Reynolds–Nijenhuis Associative Algebras

• Reynolds–Nijenhuis associative algebras consist of an associative algebra endowed with a linear operator satisfying both Reynolds and Nijenhuis identities, defining a new associative product.
• They serve as a bridge between classical Rota-type operators and modified Rota–Baxter operators, offering robust frameworks for representation, cohomology, and deformation theory.
• Applications include deformation analysis, algebraic renormalization, and operator splitting in physics, providing concrete tools for classifying and modeling hybrid algebraic structures.

A Reynolds–Nijenhuis associative algebra is an associative algebra equipped with a linear endomorphism that simultaneously satisfies the operator identities defining both Reynolds and Nijenhuis operators. Such structures arise naturally as hybrids of classical Rota-type operators used in deformation theory, operator splitting, and algebraic structures in mathematical physics. They also admit a robust cohomology and deformation theory generalizing classical approaches in associative and Rota–Baxter algebras (Mosbahi et al., 28 Dec 2025).

1. Definitions and Foundational Properties

Let $(A,\cdot)$ be an associative algebra over a field $\mathbb{k}$ of characteristic zero.

A linear map $P:A\to A$ is a Reynolds–Nijenhuis operator if, for all $a,b\in A$, it satisfies both:

• Nijenhuis identity

$$P(a)\cdot P(b) = P\bigl( P(a)\cdot b + a\cdot P(b) - P(a\cdot b) \bigr),$$

• Reynolds identity

$$P(a)\cdot P(b) = P\bigl( a\cdot P(b) + P(a)\cdot b - P(a)\cdot P(b) \bigr).$$

Equivalently, $P$ is a Reynolds–Nijenhuis operator if it is both a Reynolds operator and a Nijenhuis operator. These conditions are distinguished by their “correction terms”; the Nijenhuis identity subtracts $P(a\cdot b)$ while the Reynolds identity subtracts $P(a)\cdot P(b)$. The operator space described by these identities forms the basis for the structure of Reynolds–Nijenhuis associative algebras (Mosbahi et al., 28 Dec 2025).

If $(A,P)$ is a Reynolds–Nijenhuis associative algebra, it admits a new associative product

$$a \star b := a \cdot P(b) + P(a) \cdot b - P(a \cdot b),$$

for which $P$ is a homomorphism $(A, \cdot) \to (A,\star)$ and a Reynolds–Nijenhuis operator for $\star$ as well (Mosbahi et al., 28 Dec 2025).

2. Connections to Rota–Baxter and Related Operators

Reynolds–Nijenhuis operators interpolate between several classes of Rota-type operators central to the structure theory of associative algebras:

• A Rota–Baxter operator of weight $\lambda$ is a linear map $R:A\to A$ such that

$$R(a)\cdot R(b) = R \big( R(a)\cdot b + a\cdot R(b) + \lambda a\cdot b \big).$$

• A modified Rota–Baxter operator of weight $\lambda$ satisfies

$$P(a\cdot b) = P(a)\cdot b + a\cdot P(b) + \lambda a\cdot b.$$

The relationship to Rota–Baxter and modified Rota–Baxter operators specializes as follows (Mosbahi et al., 28 Dec 2025):

Condition on $P$	Type	Weight
$P^2 = 0$	Rota–Baxter	0
$P^2 = P$	Rota–Baxter	$-1$
$P^2 = \pm\mathrm{Id}$	Modified R.-Baxter	$\mp1$

Substituting $P^2$ into the defining identities reduces these hybrid Reynolds–Nijenhuis conditions to classical Rota–Baxter (or modified Rota–Baxter) identities, providing a conceptual bridge between these operator classes (Mosbahi et al., 28 Dec 2025).

3. Representation Theory

Given an associative algebra $(A,\cdot)$ and an $A$-bimodule $(V,l,r)$ with structure maps $l:A\to \mathrm{End}(V)$, $r:A\to \mathrm{End}(V)$, a Reynolds–Nijenhuis representation for $(A,P)$ is a linear operator $\xi:V\to V$ satisfying (Mosbahi et al., 28 Dec 2025):

• $\xi \, l(a) = l(P(a))\,\xi$,
• $\xi\, r(a) = r(P(a))\,\xi$,
• $l(P(a))\,l(b) = l(a)\,l(P(b))$,
• $r(P(a))\,r(b) = r(b)\,r(P(a))$,

for all $a, b \in A$. This ensures compatibility of the operator $\xi$ with both the algebra action and the Reynolds–Nijenhuis structure.

One may “twist” a Reynolds–Nijenhuis representation by defining new left and right actions

$$l'(a)(v) = l(a)(\xi v) - \xi l(a)(v) + l(P(a))(v), \quad r'(v)(a) = r(\xi v)(a) - \xi r(v)(a) + r(v)(P(a)),$$

ensuring $(V, l', r', \xi)$ is again a Reynolds–Nijenhuis representation (Mosbahi et al., 28 Dec 2025).

4. Cohomology Theory

The deformation theory of Reynolds–Nijenhuis associative algebras is governed by a dedicated cohomology complex. For any $(A,P)$, define

$$C^n_{\mathrm{RNA}}(A,V) = C^n(A,V) \oplus C^{n-1}_{\mathrm{RNO}}(A,V)$$

where $C^n(A,V)$ is the space of Hochschild $n$-cochains, and $C^{n-1}_{\mathrm{RNO}}$ denotes the subspace of operator-compatible cochains. The coboundary is

$$d^n(f,g) = \big(\delta(f), -\partial(g) - \psi^n(f)\big)$$

where $\delta$ is the Hochschild differential, $\partial$ is the differential in the operator-cochain complex, and $\psi^n$ is the “correction map”

$$\psi^n(f)(a_1,\dots,a_n) = f(P(a_1),...,P(a_n)) - \sum_{i=1}^n \xi f(P(a_1),...,a_i,...,P(a_n)) + \xi^2 f(a_1,...,a_n).$$

These yield the cohomology groups $$H^n_{\mathrm{RNA}}(A,V) = \ker d^n / \operatorname{Im} d^{n-1}$$ (Mosbahi et al., 28 Dec 2025).

Low-degree groups have specific interpretations:

• $H^0_{\mathrm{RNA}}(A,V)$ describes central elements fixed by $\xi$,
• $H^1_{\mathrm{RNA}}(A,V)$ classifies compatible derivations,
• $H^2_{\mathrm{RNA}}(A,V)$ classifies infinitesimal deformations of $(A,P)$.

5. Formal Deformation Theory

A one-parameter formal deformation of a Reynolds–Nijenhuis algebra is given by power series

$$\nu_t = \sum_{i\ge 0} \nu_i t^i, \quad P_t = \sum_{i\ge 0} P_i t^i,$$

where $\nu_0(a,b)=a\cdot b$, $P_0=P$, and all other $\nu_i$, $P_i$ are $\mathbb{k}$-linear. The identities to order $t^n$ enforce:

• Associativity: $$\sum_{i+j=n} \nu_i(\nu_j(a,b),c) = \sum_{i+j=n} \nu_i(a,\nu_j(b,c))$$.
• Nijenhuis: $$\sum_{i+j+k=n} \nu_i(P_j(a),P_k(b)) = \sum_{i+j+k=n} P_i(\nu_j(P_k(a), b)) + P_i(\nu_j(a, P_k(b))) - P_i(P_j(\nu_k(a,b)))$$.
• Reynolds: $$\sum_{i+j+k=n} \nu_i(P_j(a),P_k(b)) = \sum_{i+j+k=n} P_i(\nu_j(a, P_k(b))) + P_i(\nu_j(P_k(a), b)) - \sum_{i+j+k+\ell=n} P_i(\nu_j(P_k(a), P_\ell(b)))$$.

The infinitesimal $(\nu_1, P_1)$ of any deformation is a 2-cocycle in the Reynolds–Nijenhuis cohomology, $d^2(\nu_1,P_1) = 0$. Two deformations are equivalent if there is a formal automorphism $\phi(t)=\mathrm{Id}+\sum_{i\ge 1}\phi_i t^i$ with $\phi_i\in \mathrm{End}(A)$ such that

$$\phi(t)\circ\nu_t' = \nu_t \circ(\phi(t)\otimes \phi(t)), \quad \phi(t) \circ P_t' = P_t\circ \phi(t)$$

and the infinitesimal deformations differ by a coboundary:

$(\nu_1',P_1')-(\nu_1,P_1)=d^1(\phi_1).$

If $H^2_{\mathrm{RNA}}(A,A)=0$ then all deformations are trivial (rigidity) (Mosbahi et al., 28 Dec 2025).

6. Classification and Examples

For the standard null-filiform algebra $A_n$ (with basis $\{e_1,\ldots, e_n\}$ and $e_i\cdot e_j = e_{i+j}$ if $i+j\le n$, $0$ otherwise), homogeneous Reynolds and Nijenhuis operators are classified by their degree and scalar actions (Karimjanov et al., 2018):

• Reynolds operators of degree zero: Are either supported on direct sum powers $A^t\oplus A^{2t}\oplus \cdots$ for some $t$, or are rank one, supported only on a single basis vector beyond $n/2$.
• Nijenhuis operators: Scalar multiples of identity for degree $0$; for degree $k\ge \lfloor n/2\rfloor$, arbitrary on the lowest $n-k$ degrees, zero beyond.

Example: On the 3-dimensional null-filiform algebra, every degree-zero Nijenhuis operator is $N(e_i)=a e_i$; every degree-zero Reynolds operator includes the family $R(e_i) = \frac{c}{i-(i-1)c}e_i$ (Karimjanov et al., 2018).

As a concrete Reynolds–Nijenhuis example, for $A=\mathbb{R}^3$ with $e_1 \cdot e_3 = e_2 = e_3\cdot e_1$ and all other products zero, every RN-operator $P$ is determined by

$$P(e_1)=0, \quad P(e_3)=0, \quad P(e_2)=v\, e_1 + q\, e_3, \quad v,q\in\mathbb{R},$$

so the space of all RN-operators is 2-dimensional (Mosbahi et al., 28 Dec 2025).

7. Applications and Directions

Reynolds–Nijenhuis associative algebras and their cohomology have applications including:

• Classification of associative deformations and derivations compatible with hybrid operator structures,
• Construction of new associative products for purposes such as algebraic renormalization and operator splitting in mathematical physics,
• Explicit connections to averaging, Rota–Baxter, and modified Rota–Baxter structures in low-dimensional and null-filiform settings,
• Development of graded-algebraic models with precise deformation-theoretic control in string theory or quantum field theory, where RN-structures can organize perturbative expansions (Mosbahi et al., 28 Dec 2025, Karimjanov et al., 2018).

The rigidity or abundance of RN-operators on a given algebra reflects deep structural features, often linked to the nilpotency, grading, or dimension of the underlying associative algebra. The cohomological and deformation-theoretic approach generalizes prior frameworks, opening new avenues for classification and analysis of operator-augmented associative structures.