Shifted iYangians: Quantizing Islices & W-Algebras

• Shifted iYangians are noncommutative algebras defined via quasi-split Satake diagrams of type ADE and even spherical coweights, unifying twisted Yangian theories.
• They employ a Drinfeld-type shift and folding mechanism that yields structured PBW bases and enables truncations isomorphic to finite W-algebras.
• Their geometric realization connects representation theory and Poisson geometry through quantizations of affine Grassmannian islices and Slodowy slices.

Shifted iYangians (“shifted twisted Yangians”) constitute a family of noncommutative algebras associated to quasi-split Satake diagrams of Dynkin type ADE, together with an even spherical coweight $\mu$. These algebras generalize Olshanski’s twisted Yangians, incorporate Drinfeld-type shift and folding mechanisms, and admit well-behaved PBW monomial bases. When further truncated by a dominant coweight $\lambda$, their quotients (“truncated shifted twisted Yangians”) are isomorphic to finite $W$-algebras quantizing Slodowy slices for nilpotent elements in Lie algebras of types $\mathsf{B,C,D}$, linking representation theory, Poisson geometry, and geometric Langlands quantization. They arise fundamentally as quantizations of fixed point loci (“islices”) in the affine Grassmannian under the group involution determined by $\tau$.

1. Algebraic Definition and Relations

Let $C=(c_{ij})_{i,j\in I}$ be the Cartan matrix of a simply-laced Dynkin diagram of type ADE, and $\tau: I \to I$ a diagram involution (the Satake diagram). The algebra ${}^\imath Y_\mu$ is defined for coweights $\mu \in P^\vee$ that are even ($\langle\mu,\alpha_i\rangle \in 2\mathbb{Z}$) and spherical ($\mu = \nu + \tau\nu$ for some $\nu$). Its generators are: $$H_i^{(r)},\quad r \in \mathbb{Z},~r \geq -\langle\mu,\alpha_i\rangle,\qquad B_i^{(s)},\quad s > 0,\quad i \in I$$ with normalization $H_i^{(r)}=0$ for $r<-\langle\mu,\alpha_i\rangle$ and $H_i^{(-\langle\mu,\alpha_i\rangle)}=1$.

The defining relations include:

• Cartan commutativity:

$[H_i^{(r_1)}, H_j^{(r_2)}] = 0$

• Mixed Cartan-root Drinfeld type relations:

$$[H_i^{(r+2)}, B_j^{(s)}] - [H_i^{(r)}, B_j^{(s+2)}] = \tfrac{c_{ij} - c_{\tau i, j}}{2}[H_i^{(r+1)}, B_j^{(s)}]_+ + \tfrac{c_{ij} + c_{\tau i, j}}{2}[H_i^{(r)}, B_j^{(s+1)}]_+ + \tfrac{c_{ij} c_{\tau i, j}}{4}[H_i^{(r)}, B_j^{(s)}]$$

with $[x,y]_+ = xy + yx$.

• Root-root folded current relations:

$$[B_i^{(s_1+1)}, B_j^{(s_2)}] - [B_i^{(s_1)}, B_j^{(s_2+1)}] = c_{ij}[B_i^{(s_1)}, B_j^{(s_2)}]_+ + 2\delta_{\tau i, j}(-1)^{s_1} H_j^{(s_1 + s_2)}$$

• Folded Serre-type relations, e.g., for $c_{ij} = -1$ and $i \neq \tau i \neq j$:

$$\mathrm{Sym}_{s_1, s_2} [B_i^{(s_1)}, [B_i^{(s_2)}, B_j^{(s)}]] = 0$$

with further cases for folded nodes and modified relations when $c_{i,\tau i} = -1$; see (Lu et al., 23 Dec 2025).

The shift parameter $\mu$ only appears in the degree-bounds of the Cartan generator modes. The shift homomorphisms

$\iota^\tau_{\mu,\nu}: {}^\imath Y_\mu \to {}^\imath Y_{\mu+\nu+\tau\nu}$

preserve the defining relations and the PBW structure.

2. PBW Basis and Structural Properties

Fixing any ordering of root-vectors $B_\beta^{(r)}$, with $\beta$ a positive root, the PBW theorem asserts that ordered monomials in

$$\left\{ B_\beta^{(r)}~|~\beta \in \Delta^+,~r>0 \right\} \cup \left\{ H_i^{(2p)}~|~i \in I_0,~2p>-\langle\mu,\alpha_i\rangle\right\} \cup \left\{ H_i^{(p)}~|~i \in I_1,~p>-\langle\mu,\alpha_i\rangle\right\}$$

span the algebra and are linearly independent (Lu et al., 23 Dec 2025). For anti-dominant $\nu$, shift maps provide embeddings compatible with the PBW monomial enumeration.

The Cartan-doubled algebra ${}^\imath Y_\infty$ is filtered, and ${}^\imath Y_\mu$ is its quotient by the boundary conditions on Cartan degrees. The Gelfand–Tsetlin subalgebra, generated by Cartan coefficients, is maximal commutative in the truncated algebra (Lu et al., 23 Dec 2025).

3. Truncation and iGKLO Representations

Given a dominant $\tau$-invariant coweight $\lambda \geq \mu$, construction of representations utilizes the iGKLO (involutive GKLO) map,

$\Phi_\mu^\lambda: {}^\imath Y_\mu[\bm z] \longrightarrow A$

as an explicit homomorphism to a difference operator algebra $A$ in variables $w_{i, r}, z_{i, s}$. The image algebra,

${}^\imath Y_\mu^\lambda := \operatorname{Im} \Phi_\mu^\lambda$

is the truncated shifted twisted Yangian (TSTY). At the classical Poisson limit (setting $z_{i,s}=0$), this provides a quantization of the corresponding affine Grassmannian islice.

In type AI ($\tau = \mathrm{id}$, $g = \mathfrak{sl}_n$), a variant presentation $Y^+_{n, \ell}(\sigma)$ yields a further isomorphism

$Y^+_{n, \ell}(\sigma) \cong {}^\imath Y_{n, \mu}^{N\varpi_1^\vee}$

which matches the finite $W$-algebra of type BCD (Lu et al., 23 Dec 2025, Lu et al., 6 May 2025).

4. Geometric Realization: Affine Grassmannian Islices and Quantization

For $\mu$ even spherical, the loop group $G(\!(z^{-1})\!)$ supports the involution

$\sigma(g(z)) = \omega_\tau(g(-z))^{-1}$

with $\omega_\tau(e_i) = -f_{\tau i}$, $\omega_\tau(h_i) = -h_{\tau i}$. The islice

${}^\imath W_\mu := (W_\mu)^\sigma$

is the fixed-point locus for the slice $W_\mu$ within the affine Grassmannian, and inherits a Dirac-reduced Poisson bracket from $G(\!(z^{-1})\!)$ (Lu et al., 12 Oct 2025). The algebra ${}^\imath Y_\mu$ (filtered by loop-rotation) is isomorphic, at the associated graded level, to the coordinate ring $C[{}^\imath W_\mu]$.

For dominant $\lambda \geq \mu$ with $\tau\lambda = \lambda$, the generalized slice

$$\overline{W}_\mu^\lambda = W_\mu \cap \overline{G[z]z^\lambda G[z]}$$

is preserved by involution, and its islice

${}^\imath \overline{W}_\mu^\lambda = (\overline{W}_\mu^\lambda)^\sigma$

decomposes into leaves indexed by dominant $\nu$ with $\tau\nu = \nu$. Birational charts and difference operator realization make the quantization explicit (Lu et al., 12 Oct 2025).

In type AI, identification with Slodowy slices $$\mathcal{S}_{\pi_2} \cap \mathcal{N}_{\mathfrak{sl}_N}$$ is available via the Mirković–Vybornov theorem (Lu et al., 12 Oct 2025).

5. Identification with Finite $W$-Algebras and Applications

Truncated shifted twisted Yangians ${}^\imath Y_\mu^\lambda$ of type AI are isomorphic to finite $W$-algebras of types B, C, D associated to even nilpotent elements or two-row Jordan blocks in type D. This matches the universal equivariant quantization of the Slodowy slice for the nilpotent $e \in \mathfrak{g}$ (Lu et al., 6 May 2025, Tappeiner et al., 8 Jun 2024): ${}^\imath Y_\mu^\lambda \cong U(\mathfrak{g}, e)$ where $U(\mathfrak{g}, e)$ is the finite $W$-algebra constructed by quantum Hamiltonian reduction. This reflects the underlying Poisson geometry and conic symplectic singularity structure of the Slodowy slice.

The identification extends earlier work on untwisted shifted Yangians (Kamnitzer et al., 2012, Kamnitzer et al., 2015). In particular, functoriality and shift homomorphisms relate islices and their quantizations across different coweights and Satake folds. The center of the truncated algebra is polynomial, and the Gelfand–Tsetlin subalgebra is maximal commutative (Lu et al., 23 Dec 2025).

6. Future Directions and Connections

Shifted iYangians provide a framework for the quantization of fixed-point loci in the affine Grassmannian under diagram involutions, leading to new families of non-symplectic symplectic singularities. There is a conjectural connection to “iCoulomb branches” constructed from Satake-framed double quivers, expected to recover normalized top-dimensional components of islices (Lu et al., 12 Oct 2025). This suggests applications to quiver gauge theories and generalizations of the Brundan–Kleshchev paradigm.

An active area includes the extension to hybrid, ortho-symplectic, and non-split cases, the study of categorical and monomial crystal representations, and the geometric realization of higher-dimensional defects and quantizations (Lu et al., 23 Dec 2025, Lu et al., 12 Oct 2025).

7. Summary Table: Structural Features

Feature	Type AI Case	General Quasi-Split ADE Case
Generators	$h_{i, r}$, $b_{ij, s}$ (Tappeiner et al., 8 Jun 2024)	$H_i^{(r)}$, $B_i^{(s)}$ (Lu et al., 23 Dec 2025)
Shift Mechanism	Via symmetric shift matrix $S$ (Tappeiner et al., 8 Jun 2024)	Boundary conditions in degrees (Lu et al., 23 Dec 2025)
PBW basis	Ordered monomials in $h$, $b$ (Tappeiner et al., 8 Jun 2024)	Ordered monomials in $H$, $B$ (Lu et al., 23 Dec 2025)
Truncation / Finite W-algebra	Identification with $U(\mathfrak{g}, e)$ (Lu et al., 6 May 2025)	Isomorphism with finite W-algebra (Lu et al., 23 Dec 2025)
Geometric realization	Slodowy slice quantization (Tappeiner et al., 8 Jun 2024)	Islice quantization (Lu et al., 12 Oct 2025)

Shifted iYangians thus unify representation-theoretic, algebraic, and geometric approaches to the quantization of slices and symplectic singularities under involutive symmetry, with broad relevance for geometric representation theory, mathematical physics, and quantization theory.