Relative Rota–Baxter Groups

• Relative Rota–Baxter groups are quadruples (H, G, φ, R) where an action of G on H and a compatible map R satisfy a fundamental identity, generalizing classical Rota–Baxter structures.
• They induce skew left braces that provide set-theoretic solutions to the Yang–Baxter equation and facilitate categorical, cohomological, and extension-theoretic classifications.
• Their study leverages mixed cochain complexes and semidirect product computations to analyze extensions, compute Schur multipliers, and enumerate algorithmic classifications.

A relative Rota–Baxter group is a quadruple $(H, G, \varphi, R)$, where $H$ and $G$ are groups, $\varphi: G \to \operatorname{Aut}(H)$ is a group homomorphism describing an action of $G$ on $H$, and $R: H \to G$ is a set map, called the relative Rota–Baxter operator, satisfying the fundamental identity

$$R(h_1) R(h_2) = R\bigl( h_1\, \varphi_{R(h_1)}(h_2) \bigr)$$

for all $h_1, h_2 \in H$. This structure generalizes the concept of Rota–Baxter groups and is intimately connected to the theory of skew left braces, which underpin the set-theoretic solutions of the Yang–Baxter equation (Rathee et al., 2023, Belwal et al., 2023). The class of relative Rota–Baxter groups is now central in the study of categorical, cohomological, and extension-theoretic phenomena related to algebraic and set-theoretic Yang–Baxter structures.

1. Fundamental Structure and Examples

The defining data of a relative Rota–Baxter group include:

• An "additive" group $H$,
• An "operator" group $G$,
• An action $\varphi: G \to \operatorname{Aut}(H)$ with $$\varphi_{g_1g_2} = \varphi_{g_1} \circ \varphi_{g_2}$$,
• A map $R: H \to G$ with the Rota–Baxter compatibility.

This structure generalizes several classical cases:

• If $G = H$ and $\varphi_g(h) = g h g^{-1}$, then $R$ is an ordinary Rota–Baxter operator on $G$ (Bardakov et al., 2023, Rathee et al., 2023).
• If $\varphi$ is trivial, $R$ must be a homomorphism $H \to G$.
• If $R$ is bijective, the structure is called a bijective relative Rota–Baxter group.

The graph $$\operatorname{Gr}(R) = \{ (R(h), h) \mid h \in H \} \subset G \rtimes_\varphi H$$ is a subgroup if and only if $R$ is a relative Rota–Baxter operator. This gives an interpretation of $R$ as a splitting in the semidirect product framework (Rathee et al., 2023).

2. Correspondence with Skew Left Braces

A skew left brace is a set $H$ with two group laws $(H, +)$ and $(H, \circ)$ such that

$a \circ (b + c) = (a \circ b) - a + (a \circ c)$

for all $a, b, c \in H$, where $-a$ is the inverse in $(H, +)$.

Given a relative Rota–Baxter group $(H, G, \varphi, R)$, one defines an induced brace structure on $H$ by

$h_1 \circ_R h_2 := h_1 + \varphi_{R(h_1)}(h_2),$

making $(H, +, \circ_R)$ a skew left brace. When $R$ is bijective, there is a categorical equivalence between bijective relative Rota–Baxter groups and skew left braces [(Rathee et al., 2023), Thm. 4.6]. Conversely, every skew left brace $(H, +, \circ)$ yields a relative Rota–Baxter group via its lambda map $\lambda: (H, \circ) \to \operatorname{Aut}(H, +)$ with $R = \mathrm{id}$ [(Rathee et al., 2023), Prop. 3.8].

3. Extensions, Cohomology, and Classification

The extension theory for relative Rota–Baxter groups encompasses both abelian and central extensions. For an extension

$$1 \to (K, L, \alpha, S) \to (H, G, \varphi, R) \to (A, B, \beta, T) \to 1$$

with $K, L$ abelian and trivial $\alpha, S$, the classification is governed by the second cohomology $H^2_{RRB}(A, B; K, L)$. The cohomology is constructed via mixed group cochain complexes with four components:

• $\tau: A \times A \to K$,
• $\omega: B \times B \to L$,
• $\sigma: B \times A \to K$,
• $\chi: A \to L$, subject to cocycle relations encoding the extension and module compatibilities [(Belwal et al., 2023), §2–3].

The main result is a bijection between equivalence classes of extensions and $H^2_{RRB}$, and, for bijective cases, this cohomology coincides with the brace (second) cohomology [(Belwal et al., 2023), Thm 3.18].

Moreover, a Wells-like exact sequence organizes the relationships between derivations, extension automorphisms, compatible automorphisms of the base and the kernel, and $H^2_{RRB}$ (Belwal et al., 2024): $$0 \to \mathrm{Der}(A,K) \to \mathrm{Aut}_\mathrm{ext}(E) \to C(\nu,\mu,\sigma,f) \to H^2_{RRB}(A,K) \to 0$$ where each group is constructed as in the classical theory, now in the RRB framework.

4. Schur Multiplier, Schur Covers, and Isoclinism

The Schur multiplier of a relative Rota–Baxter group $\mathcal{A}= (A, B, \beta, T)$ is defined as $$M_{RRB}(\mathcal{A}) := H^2_{RRB}(A, \mathbb{C}^\times)$$, generalizing the group-theoretic Schur multiplier (Belwal et al., 2023). For finite $\mathcal{A}$, the exponent of $M_{RRB}(\mathcal{A})$ divides $|A|\,|B|$.

A Schur cover is a central extension with kernel isomorphic to the Schur multiplier and embedding into the commutator subgroup. Any two Schur covers of a finite bijective relative Rota–Baxter group are weakly isoclinic. This theory provides classification invariants for set-theoretic Yang–Baxter solutions arising from braces (Belwal et al., 2023).

Isoclinism of relative Rota–Baxter groups involves the equivalence of quotients by "RB-centers" and commutator (derived) subobjects, ensuring that the pairing maps agree under the isomorphisms. This concept descends to isoclinism of the corresponding induced skew left braces [(Rathee et al., 2023), Thm. 6.11].

5. Computational Methods and Algorithmic Aspects

For finite groups, an explicit algorithm for computing relative Rota–Baxter operators utilizes the structure of the semidirect product $S = G \rtimes_\varphi H$. Subgroups $A \leq S$ of order $|H|$ with suitable projections yield all possible relative RB operators, thereby enabling computational classification and enumeration via systems such as GAP [(Rathee et al., 2023), Prop. 3.9].

Each subgroup $A$ corresponding to an operator $R_A$ encodes a unique relative Rota–Baxter structure via its graph. Furthermore, equivalence under automorphisms can be imposed to count distinct associated braces.

6. Connections to Hopf Algebras, Lie Theory, and Generalizations

Relative Rota–Baxter operators extend naturally to the setting of Hopf algebras, where the group-theoretic data is replaced by compatible coalgebraic structure and module algebra actions (Bardakov et al., 2023). For Lie groups and Lie algebras, relative Rota–Baxter operators give rise to "descendent" Lie group or algebra structures, and the corresponding cohomology admits a Van Est comparison theorem, integrating algebraic and Lie-theoretic perspectives (Jiang et al., 2021).

Cohomological and extension-theoretic frameworks for relative Rota–Baxter groups interface with the cohomology and extension theories of skew left braces, ensuring that key classification and deformation phenomena coincide in both settings (Belwal et al., 2023, Belwal et al., 2023).

7. Current Directions and Open Problems

Recent advances have established the foundations of relative Rota–Baxter groups, their cohomology, extension, and isoclinism theories. Outstanding directions include the explicit classification of such structures for general finite groups, deeper understanding of weight-zero features, the development of deformation theory, further exploration of connections to non-commutative geometry and quantum groups, and refinement of computational tools for large-scale enumeration and invariants (Li et al., 2023, Gao et al., 2024).

The close relationship with skew left braces ensures that every structural breakthrough in relative Rota–Baxter theory has immediate implications for the non-degenerate, set-theoretic solutions of the Yang–Baxter equation and allied algebraic structures fundamental in quantum algebra and related fields.