Relative Rota-Baxter Operators

• Relative Rota-Baxter operators are linear maps from a representation to a Leibniz algebra that intertwine algebraic brackets to create new algebraic structures.
• They employ a Maurer–Cartan framework and graded Lie algebra cohomology, identifying 1-cocycles and obstruction classes to govern deformation behaviors.
• This framework has practical applications in integrable systems, renormalization, and perturbation theory, extending to homotopical and higher-categorical algebra.

Relative Rota-Baxter operators are a generalization of classical Rota-Baxter operators, systematically extending their action from associative or Lie algebras to algebraic objects with representations, such as Leibniz algebras, and introducing a sophisticated cohomological deformation theory. This framework organizes the deformation behavior of operators and characterizes trivial and nontrivial deformations in terms of explicit cohomology classes. The development is rooted in the Maurer–Cartan approach to graded Lie algebras and opens the door to further generalization, including homotopical and higher-categorical settings.

1. Definition and Algebraic Framework

A relative Rota-Baxter operator (abbreviated as RRBO) is formally defined for a Leibniz algebra $\mathfrak{g}$ and a representation $(V; p_\ell, p_r)$. A linear map $T : V \to \mathfrak{g}$ is a relative Rota-Baxter operator if

$$[Tv_1, Tv_2]_\mathfrak{g} = T\left( p_\ell(Tv_1) v_2 + p_r(Tv_2) v_1 \right), \quad \forall v_1, v_2 \in V.$$

This condition ensures that $T$ intertwines the bracket of $\mathfrak{g}$ with a new Leibniz algebra structure on $V$, given by

$[u, v]_T := p_\ell(Tu) v + p_r(Tv) u.$

The operator $T$ can be viewed as a Maurer–Cartan element in the graded Lie algebra $(C^*(V, \mathfrak{g}), \{\cdot, \cdot\})$, where $$C^n(V, \mathfrak{g}) = \operatorname{Hom}(V^{\otimes n}, \mathfrak{g})$$.

2. Cohomology Theory and the Maurer–Cartan Formalism

The cohomology theory associated to an RRBO $T$ is built by introducing a differential

$d_T f := \{T, f\},$

where $\{\cdot, \cdot\}$ is the graded Lie bracket on the cochain complex. The graded Jacobi identity ensures that $d_T^2 = 0$, so one can define cohomology groups

$$H^n(V, \mathfrak{g}) = Z^n(V, \mathfrak{g}) / B^n(V, \mathfrak{g}),$$

where $Z^n$ denotes $n$-cocycles (kernel of $d_T$) and $B^n$ denotes $n$-coboundaries (image of $d_T$).

This allows the description of the deformation theory in terms of elements of the cohomology: infinitesimal deformations are cohomology classes in $H^1$, and obstructions to extending finite-order deformations live in higher cohomology (e.g., $H^2$ for the first obstruction).

3. Linear and Formal Deformations

Consider a one-parameter family $T_t := T + tE$ for $t$ formal and $E \in \operatorname{Hom}(V, \mathfrak{g})$. Substituting into the RRBO condition, the first-order requirement is

$$[Tu, Ev]_\mathfrak{g} + [Eu, Tv]_\mathfrak{g} = T\bigl(p_\ell(Eu) v + p_r(Ev) u\bigr) + E\bigl(p_\ell(Tu) v + p_r(Tv) u\bigr),$$

i.e., $E$ is a 1-cocycle with respect to $d_T$. Two deformations $T_1, T_2$ are equivalent if $E_1$ and $E_2$ differ by a coboundary, $E_1 = E_2 + d_T x$ for $x \in \mathfrak{g}$, under an associated equivalence transformation.

For formal (power series) deformations,

$T_t = T + t T_1 + t^2 T_2 + \cdots,$

the coefficient $T_1$ must be a 1-cocycle, while higher order compatibility is recursively encoded via cohomological obstruction classes.

4. Nijenhuis Elements and Trivial Deformations

A Nijenhuis element $x \in \mathfrak{g}$ for $T$ satisfies several compatibility conditions:

• $[x, y]_\mathfrak{g}$ and $T$ commute appropriately,
• $$p_\ell([x, y]_\mathfrak{g}) = p_r([x, y]_\mathfrak{g}) = 0$$,
• $[x, p_\ell(x)u - T(u)]_\mathfrak{g} = 0$ for all $u \in V$.

Given $x$ with these properties, the 1-coboundary $E = d_T(x)$ yields a linear deformation $T_t = T + t d_T(x)$ which is trivial (isomorphic to $T$). Nijenhuis elements thus generate flows in the moduli space of RRBOs that correspond to equivalence classes of deformations.

5. Extendibility and Cohomological Obstructions

For deformations of order $n$,

$T_t = T + t T_1 + \dots + t^n T_n,$

the extendibility to order $n+1$ is characterized by the vanishing of a specific 2-cocycle, the obstruction class

$$\text{Ob}_T(u, v) = \sum_{i+j = n+1} T_i( p_\ell(T_j(u)) v + p_r(T_j(v)) u ),$$

with $[\text{Ob}_T] = 0$ in $H^2(V, \mathfrak{g})$ signaling that the obstruction can be resolved. The existence of a solution to $d_T(E_{n+1}) = -\text{Ob}_T$ allows the deformation to be continued to higher order.

6. Applications and Broader Context

The cohomological deformation theory of RRBOs provides a unified language for controlling algebraic perturbations in Leibniz, Lie, and associative settings. RRBOs play a structural role in the paper of integrable systems, renormalization, and moduli problems, generalizing the context of Rota–Baxter operators from classical algebras to settings involving module actions and non-antisymmetric brackets.

Results in this area connect the formalism to:

• rigidity and perturbation theory for algebraic structures,
• classification of trivial and nontrivial deformations via explicit cohomological conditions,
• deeper understanding of moduli spaces of algebraic operators,
• and potential extensions to homotopy algebras and operadic algebra via the Maurer–Cartan machinery (Tang et al., 2020).

A plausible implication is that these frameworks will generalize further to categories where higher symmetries and higher algebraic structures are present, leveraging the flexibility of the cohomological approach developed for relative Rota–Baxter operators.