebras and quantum groups realized via Springer theory.

This geometric construction ensures that the category of finite-dimensional $U^\imath_t$-modules (for $t$ not a root of unity) is stratified and controlled by the geometry of the Steinberg variety, with explicit algebraic correspondences.

5. Hall Algebra and Categorification Approaches

Quasi-split $\imath$quantum groups can also be realized via Hall algebra constructions using quivers with involution, coherent sheaves with involutive symmetry, and, more generally, in the language of semi-derived Ringel–Hall algebras of $\varrho$-complexes (Lu et al., 20 Nov 2024, Lu et al., 2022). The 2Hall algebra associated with the category $\mathcal C_\varrho(\mathcal A)$ (for $\varrho$ an involution) yields a realization of the quantum loop algebra or the $\imath$quantum group in its Drinfeld current presentation.

The Hall algebra generators correspond to root elements, constructed via Hall products reflecting the combinatorics of extensions and torsion sheaves in suitable tubes (weighted projective lines). Verifying that these Hall algebra generators satisfy the Drinfeld and current–Serre relations (see formula (10.5) in (Lu et al., 20 Nov 2024)) involves explicit computation of Hall numbers, quantum binomial coefficients, and compatibility with the underlying geometric data.

Additionally, categorifications are achieved via 2-categorical diagrammatic frameworks generalizing Khovanov–Lauda–Rouquier 2-quantum groups. These 2-$\imath$quantum groups are defined as graded 2-categories whose Grothendieck rings recover quasi-split $\imath$quantum groups, with objects corresponding to the $\imath$-weight lattice and 1-morphisms corresponding to $B_i$ operators with appropriate biadjunction and (unoriented) string diagrams (Brundan et al., 28 May 2025). Relations in these 2-categories mirror the algebraic and braid group symmetries of the quantum symmetric pair, with essential features such as the “ibraid” relation and bubble relations appearing in generating function form.

6. Canonical and Crystal Bases, Braid Group Symmetries, and Parameter Variations

Existence and stability of $\imath$-canonical and crystal bases remain intricate, particularly in quasi-split and real rank one cases (Watanabe, 2021, Watanabe, 2022). Canonical bases can sometimes be characterized as projective limits of (modified) highest-weight modules, with stability established under precise conditions and connected to the emerging theory of $\imath$crystals—a generalization of Kashiwara’s crystal theory designed to accommodate the modified action and extra symmetry.

Braid group symmetries on quasi-split $\imath$quantum groups are constructed via Hall algebras, reflection functors, and differential operator approaches (Lu et al., 2021, Fan et al., 2023). Explicit closed formulas yield automorphisms satisfying braid relations of the restricted (folded) Weyl group and are realized both algebraically and via differential operators on polynomial rings. Compatibility across the quantum group, the modified $q$-Weyl algebra, and its polynomial modules under these symmetries is established via intertwining algebra homomorphisms.

Parameter variations, the introduction of $\pi$ in quantum covering groups, and multi-parameter versions (notably, three-parameter cases for affine types) extend the unifying framework, connecting specializations to quantum supergroups, classical (Lusztig) quantum groups, and new monoidal categorifications (Chung, 2019, Luo et al., 28 Sep 2025).

7. Invariant Theory, Representation Decomposition, and Further Directions

Invariant theory for quasi-split $\imath$quantum groups, especially of type AIII, is developed analogously to classical and type A quantum invariant theory (Luo et al., 2023). The lack of a full Hopf algebra structure (no coproduct or $R$-matrix) is circumvented by considering twisted tensor space modules, and invariants are described via balanced module conditions. The first and second fundamental theorems (FFT and SFT) establish generating sets and explicit relations among invariants, with SFT identifying quantum $n$-minors as generators of the relations (ideals).

The resulting modules decompose multiplicity-free under $\imath$-Howe duality, with irreducibles indexed by suitable partitions and highest-weight maps constructed via explicit tensor space combinatorics.

A plausible implication is that as new Drinfeld-type presentations and geometric/categorical realizations continue to be developed across broader classes, further connections to affine Hecke algebras, quantum symmetric pairs, and their applications in mathematical physics and categorification will be systematically uncovered. Techniques in Hall algebras, K-theory, and derived categories will remain central to future progress.