Non-Abelian Relative Cohomology Theory

• Non-Abelian Relative Cohomology Theory is a framework that generalizes classical abelian cohomology to settings with non-commutative coefficients, focusing on extensions and deformations.
• It employs structures such as Lie algebras, dg-Lie algebras, and Rota–Baxter algebras to establish cocycle conditions, Deligne groupoid formulations, and explicit exact sequences.
• The theory offers practical insight by unifying algebraic and topological methods to classify moduli, torsors, and representations, with applications in geometry, representation theory, and homotopy.

Non-Abelian Relative Cohomology Theory synthesizes the extension and classification of algebraic and topological structures via cohomology in settings where coefficients and symmetry groups are not abelian. It incorporates developments from Lie algebras, topological groups, Rota-Baxter structures, and singular cohomology, providing a uniform operadic and dg-Lie approach, as well as exact sequences and harmonic decompositions, for understanding moduli of extensions, torsors, and representations. This framework generalizes abelian relative cohomology, enabling classification and deformation theory in diverse categories, and supporting applications in geometry, representation theory, and homotopy theory.

1. Non-Abelian Extensions and Cohomology: Lie Algebraic Foundations

Let $\mathfrak{g}$ and $\mathfrak{h}$ be Lie algebras over a field of characteristic zero. A non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ is defined by a short exact sequence: $$0 \to \mathfrak{h} \xrightarrow{i} \mathfrak{e} \xrightarrow{p} \mathfrak{g} \to 0,$$ together with a splitting $s:\mathfrak{g}\to\mathfrak{e}$ such that $p \circ s = \mathrm{id}_\mathfrak{g}$. The vector space $\mathfrak{e}$ is identified with $\mathfrak{h} \oplus \mathfrak{g}$ via $s$, and its Lie bracket is decomposed by components:

• $$[\cdot,\cdot]_\mathfrak{e}|_{\mathfrak{g} \wedge \mathfrak{g}} = [\cdot,\cdot]_\mathfrak{g} + \chi$$,
• $$[\cdot,\cdot]_\mathfrak{e}|_{\mathfrak{g} \otimes \mathfrak{h}} = \psi$$,
• $$[\cdot,\cdot]_\mathfrak{e}|_{\mathfrak{h} \otimes \mathfrak{g}} = -\psi$$,
• $$[\cdot,\cdot]_\mathfrak{e}|_{\mathfrak{h} \wedge \mathfrak{h}} = [\cdot,\cdot]_\mathfrak{h}$$.

Here, $$\chi \in \mathrm{Hom}(\wedge^2 \mathfrak{g}, \mathfrak{h})$$ and $$\psi \in \mathrm{Hom}(\mathfrak{g}, \mathrm{Der}(\mathfrak{h}))$$. The Jacobi identity imposes two cocycle conditions:

1. $$[\psi(a), \psi(b)]_{\mathrm{Der}} = \psi([a,b]_\mathfrak{g}) + \mathrm{ad}_\mathfrak{h}(\chi(a \wedge b))$$,
2. $$\sum_\mathrm{cyclic} \left(\psi(a)\cdot \chi(b \wedge c) - \chi([a,b] \wedge c)\right) = 0$$.

The set of pairs $(\chi,\psi)$ in $Z^2_{nab}(\mathfrak{g},\mathfrak{h})$ modulo the equivalence induced by $\beta \in \mathrm{Hom}(\mathfrak{g},\mathfrak{h})$ yields the non-abelian cohomology group $H^2_{nab}(\mathfrak{g},\mathfrak{h})$ (Fregier, 2013).

2. Differential Graded Lie Algebras and the Deligne Groupoid Formalism

The Chevalley–Eilenberg complex $C^*(V, V)$ for $V = \mathfrak{g} \oplus \mathfrak{h}$, together with the Nijenhuis–Richardson bracket and the differential $$d = \mathrm{ad}_{\rho_\mathfrak{g} + \rho_\mathfrak{h}}$$, provides a dg-Lie algebra framework for extensions. The subspace: $$L = \bigoplus_{m \geq 1, n \geq 0} \mathrm{Hom}(\wedge^m \mathfrak{g} \otimes \wedge^n \mathfrak{h}, \mathfrak{h})$$ is graded by $$L^i = \bigoplus_{m+n=i+1} \mathrm{Hom}(\wedge^m \mathfrak{g} \otimes \wedge^n \mathfrak{h}, \mathfrak{h})$$. Maurer–Cartan elements $\alpha \in L^1$ satisfy $d\alpha + \frac{1}{2}[\alpha,\alpha] = 0$.

The Deligne groupoid $\mathrm{Del}(L)$ organizes these data:

• Objects: Maurer–Cartan elements.
• Morphisms: gauge transformations $\alpha' = e^{\mathrm{ad}_\beta} \alpha + g_\beta$ for $\beta \in L^0$.

The fundamental theorem establishes a natural bijection $$H^2_{nab}(\mathfrak{g},\mathfrak{h}) \cong \pi_0(\mathrm{Del}(L))$$, identifying non-abelian cohomology classes with connected components of the Deligne groupoid (Fregier, 2013).

3. Non-Abelian Relative Cohomology in Rota-Baxter Lie Algebras

Generalizing to relative Rota–Baxter Lie algebras, let $$A = ((\mathfrak{g}, [\cdot,\cdot]_\mathfrak{g}), (\mathfrak{h}, \rho), T)$$ and $$B = ((\mathfrak{b}, [\cdot,\cdot]_\mathfrak{b}), (\mathfrak{m}, \nu_m), S)$$. Non-abelian 2-cocycles $(\omega, \varpi, \chi, \mu, \rho_B, \rho_M)$ satisfy a system of nine compatibility equations (L1–L9), encoding the relative algebraic structure and extension behavior. The corresponding cohomology group $H^2_{nab}(A;B)$ is defined as the quotient of cocycles by a detailed equivalence relation involving linear maps $\zeta$ and $\eta$ (Sun et al., 2024).

Non-abelian extensions of $A$ by $B$ classify to $H^2_{nab}(A;B)$, and are described by explicit formulas for bracket and operator in $A \oplus_{(\omega, \varpi, \chi)} B$. Examples, such as the Heisenberg extension, demonstrate the reduction of the general construction to classical cases when $A$ and $B$ are abelian (Sun et al., 2024).

4. Relative Lie Algebra Cohomology: Parabolic Subalgebras and Harmonic Theory

For pairs of nested parabolic subalgebras $p \subset g$ acting on a $g$-module $V$, non-abelian relative cohomology is constructed via the Chevalley–Eilenberg complex for the quotient $g_{-} = g/p$. The $p$-equivariant cochains $C^k(g,p;V)$ carry a differential $d_{\rm rel}$ and admit a Laplacian $$\square_{\rm rel} = d_{\rm rel} \delta_{\rm rel} + \delta_{\rm rel} d_{\rm rel}$$.

Kostant’s relative theorem describes the cohomology $H^k(g,p;V)$ as a completely reducible $p_0$-module, composed of irreducible summands indexed by the relative Hasse diagram: $$W_{\rm rel} = \{ w \in W(g) : \Delta^+ \cap w^{-1}\Delta^- \subset \Delta^+(p_+) \}.$$ A Hodge decomposition holds: $C^* \cong \im(d_{\rm rel}) \oplus \ker(\square_{\rm rel}) \oplus \im(\delta_{\rm rel}),$ with $H^*(g,p;V) \cong \ker(\square_{\rm rel})$ realized as harmonic forms (Cap et al., 2015).

5. Topological and Singular Cohomology: Non-Abelian Relative Frameworks

Non-abelian cohomology for topological spaces and groups is developed in dimensions 0 and 1 via crossed complexes of cochains, with coefficients in arbitrary (not necessarily abelian) groups $G$. For a pair $(X, A)$:

• $C^0(X, A; G)$ consists of maps vanishing on $A$,
• $C^1(X, A; G)$ consists of cochains on paths vanishing on paths in $A$.

The six-term exact sequence generalizes the classical Mayer–Vietoris sequence: $$1 \to H^0(X, A; G) \to H^0(X; G) \to H^0(A; G) \xrightarrow{\partial} H^1(X, A; G) \to H^1(X; G) \to H^1(A; G),$$ with “exactness” interpreted as in classical cohomology but in pointed sets under group actions. The universal property $H^1(X; G) \cong \mathrm{Hom}(\pi_1(X, b), G)$ holds, providing group-theoretic interpretations and enabling proofs of the Seifert–van Kampen theorem, its Crowell-Fox and Brown–Salleh variants, without abelianity assumptions (Ivanov, 2023).

6. Exact Sequences and Classification: Wells-Type Structures

Wells-type exact sequences generalize classical automorphism and derivation exact sequences to the non-abelian relative setting. For non-abelian extensions in Rota–Baxter Lie algebras:

• The automorphism sequence: $$1\to \mathrm{Aut}_B^A(\widehat{A}, \widehat{V}, \widehat{T}) \to \mathrm{Aut}_B(\widehat{A}, \widehat{V}, \widehat{T}) \xrightarrow{K} \mathrm{Aut}(A)\times\mathrm{Aut}(B) \xrightarrow{W} H^2_{nab}(A; B),$$
• The derivation sequence (abelian case): $0\to \mathcal{Z}^1(A; B) \to \mathrm{Der}_B(\widehat{A}, \widehat{V}, \widehat{T}) \xrightarrow{\Digamma} \mathfrak{g}(A,B) \xrightarrow{W} H^2(A; B).$

Concrete instantiations, such as automorphisms and derivations of the Heisenberg-type Lie algebra, yield explicit kernels and classification results, recovering the structure of infinitesimal deformations (Sun et al., 2024).

7. Synthesis, Generalizations, and Applications

Non-abelian relative cohomology provides a comprehensive framework unifying diverse notions of extensions, moduli, torsors, and representations in algebraic and topological contexts. The dg-Lie and operadic viewpoint supports the inclusion of $L_\infty$-algebras, Lie algebroids, and homotopy-type structures. The theory captures non-commutative phenomena through compatibility conditions, gauge equivalence, and harmonic decomposition, subsuming abelian cases as tangent structures at fixed module actions (Fregier, 2013, Cap et al., 2015, Sun et al., 2024).

Applications include:

• Construction of invariant differential operators and BGG sequences,
• Classification and deformation of algebraic structures,
• Geometric and rigidity results in parabolic differential geometry,
• Unified treatment of low-dimensional homotopy and cohomology theories.

A plausible implication is the extension of these methodologies to moduli of higher or derived extensions, equivariant cohomology in quantum algebra, and operadic models in categorical representation theory.