Shifted Twisted Yangian

• Shifted twisted Yangian is a noncommutative, filtered algebra defined via deformed symmetric pair relations with twist parameters and shift data.
• It admits a PBW basis and Drinfeld-type presentations that enable explicit computations and geometric realizations through fixed-point loci and Slodowy slices.
• Its representation theory bridges Coulomb branch algebras, finite W-algebras, and mirror symmetry in 3d gauge theory, offering insights into quantum symmetric pairs.

A shifted twisted Yangian is a noncommutative, filtered algebraic structure arising as a deformation and coideal quantization of classical symmetric pairs, with deep connections to quantized Coulomb branch algebras, representation theory, and symplectic singularity theory. These objects generalize the notion of Yangians to settings with involutive diagram automorphisms (twist), and additional "shift" parameters tied to coweight data. They admit Drinfeld-type presentations, PBW bases, difference-operator realizations, and geometric correspondences with Slodowy slices and affine Grassmannian fixed-point loci, subsuming classical quantum symmetric pairs under the umbrella of 3d mirror symmetry and Coulomb branch constructions.

1. Algebraic Definition and Presentations

A shifted twisted Yangian, denoted ${}^\imath Y_\mu$ or $Y^+_{n,\ell}(\sigma)$ in type AI, is constructed from a simply-laced Cartan matrix $C=(c_{ij})$ with a Dynkin diagram involution $\tau$ and an "even spherical" coweight $\mu$. The algebra is generated by Cartan generators $H_i^{(r)}$ and root generators $B_i^{(s)}$ (where $i$ ranges over $I$ and $r,s$ are integers subject to shift-dependent bounds), subject to relations encoding the reflection equations and symmetry of twisted Yangians:

• Cartan commutativity: $[H_i^{(r_1)}, H_j^{(r_2)}] = 0$, with symmetry $H_i^{(r)} = (-1)^{r+1} H_{\tau(i)}^{(r)}$.
• Mixed Cartan-root relations and root-root commutators, involving both Lie-theoretic structure constants $c_{ij}$ and combined "twist" parameters $c_{\tau(i), j}$.
• Serre-type relations adapted to the twisted setting, with additional terms when roots are $\tau$-fixed or associated to symmetric pairs.
• Shift-vanishing and normalization: $H_i^{(r)} = 0$ for $r < -\langle\mu, \alpha_i\rangle$, $H_i^{(-\langle\mu, \alpha_i\rangle)} = 1$.

These relations are formulateable in terms of generating series $H_i(u) = \sum_r H_i^{(r)} u^{-r}$ and $B_i(u) = \sum_s B_i^{(s)} u^{-s}$, expressing deformation conditions on currents reminiscent of quantum algebras and reflection equations (Lu et al., 12 Oct 2025, Lu et al., 23 Dec 2025, Shen et al., 14 Oct 2025, Tappeiner et al., 2024, Lu et al., 6 May 2025, Brown, 2016).

The "shifted" form arises by restricting generators to those exceeding certain shift-matrix bounds, resulting in truncated or "cut-off" current components. This enforces the inclusion of shift data $\sigma=(s_{ij})$, which is tied to pyramid combinatorics or dimension-vector assignments.

2. PBW Bases and Structural Properties

The shifted twisted Yangians admit a filtered PBW basis indexed by root vectors and Cartan powers. For each positive root $\beta \in \Delta^+$ and admissible shift parameter $r>0$, the algebra includes "root vectors" $B_\beta^{(r)}$ recursively built as nested commutators. The set of ordered monomials

$$\prod_{\beta \in \Delta^+} \prod_{r>0} (B_\beta^{(r)})^{k_{\beta, r}} \times \prod_{i \in I_0} \prod_{p > -\langle\mu, \alpha_i\rangle} (H_i^{(p)})^{m_{i,p}} \times \prod_{i \in I_1} \prod_{p > -\langle\mu, \alpha_i\rangle} (H_i^{(p)})^{n_{i,p}}$$

spans the algebra, and forms a basis by PBW theorem arguments (Lu et al., 12 Oct 2025, Tappeiner et al., 2024, Lu et al., 6 May 2025). The proof employs filtrations (e.g., degree in root generators) and exploits isomorphism to the untwisted shifted Yangian in associated graded, often relying on shift-injection arguments or deformation-theory comparisons.

Truncation at a level $\ell$ produces a quotient algebra with finitely many generators and relations, yielding a finite-dimensional structure (Lu et al., 6 May 2025, Tappeiner et al., 2024). The truncated version coincides, in associated graded, with Poisson algebras of Slodowy slices attached to nilpotent orbits.

3. Parabolic and Drinfeld Presentations; Geometric Realizations

Shifted twisted Yangians admit presentations interpolating between the RTT (R-matrix) and Drinfeld (current) forms. For type AI and AII, one organizes the $S(u)$ matrix into block form via a parabolic decomposition $p = (p_1, \dots, p_n)$. The Gauss decomposition yields diagonal and off-diagonal blocks:

• Diagonal: $H_{a; i, j}^{(r)}$.
• Off-diagonal: $B_{a; i, j}^{(r)}$ (as $E$ or $F$-type generators).

Relations among these blocks, symmetries, and truncations in shift-parameters encode the full algebraic structure; the choice of parabolic shape is shown to be irrelevant (the algebra depends only on the shift-matrix).

Geometrically, these algebras quantize fixed-point loci in affine Grassmannian slices under involutions, denoted ${}^\imath W_\mu$ for the fixed locus for $g(z) \mapsto g(-z)^\tau$. The associated graded algebra of the shifted twisted Yangian coincides with the coordinate ring of this Poisson variety, establishing a strong link between algebraic and geometric representation-theory (Lu et al., 12 Oct 2025, Lu et al., 23 Dec 2025, Lu et al., 6 May 2025).

4. iGKLO Representations, Difference Operators, and Truncations

To each dominant coweight $\lambda \geq \mu$, there is a commutative (difference-operator) algebra $A$ constructed from highest-weight variables $w_{i, r}$ and shift operators $\eth_{i, r}$. The iGKLO homomorphism realizes current generators $H_i(u)$ and $B_i(u)$ as explicit rational functions and shifts in $A$, compatible with all algebraic relations and parity constraints (Lu et al., 23 Dec 2025, Lu et al., 12 Oct 2025, Shen et al., 14 Oct 2025, Wang, 30 Dec 2025). The image is the truncated shifted twisted Yangian ${}^\imath Y_\mu^\lambda$.

This operator-theoretic viewpoint links representation theory to explicit modules over quantized Coulomb branch algebras, and provides a powerful computational tool for studying the spectrum and modules of these noncommutative algebras (Shen et al., 14 Oct 2025, Wang, 30 Dec 2025).

5. Connections to Finite W-Algebras and Slodowy Slices

A central development is the identification, via isomorphism, of truncated shifted twisted Yangians and finite W-algebras quantizing Slodowy slices associated to nilpotent elements in types B, C, D. For type AI with truncated shift, one has

$Y^\pm_{N, \ell}(\sigma) \cong U(\mathfrak{g}, e)$

for the finite W-algebra $U(\mathfrak{g}, e)$ attached to the even nilpotent $e$ and shift data $(\sigma, \ell)$ (Lu et al., 6 May 2025, Lu et al., 12 Oct 2025, Tappeiner et al., 2024). The Slodowy slice $\mathcal{S} = x + \kappa(\mathfrak{g}^f)$ corresponds to the locus transverse to the nilpotent orbit of $e$, and the Poisson structure induced in the associated graded is matched by that of the Yangian via filtered deformation.

For type D, conjectural extensions assert a polynomial center with Pfaffian generators in two-block / even-nilpotent cases.

6. Quivers with Involutions, Coulomb Branches, and Mirror Symmetry

Shifted twisted Yangians also arise as quantizations of Coulomb branch algebras for 3d $\mathcal{N}=4$ quiver gauge theories with involution, as constructed by Braverman–Finkelberg–Nakajima and developed in (Shen et al., 14 Oct 2025, Wang, 30 Dec 2025). The diagram involution $\tau$ and associated shift parameter $\mu$ are derived from dimension vector data and quiver symmetries.

There is a canonical algebra homomorphism $\Psi$ from the shifted twisted Yangian to the quantized Coulomb branch algebra, implemented via difference operators encoding monopole and Cartan actions. This realization yields, in type AIII and other Satake types, explicit geometric deformation modules and perverse sheaf actions, and underpins a new class of 3d mirror symmetry frameworks linking quantum symmetric pairs to the twisted Yangian theory (Shen et al., 14 Oct 2025, Lu et al., 12 Oct 2025, Wang, 30 Dec 2025).

7. Impact, Applications, and Future Directions

Shifted twisted Yangians unify and generalize the algebraic structures underlying quantum symmetric pairs, twisted Yangians, and quantization of symplectic singularities (especially Slodowy slices), providing PBW bases, explicit presentations, and correspondence with Coulomb branch constructions. Their representation theory informs geometric Langlands duality, fixed-point phenomena in affine Grassmannian slices, and connections to finite W-algebras, influencing both mathematical physics and categorical representation theory (Lu et al., 6 May 2025, Lu et al., 12 Oct 2025, Lu et al., 23 Dec 2025, Shen et al., 14 Oct 2025).

Ongoing research includes $K$-theoretic generalizations (seeking homomorphisms from quantum affine algebras into $K$-theoretic Coulomb branches), construction of canonical bases and Hall algebra analogues, and full case-by-case classification for the remaining nilpotent orbits and Satake types, particularly in type D. A plausible implication is that shifted twisted Yangians will continue to serve as foundational tools in the normalization and analysis of singular affine Poisson varieties, with broad representation-theoretic and geometric applications.