Second Cohomology Groups: Extensions & Classifications

• Second cohomology groups are invariants that classify central and abelian extensions in diverse algebraic, topological, and quantum settings.
• They are computed using explicitly defined cochain complexes with measurable, continuous, or polynomial constraints, ensuring accurate classification.
• Applications range from determining deformation classes in Lie algebras to classifying tensor autoequivalences in quantum groups and beyond.

The second cohomology group is a central invariant in the homological, group-theoretic, and algebraic paper of objects ranging from topological groups to Lie algebras, algebraic groups, tensor categories, and generalized algebraic structures. It plays a fundamental classification role: it governs equivalence classes of extensions, central extensions, and deformation classes, and encodes obstructions to splitting and lifting problems. Across diverse settings, the explicit structure, computation, and applications of second cohomology reflect the algebraic and geometric complexity of the objects under analysis.

1. Second Cohomology as an Extension Classifier

The classical interpretation of the second cohomology group $H^2$ is as a parameter space for equivalence classes of (central or more general) extensions. For a group $G$ with coefficients in a $G$-module $A$, the collection of equivalence classes of extensions

$1 \to A \to E \to G \to 1$

is typically parametrized by $H^2(G,A)$. In the measurable-topological context, for $G$, $A$ locally compact, second-countable topological groups and $A$ a continuous $G$-module, the second locally continuous measurable cohomology group $H^2_{\lcm}(G,A)$ gives a precise bijection: $H^2_{\lcm}(G,A) \longleftrightarrow \{\text{equivalence classes of extensions } 1\to A\to E\to G\to 1,\ E\text{ admits a continuous local section}\}$ where the cochains are Borel measurable and continuous in a neighborhood of identity. This generalizes the purely algebraic case to accommodate the additional structure and subtleties of the topological category (Khedekar et al., 2010).

In the context of algebraic groups, $H^2$ for finite groups of Lie type with simple modules as coefficients likewise parametrizes extension classes of finite Chevalley groups by irreducible representations, and exact knowledge of $H^2$ detects new possibly nontrivial central extensions endemic to characteristic $p$ phenomena (Boe et al., 2011). For objects such as relative Rota–Baxter groups, the second cohomology group with appropriate coefficients classifies abelian extensions of such groups, extending the parallel with classical group cohomology (Belwal et al., 2023). In cycle set and brace theory, $H^2$ also controls abelian extensions and isomorphic structures between non-standard algebraic systems (Guccione et al., 14 Jun 2025).

2. Explicit Constructions: Complexes and Cocycles

The construction of $H^2$ always depends on an explicit cochain complex with carefully tailored group, module, measurable, or algebraic structure:

• For topological groups,

$C^n_{\lcm}(G,A) = \{f:G^n\to A \mid \text{Borel measurable, continuous near } (e,\dots,e)\}$

with the group coboundary operator as in classical cohomology, but with local Borel and continuity constraints (Khedekar et al., 2010).

• In group-theoretic settings for two-step nilpotent (T-) groups, the relevant cochains are built from polynomial functions in the Mal'cev coordinates, and $H^2$ admits a canonical decomposition reflecting the underlying associated graded Lie algebra:

$$H^2(G,M) \cong \mathrm{Coker}(c^*:\mathrm{Hom}(L_2,M)\to\mathrm{Hom}(\Lambda^2L_1,M)) \oplus \mathrm{Hom}(L_1\otimes L_2/S, M)$$

with explicit cocycle representatives in terms of tensor and wedge constructions on $L_1,G/V(\gamma_2(G))$, $L_2= [G,G]$, and $S$ generated by triple commutator relations (Dekimpe et al., 2014).

• For relative Rota–Baxter groups, the mixed cohomology is defined via a total complex

$$C^2 = C^2(A,K) \oplus C^2(B,L) \oplus C(A\times B,K) \oplus C(A,L)$$

with eight explicit polynomial cocycle identities linking the components (Belwal et al., 2023).

• In the theory of linear cycle sets with finite abelian adjoint group, $H^2$ emerges as the quotient of cocycles $(\beta,f)$ in a filtered double complex, with explicit, canonical generators constructed recursively in terms of the group structure and derivation data; this provides a systematic method to enumerate all possible abelian extensions (Guccione et al., 14 Jun 2025).
• For BBW parabolics in classical Lie superalgebras, the second Lie superalgebra cohomology $H^2(\mathfrak{n}^+,\mathbb{C})$ is computed via the Hochschild–Serre spectral sequence, with identification of $E_2$-page and explicit collapse yielding the weight distribution and dimension formula for $H^2$ (Galban, 2021).

3. Second Cohomology in Quantum Groups and Tensor Categories

For compact connected groups $G$, the dual object $\hat{G}$ (in the sense of quantum groups) possesses a second cohomology group of invariant unitary 2-cocycles $H^2_u(\hat{G};\mathbb{T})$, defined via the operator-algebraic framework: $$F \in W^*(G)\otimes W^*(G),\ F \text{ unitary},\ (1\otimes \Delta)(F)\cdot F_{12} = (\Delta\otimes 1)(F)\cdot F_{23}$$ modulo the natural cohomology equivalence. For compact connected $G$,

$$H^2_u(\hat{G};\mathbb{T}) \cong H^2(\widehat{Z(G)};\mathbb{T})$$

identifies this space with central data: cocycles on the dual of the center. This parameterizes full-multiplicity ergodic actions and classifies tensor autoequivalences of $\mathrm{Rep}\,G$, up to the action of $\mathrm{Out}(G)$: $$1 \to H^2(\widehat{Z(G)};\mathbb{T}) \to \mathrm{Aut}_\otimes(\mathrm{Rep}\,G) \to \mathrm{Out}(G) \to 1$$ (Neshveyev et al., 2010). This framework underpins both categorical monoidal equivalence of compact groups and the internal symmetry group of representation categories.

4. Computations and Representation-Theoretic Significance

Explicit dimension formulas and decompositions of $H^2$ are central in several contexts:

• For the second cohomology of nilpotent orbits $O_X$ in real simple Lie algebras, $\dim H^2(O_X)$ is governed by partition data and block multiplicities from the $\mathfrak{sl}_2$-module decomposition. For example, in type $\mathfrak{sl}_n(\mathbb{R})$, $H^2$ is generically zero except for orbits with a unique odd block of multiplicity 2, where $\dim H^2 = 1$. Analogous explicit combinatorial formulas exist for other classical real types, involving signatures and signed diagrams (Biswas et al., 2016).
• For finite groups of Lie type, $H^2(G(\mathbb{F}_q), L(\lambda))$ is generically isomorphic (under explicit constraints) to $H^2(G, L(\lambda))$, and the latter is computed via vanishing theorems for extensions and explicit spectral sequences. Exceptional "problem weights" lead to concrete, nontrivial one-dimensional cohomology classes for small characteristic and field size, characterizing new central extension phenomena (Boe et al., 2011).
• In the setting of handlebody Torelli groups, $H^2$ is determined as the image of the cup product on explicit abelian quotients (Johnson and symplectic modules) and is decomposed algebraically via SL-generic module representations, with a complete prescription for the kernel and image of the cup product (Holden, 3 Sep 2025).

5. Classification of Extensions and Structural Applications

Second cohomology realizes a unifying thread: it fully encodes the data of possible central or abelian extensions—modulo equivalence—of the algebraic structure under consideration. In particular:

• In topological groups, only locally split extensions (i.e., extensions with a continuous local section) are captured by $H^2_{\lcm}$, reflecting the sensitivity to topology and measurability.
• For linear cycle sets and left braces, $H^2$ organizes all equivalence classes of central extensions, with the additional structure (e.g., compatibilities between sum and operation) fully incorporated into the cocycle identities (Guccione et al., 14 Jun 2025). For bijective relative Rota–Baxter groups, the $H^2$ of the group and the skew-left-brace coincide in dimension two (Belwal et al., 2023).
• The cohomological invariants tie directly into deformation theory, representation theory, and quantum symmetry. For instance, in compact quantum groups, $H^2$ dictates twisting data and advances the classification of monoidal equivalence classes and categorical symmetries (Neshveyev et al., 2010).

6. Cohomological Comparison, Exactness, and Further Directions

The interaction between various cohomology theories is codified via restriction maps and exact sequences. For topological groups: $H^n_{\mathrm{cont}}(G,A) \to H^n_{\lcm}(G,A) \to H^n_\mathrm{Moore}(G,A)$ forms a progression interpolating between continuous, locally continuous measurable, and Moore's measurable cohomology. These maps preserve low-degree isomorphisms and, crucially, $H^*_{\lcm}$ admits long exact sequences for any locally split short exact sequence of $G$-modules—remedying a primary deficiency of continuous cohomology (Khedekar et al., 2010). In favorable classes (e.g., Lie groups acting on finite-dimensional vector spaces, or profinite groups with discrete modules), all three theories coincide in all degrees.

In summary, the second cohomology group $H^2$ functions as a powerful invariant for the analysis of extensions and the classification of algebraic, topological, and quantum group structures. Its explicit computation, functorial properties, and categorical interpretations form foundational tools across modern algebra, geometry, and mathematical physics.