Deformed Drinfeld Coproduct

• Deformed Drinfeld coproduct is a controlled alteration of the standard coproduct via twists or dressing factors, preserving key algebraic structures.
• It is applied across diverse settings including Hopf algebras, monoidal Hom-bialgebras, and quantum affine algebras to manage coassociativity and tensor codomains.
• These constructions employ invertible elements, modified Cartan series, and categorical adjustments to transfer deformation while maintaining algebraic integrity.

A deformed Drinfeld coproduct is a coproduct obtained by modifying an existing coalgebraic structure in a manner modeled on Drinfeld’s twist formalism, but the precise construction depends strongly on the ambient category. In ordinary Hopf or bialgebra settings it is the familiar twist-conjugated coproduct $\Delta^F=F\Delta F^{-1}$; in monoidal Hom-bialgebras it takes the form $\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$; in Hopf algebroid and deformed phase-space settings it is defined in a modified tensor-product codomain; in current realizations it may appear as a factorized coproduct on modified Drinfeld–Cartan series rather than on the original generators; and in geometric Hall-algebra constructions it becomes a meromorphic or vertex coproduct that recovers Drinfeld’s Yangian coproduct in ADE type (Borowiec et al., 2016, Zhang et al., 2014, Milot, 10 Mar 2026, Jindal et al., 23 Mar 2026).

1. Formal profiles of the construction

The common core is a deformation of the coproduct while retaining a recognizable algebraic backbone. In the ordinary twist setting one starts from an invertible $F\in H\otimes H$ satisfying the 2-cocycle condition and normalization, and defines

$\Delta^F(h)=F\Delta(h)F^{-1}.$

In the monoidal Hom-bialgebra setting one instead uses an invertible $\sigma\in H\otimes H$ subject to $\alpha$-invariance, normalization, and a Hom-adapted 2-cocycle identity, and defines

$$\Delta^\sigma(x)=x_{[1]}\otimes x_{[2]}=(\sigma\Delta(x))\varrho,\qquad \varrho=\sigma^{-1}.$$

In Yangian and quantum affine current realizations, by contrast, the Hopf coproduct may remain the standard one while the generators are replaced by modified Drinfeld–Cartan series such as $S_i(z)$ or $T_i(z)$, whose coproducts factor through explicit dressing terms $\Theta_i(z)$. In CoHA constructions the coproduct is a cofield

$\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$0

rather than a map into an ordinary tensor square (Borowiec et al., 2016, Zhang et al., 2014, Milot, 10 Mar 2026, Jindal et al., 23 Mar 2026).

Two structural distinctions recur throughout the literature. First, some constructions genuinely deform the coproduct on a fixed algebra, whereas others keep the coproduct and change the generators to which it is applied. Second, the codomain can cease to be the ordinary tensor product $\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$1: Hopf algebroids require balanced or quotient tensor products, coideal algebras require one-sided tensor targets, and vertex-coalgebra constructions require Laurent-series completions in a spectral parameter $\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$2 (Jurić et al., 2014, Borowiec et al., 2016, Przezdziecki et al., 2024, Jindal et al., 23 Mar 2026).

2. Twist deformation in bialgebras and monoidal Hom-bialgebras

In the ordinary bialgebra or Hopf algebra setting, a normalized cocycle twist $\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$3 satisfies

$\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$4

and deforms the coproduct by

$\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$5

The algebra multiplication in $\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$6 does not change; the paper on twisted bialgebroids explicitly emphasizes that “twist deformation modifies coalgebraic sector only” (Borowiec et al., 2016).

The Hom-bialgebra variant makes this mechanism more rigid. A monoidal Hom-bialgebra $\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$7 is deformed by a Drinfeld twist $\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$8 satisfying

$\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$9

together with the Hom-2-cocycle identity

$F\in H\otimes H$0

The deformed coproduct is

$F\in H\otimes H$1

and only the coproduct is changed: $F\in H\otimes H$2 The multiplication, unit, counit, and Hom-structure map $F\in H\otimes H$3 remain unchanged. The central theorem is that $F\in H\otimes H$4 is again a Hom-bialgebra; if $F\in H\otimes H$5 is quasitriangular, the $F\in H\otimes H$6-matrix is transported by

$F\in H\otimes H$7

if $F\in H\otimes H$8 is Hom-Hopf, there is also a deformed antipode $F\in H\otimes H$9. The representation categories $\Delta^F(h)=F\Delta(h)F^{-1}.$0 and $\Delta^F(h)=F\Delta(h)F^{-1}.$1 are isomorphic as monoidal categories, and braided isomorphic in the quasitriangular case (Zhang et al., 2014).

A notable convention issue arises here: when $\Delta^F(h)=F\Delta(h)F^{-1}.$2, the Hom-bialgebra definition used in this work is inverse to Drinfeld’s usual convention. This matters when comparing formulas across ordinary Hopf-algebra sources and Hom-type sources, because the paper’s $\Delta^F(h)=F\Delta(h)F^{-1}.$3 corresponds to the inverse of the standard twist parameterization (Zhang et al., 2014).

3. Hopf algebroids, realizations, and deformed phase-space coproducts

In $\Delta^F(h)=F\Delta(h)F^{-1}.$4-deformed phase space and related noncommutative spacetimes, the full Weyl algebra cannot be made into an ordinary Hopf algebra because one cannot define $\Delta^F(h)=F\Delta(h)F^{-1}.$5 satisfactorily in the usual $\Delta^F(h)=F\Delta(h)F^{-1}.$6 codomain while preserving the algebra structure. The remedy is a Hopf algebroid: the coproduct lands in a quotient algebra such as

$\Delta^F(h)=F\Delta(h)F^{-1}.$7

where $\Delta^F(h)=F\Delta(h)F^{-1}.$8 is a two-sided ideal inside a suitable subalgebra $\Delta^F(h)=F\Delta(h)F^{-1}.$9. In this setting the Drinfeld-type deformation still has the formal shape

$\sigma\in H\otimes H$0

but the formula must be understood in the modified tensor-product codomain appropriate to the algebroid structure (Jurić et al., 2014).

The realization approach makes the same point from the phase-space side. A noncommutative coordinate system is embedded into an undeformed Heisenberg algebra by a realization

$\sigma\in H\otimes H$1

or, in the linear case,

$\sigma\in H\otimes H$2

From the realization one computes the deformed plane-wave composition law $\sigma\in H\otimes H$3, and then defines the coproduct of momenta by

$\sigma\in H\otimes H$4

Equivalently,

$\sigma\in H\otimes H$5

For linear realizations one obtains the explicit twist

$\sigma\in H\otimes H$6

and a compact momentum coproduct

$\sigma\in H\otimes H$7

The same framework yields the star product and deformed addition law of momenta; coassociativity of the coproduct is equivalent to associativity of the star product (Juric et al., 2015, Meljanac et al., 2021).

This literature also draws a sharp distinction between Lie-type and non-Lie-type deformations. For Lie-type spaces, especially $\sigma\in H\otimes H$8-Minkowski, the twist may satisfy the usual cocycle condition and define a genuine Drinfeld twist. For Snyder space, however, the paper on symmetric ordering and Weyl realizations states that there exists symmetric ordering but no Weyl realization; the star product is nonassociative, the coproduct is noncoassociative, and the corresponding twist does not satisfy the cocycle condition. The same source therefore treats the relevant twists in the Hopf algebroid sense rather than the ordinary Hopf-algebra sense (Meljanac et al., 2022).

A further categorical refinement is that twisting a bialgebroid directly or constructing a bialgebroid from a twisted bialgebra lead to the same result for a normalized cocycle twist. In the smash-product setting the twisted coproduct is

$\sigma\in H\otimes H$9

and the comparison theorem identifies

$\alpha$0

as bialgebroids (Borowiec et al., 2016).

4. Current presentations, modified series, and mode-by-mode deformations

In Drinfeld current realizations of Yangians and quantum affine algebras, the deformation may be shifted from the coproduct itself to the Cartan generators on which the coproduct is evaluated. For $\alpha$1, the modified Drinfeld–Cartan series $\alpha$2 satisfy

$\alpha$3

and the crucial explicit formula is

$\alpha$4

Hence

$\alpha$5

For $\alpha$6, the modified Cartan $\alpha$7-series satisfy

$\alpha$8

with $\alpha$9 and $$\Delta^\sigma(x)=x_{[1]}\otimes x_{[2]}=(\sigma\Delta(x))\varrho,\qquad \varrho=\sigma^{-1}.$$0 expressed as products of commuting $$\Delta^\sigma(x)=x_{[1]}\otimes x_{[2]}=(\sigma\Delta(x))\varrho,\qquad \varrho=\sigma^{-1}.$$1-exponentials in root-current modes. The paper is explicit that this does not define a new Hopf structure: the coproduct is still the standard coproduct, while the generators are modified so that the factorization becomes tractable (Milot, 10 Mar 2026).

A distinct pattern appears in the super Yangian of $$\Delta^\sigma(x)=x_{[1]}\otimes x_{[2]}=(\sigma\Delta(x))\varrho,\qquad \varrho=\sigma^{-1}.$$2. Here the coproduct is given on the minimal generators by primitive formulas at level $$\Delta^\sigma(x)=x_{[1]}\otimes x_{[2]}=(\sigma\Delta(x))\varrho,\qquad \varrho=\sigma^{-1}.$$3,

$$\Delta^\sigma(x)=x_{[1]}\otimes x_{[2]}=(\sigma\Delta(x))\varrho,\qquad \varrho=\sigma^{-1}.$$4

but is deformed at level $$\Delta^\sigma(x)=x_{[1]}\otimes x_{[2]}=(\sigma\Delta(x))\varrho,\qquad \varrho=\sigma^{-1}.$$5 by the half Casimir: $$\Delta^\sigma(x)=x_{[1]}\otimes x_{[2]}=(\sigma\Delta(x))\varrho,\qquad \varrho=\sigma^{-1}.$$6

$$\Delta^\sigma(x)=x_{[1]}\otimes x_{[2]}=(\sigma\Delta(x))\varrho,\qquad \varrho=\sigma^{-1}.$$7

Higher Drinfeld generators are then obtained recursively. The paper explicitly presents this as the closest analogue, within its framework, of a deformed Drinfeld coproduct (Lin et al., 30 Apr 2025).

The super-Virasoro case provides an explicit Jordanian example at the level of infinite-mode generators. Starting from a twist $$\Delta^\sigma(x)=x_{[1]}\otimes x_{[2]}=(\sigma\Delta(x))\varrho,\qquad \varrho=\sigma^{-1}.$$8 built from $$\Delta^\sigma(x)=x_{[1]}\otimes x_{[2]}=(\sigma\Delta(x))\varrho,\qquad \varrho=\sigma^{-1}.$$9 and $S_i(z)$0 with $S_i(z)$1, the coproduct is deformed by

$S_i(z)$2

yielding explicit series formulas for $S_i(z)$3 and $S_i(z)$4 involving powers of $S_i(z)$5, shifted modes, and coefficients $S_i(z)$6, $S_i(z)$7. The algebra structure remains that of $S_i(z)$8; the coalgebra is deformed into a noncocommutative Hopf superalgebra (Yang, 2010).

A broader generalization is the slope-dependent family $S_i(z)$9 on general quantum loop algebras. These “new new” coproducts are topological coproducts on Borel-like half algebras indexed by $T_i(z)$0. In the affine case $T_i(z)$1, the paper proves that $T_i(z)$2 coincides with the standard Drinfeld–Jimbo coproduct under the Drinfeld–Beck isomorphism. The deformation here is therefore not a one-parameter twist but a slope-dependent generalization of the old coproduct constructed inside the current/shuffle formalism (Neguţ, 1 Feb 2026).

5. Boundary and coideal analogues

Quantum symmetric pairs replace Hopf subalgebras by coideal subalgebras, so the analogue of a deformed Drinfeld coproduct is no longer an internal map $T_i(z)$3. In the split affine case of types $T_i(z)$4, the boundary Drinfeld–Cartan currents $T_i(z)$5 satisfy the factorization

$T_i(z)$6

and the coproduct-like statement

$T_i(z)$7

The paper presents this as approximate group-likeness rather than as a full Hopf coproduct, and uses it to establish compatibility of a boundary $T_i(z)$8-character map with the ordinary $T_i(z)$9-character map (Przezdziecki et al., 2024).

In the quasi-split affine type $\Theta_i(z)$0, the relevant object is the renormalized current

$\Theta_i(z)$1

Its factorization takes the form

$\Theta_i(z)$2

and the corresponding coideal coproduct satisfies

$\Theta_i(z)$3

The underlying coideal coproduct on generators is

$\Theta_i(z)$4

This work explicitly states that it does not construct a Hernandez-style deformed Drinfeld coproduct on the full current algebra; what it provides is a boundary or coideal analogue for the Cartan sector (Li et al., 5 Jan 2026).

These coideal results clarify a frequent source of ambiguity. In this setting, “deformed Drinfeld coproduct” does not mean a new Hopf coproduct on the whole boundary current algebra. It means a Cartan-level, approximately group-like coproduct law in which exact equalities are replaced by congruences modulo the positive Drinfeld half, and in which the ambient Drinfeld currents $\Theta_i(z)$5 appear only after embedding the coideal algebra into the ambient quantum affine algebra (Przezdziecki et al., 2024, Li et al., 5 Jan 2026).

6. Drinfeld doubles, vertex coproducts, and geometric reformulations

One line of generalization deforms the Drinfeld double datum itself. The deformed half algebra $\Theta_i(z)$6 modifies Lusztig’s Serre relations by a bicharacter $\Theta_i(z)$7, and is twist-equivalent to Lusztig’s $\Theta_i(z)$8 after changing the multiplication by a bicharacter $\Theta_i(z)$9. The full double $\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$00 is then constructed from positive and negative halves with toral actions depending on $\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$01 and $\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$02, and has explicit coproduct

$\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$03

This pattern specializes to two-parameter, multiparameter, and super quantum groups after suitable quotienting or identification of the extra group-like elements $\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$04 (Fan et al., 2019).

A second line is geometric. For the critical CoHA of a quiver with potential, the Joyce–Liu coproduct is a vertex coproduct

$\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$05

It satisfies vertex coassociativity rather than ordinary coassociativity, forms a vertex bialgebra together with the CoHA product, and after a vertex-theoretic analogue of Majid–Radford bosonisation one obtains an extended CoHA containing a Cartan part. In ADE type, the resulting extended Joyce–Liu vertex coproduct is identified with Drinfeld’s meromorphic coproduct on the Yangian: $\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$06

$\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$07

The paper therefore interprets Drinfeld’s deformed Yangian coproduct as the natural geometric coproduct arising from direct sum, $\Delta^\sigma(x)=(\sigma\Delta(x))\sigma^{-1}$08-translation, and the Ext complex in critical CoHA theory (Jindal et al., 23 Mar 2026).

Taken together, these constructions show that the term “deformed Drinfeld coproduct” is not tied to a single universal formula. It can denote twist-conjugation of an ordinary coproduct, a Hom-corrected coproduct, a phase-space coproduct extracted from realizations and star products, a factorized current coproduct on modified Cartan series, a coideal approximation to group-likeness, a slope-dependent topological coproduct, or a vertex coproduct recovering Yangian structure. What remains common is that the coproduct is altered in a controlled Drinfeld-type manner and that the deformation is encoded either by a twist, a dressing factor, a Casimir correction, an Ext kernel, or a categorical change of tensor product (Zhang et al., 2014, Borowiec et al., 2016, Milot, 10 Mar 2026, Jindal et al., 23 Mar 2026).