Shifted Twisted Yangians

• Shifted twisted Yangians are associative algebras that generalize classical Yangians by incorporating diagram involutions and shift parameters into their defining relations.
• They feature a robust structure with PBW bases, Drinfeld–type presentations, and truncations that relate to finite W-algebras and Slodowy slices in classical Lie theory.
• Their quantum difference operator realizations connect Poisson geometry with 3d gauge theory quantizations, highlighting applications in Coulomb branch studies.

A shifted twisted Yangian is a family of associative algebras generalizing the classical Yangian, incorporating both a diagram involution (yielding a "twisted" structure) and a shift parameter in their defining relations. These algebras arise in the representation theory of Lie algebras and quantum groups, the geometry of nilpotent orbits and Slodowy slices in types B, C, D, as well as the quantization of fixed-point loci in affine Grassmannians and Coulomb branches of 3d $\mathcal{N}=4$ quiver gauge theories with involution.

1. Algebraic Structure and Drinfeld–Type Presentation

Given a simple simply-laced Lie algebra $g$ (ADE type), let $\tau$ be a diagram involution ($\tau^2=1$), and let $\mu$ be an even spherical coweight: $\mu = \nu + \tau\nu$, with $\langle\mu, \alpha_i\rangle \in 2\mathbb{Z}$ for all simple roots. The shifted twisted Yangian $^\imath Y_\mu$ is the unital associative algebra generated by modes $H_i^{(r)}$ (Cartan generators) and $B_i^{(s)}$ (raising operators), where $g$0 for $g$1, $g$2, $g$3, $g$4, $g$5 ranging over the nodes of the Dynkin diagram. The key relations are:

• Commutativity of the Cartan subalgebra: $g$6.
• Drinfeld–type Cartan-raising/lowering and raising-raising relations incorporating both the Cartan matrix $g$7 and the involution $g$8:

$g$9

Further relations for $\tau$0 and a hierarchy of Serre-type relations for various patterns of $\tau$1 with respect to $\tau$2.

• Symmetry and vanishing conditions stem from the shift by $\tau$3 and the structure of $\tau$4.

For split types ($\tau$5), these relations reduce to those of shifted Yangians in type A, while quasi-split types admit additional quasi-split Serre relations (Lu et al., 23 Dec 2025, Lu et al., 12 Oct 2025, Shen et al., 14 Oct 2025).

2. PBW Theorems and Structural Properties

The existence of a Poincaré–Birkhoff–Witt (PBW) basis is a fundamental property of shifted twisted Yangians. Root-vector constructions, paralleling the standard triangular decomposition, are used for PBW monomials: $\tau$6 for each positive root $\tau$7. Ordered monomials in the $\tau$8 and appropriate Cartan generators span $\tau$9, and these monomials form a $\tau^2=1$0-basis (Lu et al., 23 Dec 2025, Lu et al., 6 May 2025). The shift homomorphisms $$^\imath Y_\mu \to {}; ^\imath Y_{\mu+\nu+\tau \nu}$$\tau^2=1$1\nu$) are injective, further reinforcing this structure.

In types AI and AII, parabolic presentations interpolate between the Drinfeld and R-matrix forms, yielding finer control over block decompositions and giving rise to baby comultiplications and block-Gauss factorizations (Lu et al., 6 May 2025).

3. Truncations, Central Structure, and Relation to Finite $\tau^2=1$2-Algebras

Given a symmetric shift matrix $\tau^2=1$3 (in types AI/AII), truncations of shifted twisted Yangians are constructed by quotienting out by higher modes or imposing level conditions related to the shape of associated Young or pyramid diagrams. These truncated shifted twisted Yangians $\tau^2=1$4 admit explicit PBW bases and their centers are generated by even-degree coefficients of the Sklyanin determinant, with possible "Pfaffian generators" in even cases (Lu et al., 6 May 2025, Tappeiner et al., 2024).

Crucially, in a wide range of cases (all even nilpotent orbits in types B, C, and certain cases in D), these truncated algebras are isomorphic as filtered algebras to finite $\tau^2=1$5-algebras of the corresponding type, quantizing Slodowy slices for nilpotent elements. This identification is established via explicit presentation matching, dimension counts, and relates natural algebraic filtrations (canonical, Kazhdan) on both sides (Lu et al., 6 May 2025, Tappeiner et al., 2024, Lu et al., 23 Dec 2025).

The following table summarizes the correspondence:

Algebra	Truncation Parameter	Geometric/Algebraic Object Quantized
$\tau^2=1$6 (type AI)	$\tau^2=1$7, pyramid, shift matrix	$\tau^2=1$8, Slodowy slice
$\tau^2=1$9 (type AII)	$\mu$0, even admissible	$\mu$1, Slodowy slice
$\mu$2 (quasi-split ADE)	coweight $\mu$3	Quantizes affine Grassmannian islice

4. Difference-Operator/Q-Torus Realizations (iGKLO Type)

Shifted twisted Yangians admit faithful representations via quantum difference operators (iGKLO representations). This construction introduces quantum torus variables $\mu$4, shifted by operators $\mu$5, with

$\mu$6

and the remaining generators acting as explicit rational functions or translation/difference operators. For types AI/AII, these formulas recover the Drinfeld–Gelfand–Zeitlin (GZ) operators, and for quasi-split types, the iGKLO map gives surjective homomorphisms to explicitly constructed difference algebras, whose images yield the truncated shifted twisted Yangians (Lu et al., 23 Dec 2025, Lu et al., 12 Oct 2025).

Matching of Cartan and raising operator generating series under these representations establishes both the faithfulness of the realization and compatibility with geometric constructions, such as Poisson reductions and quiver gauge-theory Coulomb branches.

5. Poisson Geometry, Dirac Reduction, and Fixed Loci

The semiclassical (degeneration $\mu$7) limits of shifted twisted Yangians are Poisson algebras. For type AI, these coincide with Dirac reductions of the shifted Yangian for $\mu$8 by the Cartan involution, and their coordinate rings quantize the fixed-point loci in loop symmetric spaces: $\mu$9 where $\mu = \nu + \tau\nu$0 is the Cartan involution automorphism. This Poisson structure matches the reduction of the Yangian current algebra to the twisted setting. Truncations of these semiclassical algebras yield Poisson presentations of the Slodowy slices for even nilpotent orbits in types B, C, D (Tappeiner et al., 2024).

More generally, for quasi-split ADE types, Poisson Dirac reductions correspond to fixed-point subvarieties in affine Grassmannian slices, and the quantized shifted twisted Yangians serve as their quantum coordinate rings (Lu et al., 12 Oct 2025).

6. Coulomb Branches, Quivers with Involution, and 3d Gauge Theory

Shifted twisted Yangians appear as quantizations of Coulomb branches for 3d $\mu = \nu + \tau\nu$1 quiver gauge theories with involutions on the quivers. The construction details a correspondence between:

• The involution-fixed part of the gauge theory's data $\mu = \nu + \tau\nu$2 and the algebraic structure of the shifted twisted Yangian (node-by-node Cartan data, involution-induced relations).
• Difference-operator algebras generated by minuscule monopole operators and Cartan classes realize the quantized Coulomb branch, with the algebra homomorphism from the shifted twisted Yangian mapping generators to minuscule monopoles and Cartan Gelfand–Tsetlin subalgebra (Shen et al., 14 Oct 2025, Wang, 30 Dec 2025).

In particular, this establishes new instances of 3d mirror symmetry, relating the representation categories of shifted twisted Yangians to categories of sheaves or modules over perverse sheaves on fixed-point quiver varieties, and folded quantum group structures (Shen et al., 14 Oct 2025).

7. Applications, Examples, and Generalizations

In type AI, shifted twisted Yangians classify all even nilpotent orbits in classical types, providing explicit presentations for both truncated quantum and Poisson algebras attached to Slodowy slices. In higher rank and more general types (quasi-split ADE, Satake diagrams), this framework extends to cover quantization of generalized affine Grassmannian islices, their fixed loci, and the symplectic singularities in Coulomb branches of quiver gauge theories with symmetry.

Parabolic and Drinfeld-type presentations allow flexibility in applications, including baby comultiplications, knowledge of the center via the Sklyanin determinant, and connection to ortho-symplectic Coulomb branches and hybrid constructions in split/nonsplit Satake settings (Lu et al., 12 Oct 2025, Lu et al., 6 May 2025).

Representative examples include explicit calculations for rank 1 (recovering the classical $\mu = \nu + \tau\nu$3 twisted Yangian) and type AIII/AII quivers, illustrating the underlying algebraic and geometric correspondences (Lu et al., 23 Dec 2025, Shen et al., 14 Oct 2025).

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References

• "Shifted twisted Yangians and finite W–algebras of classical type" (Lu et al., 6 May 2025)
• "Shifted twisted Yangians and Slodowy slices in classical Lie algebras" (Tappeiner et al., 2024)
• "Shifted twisted Yangians of quasi-split ADE types" (Lu et al., 23 Dec 2025)
• "Shifted twisted Yangians and affine Grassmannian islices" (Lu et al., 12 Oct 2025)
• "Quivers with Involutions and Shifted Twisted Yangians via Coulomb Branches" (Shen et al., 14 Oct 2025)
• "Quivers with Involutions and Shifted Twisted Yangians via Coulomb Branches II" (Wang, 30 Dec 2025)
• "A Drinfeld presentation for the twisted Yangian $\mu = \nu + \tau\nu$4" (Brown, 2016)