Twisted Affine Yangian of Type D

• The twisted affine Yangian is a quantum group defined by a finite Drinfeld-type presentation with explicit commutator relations and a PBW basis for type D Lie algebras.
• It bypasses the infinite generator lists of classical RTT presentations by using finite sets, enabling concrete constructions of coideal subalgebras and links to rectangular W-algebras.
• The structure supports a PBW theorem and highest weight representation theory, facilitating integrable models and reflection symmetries in quantum groups.

The twisted affine Yangian is a quantum group associated to symmetric pairs of affine Lie algebras, here specifically of type $D_n^{(1)}$ (i.e., relating $\widehat{\mathfrak{sl}(2n)}$ with $\widehat{\mathfrak{so}(2n)}$ via involution). It arises as a flat deformation and quantum integrable analogue of the central extension of twisted current algebras, providing a highly structured and minimal Drinfeld-type presentation, PBW basis, and strong links to both representation theory and integrable boundary conditions. The recent presentations use finite sets of generators and explicit commutator relations to bypass the infinite generator lists of classical RTT or J-presentations, facilitating applications to coideal subalgebras, rectangular $W$-algebras, and reflection algebras (Harako et al., 1 Jul 2025, Ueda, 6 Dec 2025).

1. Drinfeld-Type Presentations: Finite and Affine Twisted Yangians

In the finite case, the twisted Yangian of type $D$ for $n\ge 4$ is generated by

• Cartan elements $H_{i,0}$ and $H_{i,1}$ ($1\le i\le n$)
• Chevalley elements $X_{i,0}^{\pm}, X_{i,1}^{\pm}$ ($1\le i\le n$)

with the following key commutator relations:

• $[H_{i,r}, H_{j,s}] = 0$ (Cartan subalgebra for $r,s\in\{0,1\}$)
• $$[H_{i,0}, X^{\pm}_{j,0}] = \pm a_{ij}\, X^{\pm}_{j,0}$$; $[X^+_{i,0}, X^-_{j,0}] = \delta_{ij}\,H_{i,0}$
• $$[H_{i,1}-(\hbar/2)H_{i,0}^2, X^{\pm}_{j,0}] = \pm a_{ij}X^{\pm}_{j,1}$$
• Deformed mode commutation: $$[X_{i,1}^{\pm}, X_{i,0}^{\pm}] = \hbar(X_{i,0}^{\pm})^2$$
• Serre-type relations and special mixed-mode, quartic $\hbar$–corrected commutators, especially at branch nodes $i=n-2,n-1,n$.

This presentation extends to the twisted affine Yangian $\mathrm{TY}_\hbar(\widehat{\mathfrak{so}(2n)})$ by adding one extra node ($i=0$), resulting in similar generators and relations indexed by $i=0,1,\dots,n$. All defining relations are encompassed by shifts in the affine Cartan matrix, with all Serre/reflection relations at the new node mirroring those at the $n$th node (Harako et al., 1 Jul 2025).

Recent work has introduced a two-parameter generalization $$\mathrm{TY}_{\hbar,\varepsilon}(\widehat{\mathfrak{so}(2n)})$$ and provided a “minimalistic” finite set of defining relations, with explicit mode-by-mode commutators and identification of all $\hbar$–corrections (Ueda, 6 Dec 2025).

2. Structural Results: PBW Theorem, Classical Limit, and Universal Central Extension

The Drinfeld-filtration assigns degrees $\deg X_{i,r}^{\pm} = r$, $\deg H_{i,r} = r$, and the associated graded algebra is isomorphic to the universal enveloping algebra $U(\mathfrak{sl}(2n)[u]^\tau)$, the current algebra fixed by the involution whose fixed points correspond to $\mathfrak{so}(2n)$ (Harako et al., 1 Jul 2025).

An explicit homomorphism, preserving filtration degrees, identifies the twisted Yangian of type $D$ with Olshanskii’s RTT-based presentation, and establishes the PBW property: ordered monomials in the generators yield a basis. The twisted affine extension enjoys the same properties.

In the classical ($\hbar=\varepsilon=0$) limit, $$\mathrm{TY}_{\hbar,\varepsilon}(\widehat{\mathfrak{so}(2n)})$$ coincides precisely with the universal enveloping algebra of the $\tau$–fixed subalgebra of the universal central extension of $\mathfrak{sl}(2n)[u^{\pm 1},v]$ (Ueda, 6 Dec 2025).

Classical Limit Table

Presentation	Generators	Classical Limit
Twisted finite Yangian	$H_{i,r}, X_{i,r}^\pm$	$U(\mathfrak{sl}(2n)[u]^\tau)$
Twisted affine Yangian	$H_{i,r}, X_{i,r}^\pm, i=0$	UEA of UCE of $\mathfrak{sl}(2n)[u^{\pm1},v]^\tau$

3. Relations to RTT, J-Presentations, and Quantum Symmetric Pairs

The new finite Drinfeld-type presentation offers a concrete alternative to the infinite generator count of classical RTT and J-presentations in Yangian theory. For type $A$, Drinfeld and Guay–Nakajima–Wendlandt’s affine presentations previously yielded explicit Hopf structure; only recently have similar explicit finite-mode presentations been achieved for type $D$ (Harako et al., 1 Jul 2025).

The extension to quasi-split and split types via degenerations from affine $\imath$-quantum groups, and Gauss–decomposition techniques, realizes many twisted Yangians as graded limits or as reflection algebras in R-matrix language. In type AI (untwisted $A$), these presentations coincide (Lu et al., 13 Aug 2024, Lu et al., 7 Jun 2024).

Twisted Yangians naturally appear as quantum symmetric pairs, i.e., coideal subalgebras inside Yangians of classical or affine type, and are crucial for boundary conditions and reflection symmetries in integrable models (Bao, 2023).

4. Representation Theory, W-Algebra Realizations, and Applications

Twisted affine Yangians admit a highest weight representation theory modeled on Drinfeld polynomials at the branch nodes, paralleling the untwisted story. There exist surjective homomorphisms from the twisted affine Yangian to rectangular $W$-algebras of classical types, furnishing large families of modules and enabling connection with integrable spin-chain and quantum cohomology models (Ueda, 6 Dec 2025, Ueda, 2021).

Explicit formulas for R-matrices with reflection, Bethe-ansatz for open spin chains with orthogonal symmetry, and deformations of shifted current algebras in type $D$ are tractable within the finite-mode Drinfeld framework.

A construction for $n=4$ relates $$\mathrm{TY}_{\hbar,\varepsilon}(\widehat{\mathfrak{so}(8)})$$ surjectively to the UEA of the rectangular W-algebra $\mathcal{W}^k(\mathfrak{sp}(16),(2^8))$ via explicit algebra maps for all Drinfeld generators; extension to higher $n$ is an open direction (Ueda, 6 Dec 2025).

5. Conceptual Framework and Comparative Motivations

Finite-mode Drinfeld-style presentations enable a concrete construction of twisted Yangians with strong filtration, PBW, and triangular decomposition properties, and pave the way for explicit coideal, boundary, and reflection algebra constructions. Compared to classical approaches, the new presentations offer:

• Uniform indexing and finite listing of all relations
• Direct isomorphism with RTT-based twisted Yangians in type $D$
• Ease in extending coproducts, coideal subalgebra arguments, and in building representation theory and W-algebra links (Harako et al., 1 Jul 2025).

Twisted affine Yangians and their quasi-split/split analogues provide the algebraic backbone for integrable models with boundary conditions and for the quantum geometry and representation theory of symmetric spaces (Lu et al., 13 Aug 2024, Bao, 2023).

6. Open Questions and Future Directions

Open problems include construction of explicit coproducts and the full Drinfeld–Rosso PBW proof for the newly defined twisted affine Yangians. The extension and comparison of alternative presentations (e.g., Harako–Ueda), classification of finite-dimensional modules, and the realization of universal $K$-matrices for boundary integrability remain central questions (Ueda, 6 Dec 2025).

Extending the surjective W-algebra homomorphism beyond rank $n=4$ and connecting twisted affine Yangians with the full family of rectangular classical $W$-algebras constitutes a significant ongoing direction.

• "New presentation of the twisted Yangian of type $D$" (Harako et al., 1 Jul 2025)
• "Suggestion of a definition of the twisted affine Yangian of type $D$" (Ueda, 6 Dec 2025)
• "A Drinfeld type presentation of twisted Yangians of quasi-split type" (Lu et al., 13 Aug 2024)
• "Affine $\imath$quantum groups and twisted Yangians in Drinfeld presentations" (Lu et al., 7 Jun 2024)
• "Twisted Affine Yangian and Rectangular $W$-algebra of type $D$" (Ueda, 2021)
• "More on Affine Dynkin Quiver Yangians" (Bao, 2023)