Quasi-Split iQuantum Groups

• Quasi-split iquantum groups are coideal subalgebras defined by quantum symmetric pairs with Satake diagrams having no black nodes, which simplify quantum group symmetries.
• They incorporate Serre-type and current presentations using divided powers and braid group symmetries, bridging advanced representation theory and categorification.
• Their Hall algebra and cluster realizations provide geometric insights and support the development of canonical bases and categorical quantum algebra.

Quasi-split $\imath$quantum groups are coideal subalgebras of Drinfeld–Jimbo quantum groups defined via quantum symmetric pairs, with their quasi-split property indicated by Satake diagrams containing no black nodes. These algebras generalize the foundational structures in quantum group theory, incorporating diagram automorphisms and involutions to encode symmetries originating from Lie theory, and play central roles across algebraic, representation-theoretic, and categorification contexts.

1. Foundational Definition and Quasi-split Condition

A quantum symmetric pair consists of a Drinfeld–Jimbo quantum group $\mathbf{U}$ and a coideal subalgebra $\mathbf{U}^\imath$, the $\imath$quantum group. The quasi-split condition is characterized by the Satake diagram having no black nodes, indicating the simplest symmetry—either a trivial or non-trivial diagram involution—but precluding additional twisting present in non-quasi-split (i.e., black-node) cases (Chen et al., 2018, Chung, 2019). The generic generator structure involves

$B_i = F_i + S_i E_{\tau i} K_i^{-1}$

for all $i$ in the index set, with $F_i$, $E_i$ the standard Chevalley generators, $K_i$ Cartan elements, $S_i$ parameters and $\tau$ the diagram involution.

For split type ($T=\mathrm{id}$), $B_i = F_i + E_i K_i^{-1}$, matching and simplifying the more general quasi-split structure (Chen et al., 2021, Lu et al., 13 Feb 2025).

2. Serre Presentation, $\imath$-Serre Relations, and Divided Powers

Quasi-split $\imath$quantum groups admit a Serre-type presentation wherein the relations generalize Lusztig's quantum group Serre relations to incorporate divided powers specific to the $\imath$ setting (Chen et al., 2018, Chung, 2019):

$$\sum_{n=0}^{1-a_{ij}} (-1)^n B_i^{(n)} B_j B_i^{(1-a_{ij}-n)} = 0,$$

where $a_{ij}$ is the Cartan entry. The $\imath$-divided powers $B_i^{(n)}$ include polynomial modifications depending on parity parameters and central elements (e.g., $J$ for super cases). Rank-one formulas specialize to

$$B^{(2a)} = [2a]!^{-1} \prod_{k=1}^a (B^2 - [2k-2]^2 J), \qquad B^{(2a+1)} = [2a+1]!^{-1} B \prod_{k=1}^a (B^2 - [2k]^2 J),$$

reflecting the detailed combinatorial structure in both even and odd scenarios (Chung, 2019).

Verification of these relations is reduced to proving universal $q$-binomial identities, such as:

$T(w,u,\ell) := \sum_{\substack{c+e+r=u\t=0}}^{\ell} (-1)^t q^{-t(\ell+u-1)+(\ell+u)(c-e)} \left[\cdots\right] = 0,$

where the sum and coefficients encode the expansion in the PBW basis (Chen et al., 2018).

3. Drinfeld-Type Presentation and Current Generators

The Drinfeld-type presentation for affine quasi-split $\imath$quantum groups is established in terms of current generators:

• Real roots: $B_{i,k}$, indexed by $k \in \mathbb{Z}$ via translation automorphisms.
• Imaginary roots: elements $O_{i,m}$ (or $H_{i,m}$ via exponentiation), capturing the loop algebra structure (Lu et al., 2020, Lu et al., 2022).

Defining relations involve both classical current commutation and genuine deformations:

$$[H_{i,m}, B_{j,l}] = m c_{ij} B_{j,l+m}\quad\text{(up to normalization)},$$

with “current Serre relations” described symmetrically, e.g.:

$$S(k_1, k_2 \mid l; i) = \operatorname{Sym}_{k_1,k_2}\left(B_{i,k_1}B_{i,k_2}B_{j,l} - [2] B_{i,k_1}B_{j,l}B_{i,k_2} + B_{j,l}B_{i,k_1}B_{i,k_2}\right) = 0,$$

where $[2]$ is a quantum integer (Lu et al., 2022). These recast the algebraic structure in generating-function form, facilitating representation-theoretic constructions and categorification.

4. Braid Group and Weyl Group Symmetries; Reflection Functors

A distinguishing feature of quasi-split $\imath$quantum groups is the explicit construction of braid group automorphisms, which generalize Lusztig's symmetries to the quantum symmetric pair context. These automorphisms are realized via closed formulas involving the generators, divided powers, and i-weights of modules (Lu et al., 2021, Wang et al., 16 Aug 2025):

• For split and diagonal types, actions are in terms of sums over idivided powers with explicit $q$-scalars and signs.
• For quasi-split (non-split) types, formulas are constructed from triple products

$$B^{(t, t+l, l)} := B_{\tau i}^{(t)} B_{i}^{(t+l)} B_{\tau i}^{(l)},$$

mirroring the combinatorics of $U(\mathfrak{sl}_3)$ (Wang et al., 16 Aug 2025).

These braid group operators satisfy the relative braid group relations, ensuring that composite symmetries

$$(\dot{T}_i' \dot{T}_j' \cdots) = (\dot{T}_j' \dot{T}_i' \cdots)$$

(with $m_{ij}$ factors as the corresponding order) hold at the algebra and module level. The automorphisms are compatible with quasi-$K$ matrices, intertwining with Lusztig symmetries (Wang et al., 16 Aug 2025, Lu et al., 2022).

Reflection functors in Hall algebraic realizations categorify these symmetries, linking the categorical module theory of quivers with involution to algebra automorphisms (Lu et al., 2021).

5. Hall Algebra and Categorical Realizations

Quasi-split $\imath$quantum groups can be realized via twisted semi-derived Ringel–Hall algebras of categories equipped with an involution (Lu et al., 20 Nov 2024). In particular:

• $i$-Hall algebras are constructed from categories of $\rho$-complexes (periodic or twisted complexes satisfying $\rho(d) \circ d = 0$).
• Homological properties analogous to 1-Gorenstein conditions support acyclicity and enable the localization and twisting that match the quantum group structure.
• The Drinfeld-type presentation of quasi-split $\imath$quantum loop algebras is realized via explicit homomorphisms:

$$\Omega: \mathrm{Dr}U_v(L\mathfrak{g}) \rightarrow \mathcal{H}(\mathbb{X}, \rho),$$

where $\mathcal{H}(\mathbb{X}, \rho)$ is a twisted semi-derived i-Hall algebra of coherent sheaves on weighted projective lines (Lu et al., 20 Nov 2024).

These categorical constructions provide geometric and homological interpretations of algebraic relations, and facilitate advanced tools for canonical bases and categorification.

6. Canonical Bases, Bar Involutions, and Cluster Realizations

A fundamental property for quasi-split $\imath$quantum groups is the existence of a bar involution given appropriate parameter conditions (Chen et al., 2018):

$$\overline{q} = q^{-1},\quad \overline{K_\mu} = K_\mu^{-1},\quad \overline{B_i} = B_i,$$

with parameter constraints

• $s_i = \overline{s_i}$ when $T(i) = i$ and off-diagonal Cartan entries,
• $s_i = s_{T(i)}$ if $T(i) \neq i$, $a_{i,T(i)} = 0$,
• $s_i = q\overline{s_i}$ for $T(i) \neq i$, $a_{i,T(i)} \neq 0$.

This involution underpins the theory of canonical (i-canonical) bases, which, through categorification, give rise to orthogonal bases with deep geometric and representation-theoretic meaning (Brundan et al., 28 May 2025).

Cluster-theoretic realizations are achieved by constructing explicit algebra homomorphisms (the "i-analogue of Feigin's map") from $\imath$quantum groups to quantum tori:

$\pi: U^\imath \rightarrow \mathcal{T}_\imath(Q),$

sending generators to monomials in cluster variables. These maps respect the (2)-Serre relations and provide bases analogous to cluster character formulas (Lu et al., 13 Feb 2025).

7. Structural Links to Representation Theory and Categorification

Quasi-split $\imath$quantum groups generalize quantum groups in several axes:

• New presentations enable BLM-type constructions and integral bases, bridging to q–Schur algebras and modular representation theory of finite symplectic groups (Du et al., 2021).
• The categorical framework developed via Hall algebras and diagrammatic 2-categories (new graded 2-categories U$^\imath$) categorifies the modified i-quantum group and its Serre relations, with objects, 1-morphisms, and 2-morphisms reflecting the involutive symmetries (Brundan et al., 28 May 2025).
• The underlying symmetry data (involution, parameters, canonical bases) elaborate the role of these structures in explicit module theory, braid and Weyl group symmetries, and categorification.

A plausible implication is that the integral and categorical versions of relative braid group automorphisms for quasi-split iquantum groups will continue to inform the structure of canonical bases and their duals in both algebraic and geometric representation theory settings.

Table: Key Features of Quasi-Split $\imath$Quantum Groups

Feature	Occurrence/Reference	Mathematical Formulation/Significance
Serre Relations	(Chen et al., 2018, Chung, 2019)	$$\sum_{n} (-1)^n B_i^{(n)} B_j B_i^{(1-a_{ij}-n)} = 0$$
Braid Symmetries	(Lu et al., 2021, Wang et al., 16 Aug 2025, Lu et al., 2022)	Automorphisms via closed formulas, compatible with quasi-$K$-matrices
Drinfeld-Type Currents	(Lu et al., 2020, Lu et al., 2022)	Currents $B_{i,l}$, $H_{i,m}$; "current Serre" relations
Hall Realization	(Lu et al., 20 Nov 2024, Chen et al., 2021)	$\imath$Hall algebras via semi-derived Ringel–Hall theory
Bar Involution	(Chen et al., 2018)	Preserved under involutive parameter conditions
Cluster Embeddings	(Lu et al., 13 Feb 2025)	Homomorphism to quantum torus: $\pi: U^{\imath} \to \mathcal{T}_\imath(Q)$
Categorification	(Brundan et al., 28 May 2025)	Graded 2-category generalizing Khovanov–Lauda–Rouquier framework

Key works referenced herein include the Serre presentations (Chen et al., 2018, Chung, 2019), Drinfeld-type presentations (Lu et al., 2020, Lu et al., 2022), Hall algebraic and categorical constructions (Lu et al., 20 Nov 2024, Chen et al., 2021), structural links to canonical bases and representation theory (Du et al., 2021), and explicit braid group symmetry formulas (Lu et al., 2021, Wang et al., 16 Aug 2025, Brundan et al., 28 May 2025, Lu et al., 13 Feb 2025).

Quasi-split $\imath$quantum groups therefore serve as a foundational and unifying structure in modern quantum algebra, with pronounced impacts on categorical representation theory, integrable models, and the ongoing development of quantum symmetric pairs and their applications.