---
title: Affine Super Yangian Overview
url: https://www.emergentmind.com/topics/affine-super-yangian
type: topic
---

# Affine Super Yangian Overview

Searching arXiv for recent and foundational papers on affine super Yangians and closely related constructions.
Search 1: affine super Yangian type A, Weyl groupoid, coproduct, evaluation.
Affine super Yangian denotes a Yangian-type associative superalgebra attached to superalgebraic current data, but the term is used in more than one established sense. In one line of work it refers to type-\(A\) algebras \(Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})\) or \(Y_{\hbar}(\widehat{sl}(m|n,\Pi))\), built from affine Kac–Moody Lie superalgebras and their current or current-superbialgebra structures. In another it refers to supersymmetric extensions of the affine Yangian of \(\mathfrak{gl}_1\), designed to model \({\cal N}=2\) \({\cal W}_{1+\infty}\), while a low-rank affine super-Yangian \(Y(\widehat{gl}_{1|1})\) supports explicit super-Schur-function technology [1911.06666, 2306.14598, 1711.07449, 2307.03150].

## 1. Terminology and scope

The phrase “affine super Yangian” is not monolithic in the literature. The main usages represented in current work are summarized below.

| Setting | Algebraic input | Representative papers |
|---|---|---|
| Type \(A\) affine Kac–Moody superalgebra | \(\widehat{\mathfrak{sl}(m|n)}\), or \(\widehat{sl}(m|n,\Pi)\) for arbitrary simple roots | [1911.06666], [2306.14598], [2510.04221] |
| Supersymmetric affine Yangian of \(\mathfrak{gl}_1\) | Two commuting affine Yangians of \(\mathfrak{gl}_1\) plus bi-minimal fermions | [1711.07449], [1910.10129] |
| Small-rank affine super-Yangian model | \(Y(\widehat{gl}_{1|1})\) in a semi-Fock representation | [2307.03150] |

Several nearby literatures are explicitly not about affine super Yangians in this sense. The characteristic-\(2\) paper on \(Y_{m|n}\) studies the ordinary RTT super Yangian of type \(A\), not an affine super Yangian; the Gelfand–Tsetlin paper studies \(Y(\mathfrak{gl}_{m|n})\) and \(U_q(\widehat{\mathfrak{gl}_{m|n}})\); and the finite \(W\)-superalgebra paper studies shifted super Yangians \(Y_{m|n}(\sigma)\), again not affine super Yangians [2602.13603, 2103.08758, 2001.08718].

This terminological plurality matters because the type-\(A\) affine-Kac–Moody constructions and the \(\mathfrak{gl}_1\)-based supersymmetric constructions share Yangian-type current formalisms, but they quantize different algebraic objects and organize their odd generators differently.

## 2. Type-\(A\) affine super Yangians attached to \(\widehat{\mathfrak{sl}(m|n)}\)

In the type-\(A\) Kac–Moody setting, the ambient finite-dimensional Lie superalgebra is \(\mathfrak{sl}(m|n)\) of type \(A(m-1,n-1)\), and its affinization is the affine Kac–Moody superalgebra \(A^{(1)}(m-1,n-1)\), written as \(\widehat{\mathfrak{sl}(m|n)}\) or \(sl^{(1)}(m|n,\Pi)\). A distinctive super feature is that one must allow arbitrary simple root systems \(\Pi=\{\alpha_0,\alpha_1,\dots,\alpha_{m+n-1}\}\), not only a distinguished Dynkin diagram. Odd simple roots are isotropic in type \(A\), so \((\alpha_i,\alpha_i)=0\) when \(\alpha_i\) is odd, and the affine root is \(\alpha_0=\delta-\theta\). The dependence on \(\Pi\) enters through the indexing of roots, the parity assignment, the Cartan matrix \(A=(a_{ij})\), and the additional odd Serre-type relations [2306.14598].

A standard standing assumption for the affine constructions is
\[
m,n\ge 2,\qquad m\ne n.
\]
In the arbitrary-\(\Pi\) one-parameter presentation, \(Y_{\hbar}(\widehat{sl}(m|n,\Pi))\) is the associative superalgebra generated by
\[
\{x^{\pm}_{\alpha_i,r},\ h_{\alpha_i,r},\ d \mid i\in\{0,1,\dots,m+n-1\},\ r\in \mathbb Z_{\ge0}\},
\]
with \(x^\pm_{\alpha_i,r}\) odd exactly when \(\alpha_i\) is an odd simple root, and with defining relations of Drinfeld type:
\[
[h_{\alpha_i,r},h_{\alpha_j,s}]=0,\qquad
[h_{\alpha_i,0},x^{\pm}_{\alpha_j,s}] =\pm a_{ij}\,x^{\pm}_{\alpha_j,s},
\]
\[
[x^+_{\alpha_i,r},x^-_{\alpha_j,s}] =\delta_{ij}\,h_{\alpha_i,r+s},
\]
together with mode-shift relations, ordinary Serre relations, the isotropic odd-root relations
\[
[x^{\pm}_{\alpha_i,r},x^{\pm}_{\alpha_i,s}]=0,
\]
and the extra odd relation
\[
\bigl[\,[x^{\pm}_{\alpha_i,0},x^{\pm}_{\alpha_{i-1},0}], [x^{\pm}_{\alpha_i,0},x^{\pm}_{\alpha_{i+1},s}]\,\bigr]=0
\]
for every odd root \(\alpha_i\). The derivation satisfies
\[
[d,h_{\alpha_i,r}]=0,\qquad [d,x^{\pm}_{\alpha_i,r}]=\pm \delta_{i0}\,x^{\pm}_{\alpha_i,r}.
\]
The super structure is encoded both in parity and in the super-anticommutator
\[
\{a,b\}=ab+(-1)^{p(a)p(b)}ba
\]
[2306.14598].

A two-parameter presentation for the standard affine Dynkin realization was constructed earlier as
\[
Y_{\varepsilon_1,\varepsilon_2}\bigl(\widehat{\mathfrak{sl}(m|n)}\bigr),
\]
generated by \(x_{i,r}^\pm,h_{i,r}\) with odd nodes \(0\) and \(m\). Its defining relations have the same current-mode pattern, but the shift relations carry both the symmetric combination \(\varepsilon_1+\varepsilon_2\) and an additional matrix \(b_{ij}\) multiplying \(\varepsilon_1-\varepsilon_2\):
\[
[h_{i, r+1}, x_{j, s}^{\pm}] - [h_{i, r}, x_{j, s+1}^{\pm}]
= \pm a_{i,j} \dfrac{\varepsilon_1 + \varepsilon_2}{2} \{h_{i, r}, x_{j, s}^{\pm}\}
- b_{i,j} \dfrac{\varepsilon_1 - \varepsilon_2}{2} [h_{i, r}, x_{j, s}^{\pm}],
\]
and similarly for the \(x^\pm\)-\(x^\pm\) relations, together with the isotropic-node relations at \(i=0,m\) [1911.06666].

## 3. Drinfeld realizations, minimalistic presentations, and recursive generation of modes

A central structural feature of the type-\(A\) affine super Yangian is the equivalence between an infinite current-mode presentation and a finite “minimalistic” presentation. In the arbitrary-\(\Pi\) one-parameter theory, \(Y_{\hbar}(\widehat{sl}(m|n,\Pi))\) is isomorphic to the associative superalgebra generated only by the degree \(0\) and \(1\) modes
\[
x^{\pm}_{\alpha_i,r},\ h_{\alpha_i,r}\qquad (r\in\{0,1\}),
\]
subject to relations such as
\[
[x^+_{\alpha_i,0},x^-_{\alpha_j,0}]=\delta_{ij}h_{\alpha_i,0},\qquad
[x^+_{\alpha_i,1},x^-_{\alpha_j,0}]=\delta_{ij}h_{\alpha_i,1}=[x^+_{\alpha_i,0},x^-_{\alpha_j,1}],
\]
\[
[x^{\pm}_{\alpha_i,0},x^{\pm}_{\alpha_i,0}]=0
\quad\text{for every odd root }\alpha_i,
\]
and the degree-zero odd Serre relation
\[
\bigl[\,[x^{\pm}_{\alpha_{i-1},0},x^{\pm}_{\alpha_i,0}], [x^{\pm}_{\alpha_i,0},x^{\pm}_{\alpha_{i+1},0}]\,\bigr]=0
\quad\text{for every odd root }\alpha_i.
\]
All higher modes are then reconstructed recursively [2306.14598].

In the one-parameter Hopf setting, the shifted Cartan element
\[
\tilde h_{i,1}=h_{i,1}-\frac{\hbar}{2}h_{i,0}^2
\]
rewrites the current relation as
\[
[\tilde h_{i,1},x_{j,r}^{\pm}]=\pm a_{ij}x_{j,r+1}^{\pm},
\]
and this yields recursive formulas
\[
x_{i,k+1}^{\pm} = \pm (\alpha_i,\alpha_{i+1})^{-1}[\tilde h_{i+1,1},x_{i,k}^{\pm}],\qquad
h_{i,k+1}=[x_{i,k}^{+},x_{i,0}^{-}].
\]
Theorem 6.2 in that work states that, for \(m\neq n\) and \(m,n\ge2\), the minimalistic presentation and the Drinfeld presentation \(Y^D_{\hbar}(\widehat{sl}(m|n,\Pi))\) are isomorphic as associative superalgebras [2510.04221].

In the two-parameter theory one uses instead
\[
\widetilde h_{i,1}=h_{i,1}-\frac{\varepsilon_1+\varepsilon_2}{2}h_{i,0}^2,
\]
so that
\[
[\widetilde{h}_{i,1}, x_{j,r}^{\pm}] = \pm a_{i,j}\left(x_{j,r+1}^{\pm}-b_{i,j}\dfrac{\varepsilon_1 - \varepsilon_2}{2} x_{j, r}^{\pm}\right),
\]
with separate recursive formulas for non-isotropic and isotropic nodes. This gives a finite-generator formulation used to prove the existence of coproducts and evaluation maps [1911.06666].

The proofs of equivalence are inductive and technically concentrated on odd isotropic roots. They reconstruct higher generators from level \(0,1\) data and then verify that the full mode relations follow. A separate PBW theorem is not proved in full generality in the arbitrary-\(\Pi\) paper; this suggests PBW-type behavior rather than establishing it there as a standalone theorem [2306.14598].

## 4. Weyl groupoid, odd reflections, and root-system independence

For ordinary affine Yangians of simple Lie algebras, different simple systems are related by a Weyl group. In the super case the correct symmetry object is a Weyl groupoid, because odd reflections do not preserve a fixed Cartan datum. This is the conceptual center of the arbitrary-\(\Pi\) theory.

The Weyl groupoid has objects given by simple root systems \(\Pi\) of affine type \(A\), and morphisms generated by even and odd simple reflections. For an odd simple root \(\alpha_i\), the superreflection is
\[
s_{\alpha_i}(\lambda)=
\begin{cases}
\lambda+\alpha_i,& \alpha_i+\lambda \text{ is a root},\\
-\alpha_i,& \lambda=\alpha_i,\\
\lambda,& \text{otherwise}.
\end{cases}
\]
The corresponding Yangian morphisms organize the comparison between different presentations \(Y_{\hbar}(\widehat{sl}(m|n,\Pi))\) and \(Y_{\hbar}(\widehat{sl}(m|n,\Pi'))\) [2306.14598].

At the algebraic level, even reflections act by braid-type operators on generators, while odd reflections send the reflected odd simple root to the opposite root generator and send adjacent roots to brackets with it. In the arbitrary-\(\Pi\) theory, Theorem 3.3 states that for every element \(s\) of the Weyl groupoid there exists an isomorphism
\[
T_s:\ Y(\mathfrak{sl}(E,\Pi,p))\longrightarrow Y(\mathfrak{sl}(E,s(\Pi),p)),
\]
and that \(T_s\) is an automorphism if and only if \(s\) is an even reflection [2306.14598].

The 2025 Hopf-theoretic extension strengthens this statement. There the Weyl groupoid acts by superalgebra and supercoalgebra isomorphisms
\[
T_s: Y_{\hbar}(\widehat{sl}(m|n,\Pi))\to Y_{\hbar}(\widehat{sl}(m|n,s(\Pi))),
\]
compatible with coproducts via
\[
(T_{\alpha_i}\otimes T_{\alpha_i})\Delta=\Delta' T_{\alpha_i}.
\]
A key identity for odd reflections is
\[
(T_{\alpha_i}\otimes T_{\alpha_i})(\Omega_+)
= x_{\beta_i,0}^{+}\otimes x_{\beta_i,0}^{-}
- x_{\beta_i,0}^{-}\otimes x_{\beta_i,0}^{+}
+\Omega_+,
\]
which produces the extra \(\hbar\)-corrections needed for coalgebra compatibility. The resulting global theorem states that \(Y_{\hbar}(\widehat{sl}(m|n,\Pi_1))\) and \(Y_{\hbar}(\widehat{sl}(m|n,\Pi_2))\) are isomorphic as Hopf superalgebras for any two simple root systems \(\Pi_1,\Pi_2\) [2510.04221].

This root-system dependence and root-system independence coexist: the presentation depends on \(\Pi\), but the underlying associative or Hopf superalgebra is invariant up to isomorphism.

## 5. Hopf structure, evaluation maps, and current-superalgebra quantization

One major development is the realization of the affine super Yangian as a Hopf superalgebra quantizing a current Lie superbialgebra. In the one-parameter formulation, the object being quantized is
\[
\bigl(\widehat{sl}(m|n,\Pi)[u],\varphi\bigr),
\]
with cobracket
\[
\varphi(a)(u,v)=\left[a(u)\otimes 1+1\otimes a(v),\frac{\Omega}{u-v}\right].
\]
A Hopf superalgebra \(\mathbb A\) is a quantization when
\[
\varphi(x_0)=\hbar^{-1}\bigl(\Delta(x)-\Delta^{op}(x)\bigr)\mod\hbar.
\]
The minimalistic coproduct is given on degree \(0\) and \(1\) generators by
\[
\Delta(a_{i,0})=a_{i,0}\otimes1+1\otimes a_{i,0},
\]
\[
\Delta(h_{i,1}) = h_{i,1}\otimes1+1\otimes h_{i,1} +\hbar [h_{i,0}\otimes1,\Omega_+],
\]
and then transferred to the Drinfeld presentation. In the induced Drinfeld form,
\[
\Delta(h_{i,1}) = h_{i,1}\otimes1+1\otimes h_{i,1} +\hbar\, h_{i,0}\otimes h_{i,0}
-\hbar \sum_{\alpha\in\Delta_+}\sum_{k_\alpha}(\alpha,\alpha_i)\,x_{-\alpha}^{k_\alpha}\otimes x_{\alpha}^{k_\alpha},
\]
and the coproduct is coassociative [2510.04221].

A closely related coproduct had already been constructed in the two-parameter theory, but only as a map into a degreewise completed tensor product, because the affine root sums are infinite. In that setting,
\[
\Delta(h_{i,0})=h_{i,0}\otimes1+1\otimes h_{i,0},\qquad
\Delta(x_{i,0}^{\pm})=x_{i,0}^{\pm}\otimes1+1\otimes x_{i,0}^{\pm},
\]
and
\[
\Delta(h_{i,1}) = h_{i,1}\otimes 1+1\otimes h_{i,1}
+(\varepsilon_1+\varepsilon_2)h_{i,0}\otimes h_{i,0}
-(\varepsilon_1+\varepsilon_2)\sum_{\alpha\in\Delta_+}\sum_{k_\alpha}(\alpha,\alpha_i)\,x_{-\alpha}^{k_\alpha}\otimes x_\alpha^{k_\alpha}.
\]
This yields a coassociative topological coproduct on \(Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})\) and on its extension with derivation \(d\) [1911.06666].

The other major structural map is the evaluation homomorphism to a completion of \(U(\widehat{\mathfrak{gl}(m|n)})\). In the two-parameter theory the map exists under
\[
c(\varepsilon_1+\varepsilon_2)=(n-m)\varepsilon_1,
\]
and sends the zero modes to the standard affine Chevalley generators while \(h_{i,1}\) and \(x_{i,1}^{\pm}\) are sent to explicit quadratic loop-current expressions involving infinite sums. The completion is essential precisely because these images are not finite sums in the ordinary enveloping algebra [1911.06666].

Ueda later proved that, provided \(\varepsilon_1\ne0\), the image of the evaluation map
\[
\ev_{\varepsilon_1,\varepsilon_2}\colon
Y_{\varepsilon_1,\varepsilon_2}\bigl(\widehat{\mathfrak{sl}(m|n)}\bigr)
\to U(\widehat{\mathfrak{gl}(m|n)})_{\rm comp}
\]
is dense. The proof first recovers the \(\widehat{\mathfrak{sl}(m|n)}\)-subalgebra from the image, then extracts the missing diagonal \(\mathfrak{gl}\)-currents from the images of \(h_{i,1}\). Through this homomorphism one obtains irreducible representations of the affine super Yangian by pullback from representations of the completed affine Lie superalgebra [2001.06398].

A further development connects these algebras to Schur–Weyl-type constructions. Recent work extends certain Schur–Weyl duality results from the super Yangian of \(\mathfrak{sl}_{m|n}\) to its affine version and introduces a deformed double current superalgebra of \(\mathfrak{sl}_{m|n}\), placing affine super Yangians inside a broader current-algebraic framework [2404.07328].

## 6. Supersymmetric \(\mathfrak{gl}_1\), gluing constructions, and the \(\widehat{gl}_{1|1}\) model

A second major usage of the term begins from the bosonic affine Yangian of \(\mathfrak{gl}_1\), whose representation theory is organized by plane partitions and whose algebra is identified with \(\mathcal W_{1+\infty}\). In that bosonic prototype the generating currents \(e(u),f(u),\psi(u)\) are governed by the structure function
\[
\varphi(u)=\frac{(u+h_1)(u+h_2)(u+h_3)}{(u-h_1)(u-h_2)(u-h_3)},
\]
and the plane-partition rule
\[
\psi_\Lambda(u)=\psi_0(u)\prod_{\Box\in\Lambda}\varphi(u-q-h_\Box)
\]
makes the Cartan subalgebra diagonal on combinatorial states [1512.07178].

The supersymmetric affine Yangian proposed for \({\cal N}=2\) \({\cal W}_{1+\infty}\) is built from two commuting bosonic affine Yangians of \(\mathfrak{gl}_1\),
\[
\mathcal Y=\langle e_r,f_r,\psi_r\rangle,\qquad
\hat{\mathcal Y}=\langle \hat e_r,\hat f_r,\hat\psi_r\rangle,
\]
together with fermionic towers
\[
x_s,\ \bar x_s,\ y_s,\ \bar y_s,\qquad s\in\tfrac12+\mathbb Z_{\ge0}.
\]
These additional generators transform in bi-minimal representations: \(x\) is minimal for one Yangian and conjugate-minimal for the other, \(\bar x\) behaves oppositely, and similarly for \(y,\bar y\). The minimal representation is controlled by
\[
\varphi_2(u)=\frac{u(u+h_2)}{(u-h_1)(u-h_3)},
\]
while the conjugate-minimal representation is encoded by the shifted inversion
\[
\varphi^{-1}(-u-\psi_0\sigma_3).
\]
The proposal is substantial but not fully closed: the \(e/f\)-fermion OPE sector is explicitly incomplete [1711.07449].

A closely related but non-isomorphic gluing construction replaces the mixed bimodules
\[
(\square,\overline{\square}),\qquad (\overline{\square},\square)
\]
by
\[
(\square,\square),\qquad (\overline{\square},\overline{\square}).
\]
Its representation space consists of pairs of plane partitions connected by a common leg whose cross-section is a Young diagram, and the authors emphasize that the resulting algebra is “similar (but non-isomorphic)” to the earlier supersymmetric gluing algebra. This construction clarifies that changing the gluing bimodule changes both the parameter matching and the representation geometry [1910.10129].

At the smallest genuinely super affine type, the affine super-Yangian
\[
\mathsf{Y}(\widehat{\mathfrak{gl}_{1|1}})
\]
admits an explicit semi-Fock representation. Its odd generators \(e_n^\pm,f_n^\pm\) and even Cartan currents \(\psi_k^\pm\) act on a crystal basis indexed by super-Young diagrams, and the representation can be bosonized by commuting variables \(p_k\) and anticommuting variables \(\theta_k\). The key operators are super-cut-and-join Hamiltonians \(\hat w^\pm\), whose joint eigenfunctions are the Super-Schur polynomials \(\mathcal S_\lambda\), characterized by
\[
\mathcal S_\lambda(p_1,p_2,\ldots,\theta_1,\theta_2,\ldots)|\varnothing\rangle=|\lambda\rangle.
\]
The same model carries a generalized hook measure \(m_\lambda\) and a Cauchy identity
\[
\exp\left[\sum_{k=1}^{\infty}\left(\frac{p_k q_k}{k}+\theta_k\xi_k\right)\right]
=
\sum_{\lambda}
\frac{(-\epsilon_1\epsilon_2)^{n_{\lambda}^-}}{m_{\lambda}}
\mathcal S_{\lambda}(p,\theta)\mathcal S_{\lambda}(q,\xi).
\]
This construction shows that affine super Yangians support not only current-algebraic and Hopf-theoretic structures, but also explicit symmetric-function technology in a genuinely super combinatorial setting [2307.03150].

Across these lineages, the common theme is not a single universal presentation but a recurring package of structures: Yangian-type currents, parity-sensitive Serre relations, combinatorial highest-weight models, and deformation of current superalgebras. The principal divide is between the type-\(A\) affine-Kac–Moody superalgebra approach and the \(\mathfrak{gl}_1\)-based supersymmetric \(\mathcal W\)-algebra approach; both are central to the modern meaning of affine super Yangian.

Source: https://www.emergentmind.com/topics/affine-super-yangian