Super Yangians in characteristic $2$

Abstract: We define the super Yangian $Y_{m|n}$ over a field $\mathbbm{k}$ of characteristic $2$, and show that the super Yangian $Y_{m|n}$ is a deformation of the super universal enveloping algebra of the current Lie algebra $\mathfrak{gl}<em>{m+n}[t]$. By employing the methods of the work of \cite{BT18}, we also give a description of the center of $Y</em>{m|n}$.

• The paper constructs the super Yangian Y_{m|n} in characteristic 2 as a quotient of the modular Yangian, using quadratic-map Lie superalgebras to preserve odd-generator structure lost by standard RTT relations.
• The paper proves a PBW theorem and establishes gr Y_{m|n} ≅ U_super(gl_{m+n}[t]), showing that the construction is a filtered deformation of the current Lie superalgebra’s enveloping algebra.
• The paper explicitly determines the center as generated by Harish–Chandra and p-center elements, proves freeness over these centers, and identifies the restricted super Yangian with the ordinary restricted Yangian.

Overview

This paper by Chang and Hu constructs the super Yangian $Y_{m|n}$ over an algebraically closed field $k$ of characteristic $p=2$ and establishes its basic structural properties. The central difficulty is well known: the standard definition of a Lie superalgebra via skew-supersymmetry and the super Jacobi identity degenerates in characteristic 2, where it collapses to that of an ordinary $\mathbb{Z}_2$-graded Lie algebra. Consequently, Nazarov's RTT presentation of $Y_{m|n}$ loses all super content when $p=2$: the sign factors in the defining relations become trivial and $Y_{m|n}$ reduces to the Yangian $Y_{m+n}$. The paper resolves this by adopting the modern framework of Lie superalgebras in characteristic 2 — a $\mathbb{Z}_2$-graded Lie algebra equipped with a quadratic map $Q: L_{\bar 1}\to L_{\bar 0}$ satisfying $k$0 and $k$1 (2602.13603) — and by defining the super Yangian as a quotient of the modular Yangian rather than through modified RTT relations.

The main results are: (i) a PBW-type theorem for $k$2; (ii) an isomorphism $k$3, where $k$4 is the current Lie algebra with its natural characteristic-2 superalgebra structure and $k$5 is the super universal enveloping algebra; and (iii) an explicit description of the center $k$6, generalizing Brundan–Topley's description of $k$7 in positive characteristic.

Lie superalgebras in characteristic 2 and the current algebra

Throughout, $k$8 carries the parity function $k$9 on basis elements $p=2$0, splitting indices into blocks of sizes $p=2$1 and $p=2$2. As a restricted Lie algebra, $p=2$3 has $p=2$4-map given by matrix squaring, $p=2$5. A key observation is that for $p=2$6,

$p=2$7

so the restriction of the $p=2$8-map to $p=2$9 satisfies precisely the axioms of the quadratic map $\mathbb{Z}_2$0. Thus $\mathbb{Z}_2$1 inherits a canonical Lie superalgebra structure in the sense of Bouarroudj–Lebedev–Leites–Shchepochkina, and the super universal enveloping algebra is

$\mathbb{Z}_2$2

The associated graded object satisfies the expected PBW form $\mathbb{Z}_2$3, consistent with the theory in characteristic $\mathbb{Z}_2$4 (cf. Etingof–Hu). The authors also isolate the subalgebra $\mathbb{Z}_2$5 generated by $\mathbb{Z}_2$6 for odd $\mathbb{Z}_2$7, which is a free polynomial algebra on the squares $\mathbb{Z}_2$8 with $\mathbb{Z}_2$9.

The center of the super enveloping algebra

The first structural theorem describes $Y_{m|n}$0, the center of $Y_{m|n}$1, which coincides with the invariant subalgebra $Y_{m|n}$2 under the adjoint action. Writing $Y_{m|n}$3 for the obvious central family, the result states:

• $Y_{m|n}$4-center: $Y_{m|n}$5 is freely generated by $Y_{m|n}$6 over even pairs $Y_{m|n}$7;
• Full center: $Y_{m|n}$8 is freely generated by $Y_{m|n}$9 together with those same even-index generators with $p=2$0.

The proof follows the filtration argument of Brundan–Topley: one first shows that the invariant algebra $p=2$1 is freely generated by $p=2$2 together with the squares of even basis elements, then lifts these generators through the loop filtration and uses the inclusion chain $p=2$3 to force equality throughout. This is a direct super analogue of [BT18, Theorem 3.4], adapted to the mixed symmetric/exterior graded algebra $p=2$4.

Definition and PBW theorem for the super Yangian

Rather than modifying the RTT relations, the authors exploit the Drinfeld presentation of the modular Yangian $p=2$5 (established over any field via Gauss decomposition). Under the natural $p=2$6-gradation of $p=2$7 induced by the RTT relations, the Drinfeld generators $p=2$8 are always even, while $p=2$9 and $Y_{m|n}$0 carry parity $Y_{m|n}$1. Letting $Y_{m|n}$2 denote the free polynomial subalgebra of the $Y_{m|n}$3-center generated by the squares of odd root elements, the super Yangian is defined as the quotient

$Y_{m|n}$4

where $Y_{m|n}$5 is the maximal ideal generated by those squares. In effect, this quotient kills $Y_{m|n}$6 and $Y_{m|n}$7 across the block boundary, restoring the exterior behavior of odd generators; the authors verify directly that the additional relations $Y_{m|n}$8 of Gow's characteristic-zero presentation then hold automatically.

Two consequences follow. First, a PBW theorem: $Y_{m|n}$9 is free as a module over $Y_{m+n}$0 with basis given by ordered supermonomials in the Drinfeld generators, so their images form a basis of $Y_{m+n}$1. Second, the loop filtration descends and the isomorphism $Y_{m+n}$2 induces

$Y_{m+n}$3

so $Y_{m+n}$4 is a filtered deformation of the super universal enveloping algebra of the current superalgebra — the precise analogue, in characteristic 2, of the classical statement valid for $Y_{m+n}$5.

The center of the super Yangian

The final main theorem generalizes Brundan–Topley's description of $Y_{m+n}$6 and the authors' earlier characteristic-$Y_{m+n}$7 work. With the Harish-Chandra center $Y_{m+n}$8 generated by the coefficients $Y_{m+n}$9 of the quantum determinant series $\mathbb{Z}_2$0, and the $\mathbb{Z}_2$1-center $\mathbb{Z}_2$2 generated by $\mathbb{Z}_2$3-coefficients together with squares of even root elements, the theorem asserts:

1. $\mathbb{Z}_2$4 is generated by $\mathbb{Z}_2$5 and $\mathbb{Z}_2$6;
2. $\mathbb{Z}_2$7 is a free polynomial algebra on its listed generators;
3. $\mathbb{Z}_2$8 is freely generated by $\mathbb{Z}_2$9 together with the squares of even root elements;
4. consequently, $Q: L_{\bar 1}\to L_{\bar 0}$0 equals the image of $Q: L_{\bar 1}\to L_{\bar 0}$1 under the quotient map.

The proof again proceeds by identifying leading terms under the loop filtration — e.g., $Q: L_{\bar 1}\to L_{\bar 0}$2 and $Q: L_{\bar 1}\to L_{\bar 0}$3 — and matching them against the free generators of $Q: L_{\bar 1}\to L_{\bar 0}$4 established earlier. Two corollaries record freeness of $Q: L_{\bar 1}\to L_{\bar 0}$5 as a module over its full center and over its $Q: L_{\bar 1}\to L_{\bar 0}$6-center, with bases given by ordered monomials in the remaining Drinfeld generators with exponents at most 1.

The paper also observes a simplification peculiar to $Q: L_{\bar 1}\to L_{\bar 0}$7: the restricted super Yangian $Q: L_{\bar 1}\to L_{\bar 0}$8, defined by further quotienting by the maximal ideal of $Q: L_{\bar 1}\to L_{\bar 0}$9, is isomorphic to the ordinary restricted Yangian $k$00, mirroring the isomorphism $k$01 of restricted enveloping (super)algebras. This reflects the fact that the quadratic map is the restriction of the $k$02-map, so no genuinely new restricted object appears.

Limitations and open questions

The paper is explicitly a "short note" and leaves several natural directions unaddressed. It does not develop a Drinfeld-type presentation intrinsic to $k$03 (the construction passes through the quotient of $k$04), nor does it treat finite $k$05-superalgebras or shifted super Yangians in characteristic 2, both of which feature in the Brundan–Topley and Goodwin–Topley programs it builds upon. Representation-theoretic consequences — highest weight theory, finite-dimensional irreducibles, or evaluation homomorphisms for $k$06 in characteristic 2 — are not pursued. The freeness results depend on the algebraic closedness of $k$07 and on the identification of the quadratic map with the restriction of the $k$08-map, which is special to the current algebra $k$09; extension to other current superalgebras in characteristic 2 remains open.

Conclusion

The paper supplies the missing characteristic-2 case in the structural theory of type $k$10 super Yangians. By combining the quadratic-map formalism for Lie superalgebras in characteristic 2 with the modular Yangian machinery of Brundan–Topley, it produces a super Yangian $k$11 that is a genuine filtered deformation of $k$12, admits a PBW basis in Drinfeld generators, and has a center described by explicit free generators comprising the Harish-Chandra center and a suitably defined $k$13-center. The methods are filtrations and associated gradings throughout, and the results complete, for $k$14, the picture previously available in characteristic zero and odd primes.