Affine Super Yangian Overview

• Affine super Yangian is a Yangian-type associative superalgebra defined using both affine Kac–Moody superalgebras and supersymmetric extensions of gl1, capturing intricate current and odd generator structures.
• It features dual presentations via infinite Drinfeld current modes and a finite minimalistic formulation, with recursive relations ensuring full recovery of higher modes.
• Advanced constructions utilize Hopf superalgebra frameworks and Weyl groupoids, enabling coherent coproducts, evaluation maps, and odd reflection symmetries in representation theory.

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 (Ueda, 2019, Stukopin et al., 2023, Gaberdiel et al., 2017, Galakhov et al., 2023).

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	(Ueda, 2019, Stukopin et al., 2023, Volkov et al., 5 Oct 2025)
Supersymmetric affine Yangian of $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$0	Two commuting affine Yangians of $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$1 plus bi-minimal fermions	(Gaberdiel et al., 2017, Li, 2019)
Small-rank affine super-Yangian model	$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$2 in a semi-Fock representation	(Galakhov et al., 2023)

Several nearby literatures are explicitly not about affine super Yangians in this sense. The characteristic-$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$3 paper on $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$4 studies the ordinary RTT super Yangian of type $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$5, not an affine super Yangian; the Gelfand–Tsetlin paper studies $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$6 and $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$7; and the finite $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$8-superalgebra paper studies shifted super Yangians $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$9, again not affine super Yangians (Chang et al., 14 Feb 2026, Lu, 2021, Peng, 2020).

This terminological plurality matters because the type-$Y_{\hbar}(\widehat{sl}(m|n,\Pi))$0 affine-Kac–Moody constructions and the $Y_{\hbar}(\widehat{sl}(m|n,\Pi))$1-based supersymmetric constructions share Yangian-type current formalisms, but they quantize different algebraic objects and organize their odd generators differently.

2. Type-$Y_{\hbar}(\widehat{sl}(m|n,\Pi))$2 affine super Yangians attached to $Y_{\hbar}(\widehat{sl}(m|n,\Pi))$3

In the type-$Y_{\hbar}(\widehat{sl}(m|n,\Pi))$4 Kac–Moody setting, the ambient finite-dimensional Lie superalgebra is $Y_{\hbar}(\widehat{sl}(m|n,\Pi))$5 of type $Y_{\hbar}(\widehat{sl}(m|n,\Pi))$6, and its affinization is the affine Kac–Moody superalgebra $Y_{\hbar}(\widehat{sl}(m|n,\Pi))$7, written as $Y_{\hbar}(\widehat{sl}(m|n,\Pi))$8 or $Y_{\hbar}(\widehat{sl}(m|n,\Pi))$9. A distinctive super feature is that one must allow arbitrary simple root systems $\mathfrak{gl}_1$0, not only a distinguished Dynkin diagram. Odd simple roots are isotropic in type $\mathfrak{gl}_1$1, so $\mathfrak{gl}_1$2 when $\mathfrak{gl}_1$3 is odd, and the affine root is $\mathfrak{gl}_1$4. The dependence on $\mathfrak{gl}_1$5 enters through the indexing of roots, the parity assignment, the Cartan matrix $\mathfrak{gl}_1$6, and the additional odd Serre-type relations (Stukopin et al., 2023).

A standard standing assumption for the affine constructions is

$\mathfrak{gl}_1$7

In the arbitrary-$\mathfrak{gl}_1$8 one-parameter presentation, $\mathfrak{gl}_1$9 is the associative superalgebra generated by

${\cal N}=2$0

with ${\cal N}=2$1 odd exactly when ${\cal N}=2$2 is an odd simple root, and with defining relations of Drinfeld type: ${\cal N}=2$3

${\cal N}=2$4

together with mode-shift relations, ordinary Serre relations, the isotropic odd-root relations

${\cal N}=2$5

and the extra odd relation

${\cal N}=2$6

for every odd root ${\cal N}=2$7. The derivation satisfies

${\cal N}=2$8

The super structure is encoded both in parity and in the super-anticommutator

${\cal N}=2$9

(Stukopin et al., 2023).

A two-parameter presentation for the standard affine Dynkin realization was constructed earlier as

${\cal W}_{1+\infty}$0

generated by ${\cal W}_{1+\infty}$1 with odd nodes ${\cal W}_{1+\infty}$2 and ${\cal W}_{1+\infty}$3. Its defining relations have the same current-mode pattern, but the shift relations carry both the symmetric combination ${\cal W}_{1+\infty}$4 and an additional matrix ${\cal W}_{1+\infty}$5 multiplying ${\cal W}_{1+\infty}$6: ${\cal W}_{1+\infty}$7 and similarly for the ${\cal W}_{1+\infty}$8-${\cal W}_{1+\infty}$9 relations, together with the isotropic-node relations at $Y(\widehat{gl}_{1|1})$0 (Ueda, 2019).

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

A central structural feature of the type-$Y(\widehat{gl}_{1|1})$1 affine super Yangian is the equivalence between an infinite current-mode presentation and a finite “minimalistic” presentation. In the arbitrary-$Y(\widehat{gl}_{1|1})$2 one-parameter theory, $Y(\widehat{gl}_{1|1})$3 is isomorphic to the associative superalgebra generated only by the degree $Y(\widehat{gl}_{1|1})$4 and $Y(\widehat{gl}_{1|1})$5 modes

$Y(\widehat{gl}_{1|1})$6

subject to relations such as

$Y(\widehat{gl}_{1|1})$7

$Y(\widehat{gl}_{1|1})$8

and the degree-zero odd Serre relation

$Y(\widehat{gl}_{1|1})$9

All higher modes are then reconstructed recursively (Stukopin et al., 2023).

In the one-parameter Hopf setting, the shifted Cartan element

$A$0

rewrites the current relation as

$A$1

and this yields recursive formulas

$A$2

Theorem 6.2 in that work states that, for $A$3 and $A$4, the minimalistic presentation and the Drinfeld presentation $A$5 are isomorphic as associative superalgebras (Volkov et al., 5 Oct 2025).

In the two-parameter theory one uses instead

$A$6

so that

$A$7

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 (Ueda, 2019).

The proofs of equivalence are inductive and technically concentrated on odd isotropic roots. They reconstruct higher generators from level $A$8 data and then verify that the full mode relations follow. A separate PBW theorem is not proved in full generality in the arbitrary-$A$9 paper; this suggests PBW-type behavior rather than establishing it there as a standalone theorem (Stukopin et al., 2023).

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-$\widehat{\mathfrak{sl}(m|n)}$0 theory.

The Weyl groupoid has objects given by simple root systems $\widehat{\mathfrak{sl}(m|n)}$1 of affine type $\widehat{\mathfrak{sl}(m|n)}$2, and morphisms generated by even and odd simple reflections. For an odd simple root $\widehat{\mathfrak{sl}(m|n)}$3, the superreflection is

$\widehat{\mathfrak{sl}(m|n)}$4

The corresponding Yangian morphisms organize the comparison between different presentations $\widehat{\mathfrak{sl}(m|n)}$5 and $\widehat{\mathfrak{sl}(m|n)}$6 (Stukopin et al., 2023).

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-$\widehat{\mathfrak{sl}(m|n)}$7 theory, Theorem 3.3 states that for every element $\widehat{\mathfrak{sl}(m|n)}$8 of the Weyl groupoid there exists an isomorphism

$\widehat{\mathfrak{sl}(m|n)}$9

and that $\widehat{sl}(m|n,\Pi)$0 is an automorphism if and only if $\widehat{sl}(m|n,\Pi)$1 is an even reflection (Stukopin et al., 2023).

The 2025 Hopf-theoretic extension strengthens this statement. There the Weyl groupoid acts by superalgebra and supercoalgebra isomorphisms

$\widehat{sl}(m|n,\Pi)$2

compatible with coproducts via

$\widehat{sl}(m|n,\Pi)$3

A key identity for odd reflections is

$\widehat{sl}(m|n,\Pi)$4

which produces the extra $\widehat{sl}(m|n,\Pi)$5-corrections needed for coalgebra compatibility. The resulting global theorem states that $\widehat{sl}(m|n,\Pi)$6 and $\widehat{sl}(m|n,\Pi)$7 are isomorphic as Hopf superalgebras for any two simple root systems $\widehat{sl}(m|n,\Pi)$8 (Volkov et al., 5 Oct 2025).

This root-system dependence and root-system independence coexist: the presentation depends on $\widehat{sl}(m|n,\Pi)$9, 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

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$00

with cobracket

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$01

A Hopf superalgebra $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$02 is a quantization when

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$03

The minimalistic coproduct is given on degree $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$04 and $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$05 generators by

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$06

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$07

and then transferred to the Drinfeld presentation. In the induced Drinfeld form,

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$08

and the coproduct is coassociative (Volkov et al., 5 Oct 2025).

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,

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$09

and

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$10

This yields a coassociative topological coproduct on $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$11 and on its extension with derivation $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$12 (Ueda, 2019).

The other major structural map is the evaluation homomorphism to a completion of $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$13. In the two-parameter theory the map exists under

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$14

and sends the zero modes to the standard affine Chevalley generators while $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$15 and $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$16 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 (Ueda, 2019).

Ueda later proved that, provided $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$17, the image of the evaluation map

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$18

is dense. The proof first recovers the $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$19-subalgebra from the image, then extracts the missing diagonal $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$20-currents from the images of $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$21. Through this homomorphism one obtains irreducible representations of the affine super Yangian by pullback from representations of the completed affine Lie superalgebra (Ueda, 2020).

A further development connects these algebras to Schur–Weyl-type constructions. Recent work extends certain Schur–Weyl duality results from the super Yangian of $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$22 to its affine version and introduces a deformed double current superalgebra of $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$23, placing affine super Yangians inside a broader current-algebraic framework (Guay et al., 2024).

6. Supersymmetric $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$24, gluing constructions, and the $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$25 model

A second major usage of the term begins from the bosonic affine Yangian of $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$26, whose representation theory is organized by plane partitions and whose algebra is identified with $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$27. In that bosonic prototype the generating currents $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$28 are governed by the structure function

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$29

and the plane-partition rule

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$30

makes the Cartan subalgebra diagonal on combinatorial states (Procházka, 2015).

The supersymmetric affine Yangian proposed for $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$31 $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$32 is built from two commuting bosonic affine Yangians of $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$33,

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$34

together with fermionic towers

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$35

These additional generators transform in bi-minimal representations: $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$36 is minimal for one Yangian and conjugate-minimal for the other, $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$37 behaves oppositely, and similarly for $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$38. The minimal representation is controlled by

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$39

while the conjugate-minimal representation is encoded by the shifted inversion

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$40

The proposal is substantial but not fully closed: the $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$41-fermion OPE sector is explicitly incomplete (Gaberdiel et al., 2017).

A closely related but non-isomorphic gluing construction replaces the mixed bimodules

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$42

by

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$43

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 (Li, 2019).

At the smallest genuinely super affine type, the affine super-Yangian

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$44

admits an explicit semi-Fock representation. Its odd generators $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$45 and even Cartan currents $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$46 act on a crystal basis indexed by super-Young diagrams, and the representation can be bosonized by commuting variables $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$47 and anticommuting variables $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$48. The key operators are super-cut-and-join Hamiltonians $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$49, whose joint eigenfunctions are the Super-Schur polynomials $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$50, characterized by

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$51

The same model carries a generalized hook measure $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$52 and a Cauchy identity

$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$53

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 (Galakhov et al., 2023).

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-$$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$54 affine-Kac–Moody superalgebra approach and the $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$55-based supersymmetric $$Y_{\varepsilon_1,\varepsilon_2}(\widehat{\mathfrak{sl}(m|n)})$$56-algebra approach; both are central to the modern meaning of affine super Yangian.