--- title: Crossed Homomorphisms on Lie 2-Groups url: https://www.emergentmind.com/topics/crossed-homomorphisms-on-lie-2-groups type: topic --- # Crossed Homomorphisms on Lie 2-Groups A crossed homomorphism on Lie 2-groups is a fundamental structure underlying weak morphisms between higher group objects in differential geometry and categorical algebra. It encodes the data needed to realize a morphism of Lie 2-groups (equivalently, a morphism of crossed modules) where the strictness of standard morphism axioms is relaxed by higher coherent maps. Such structures have decisive applications in the classification of 2-group extensions, categorical solutions to the Yang–Baxter equation, and factorization theorems in higher gauge theory. The theoretical framework is now extended to a precise correspondence with relative Rota–Baxter operators, higher group cohomology, and infinitesimal Lie 2-algebra structures [2303.13176, 1006.1262, 2602.02041]. ## 1. Lie 2-Groups and Crossed Modules A strict Lie 2-group can be described in two equivalent ways: - As a Lie groupoid $$ \mathcal{P}: \quad P_1 \rightrightarrows P_0, $$ where both $P_1$ (arrows) and $P_0$ (objects) are Lie groups, and all structure maps (source, target, identity, multiplication) are Lie group homomorphisms. - As a crossed module of Lie groups $$ (G_1 \xrightarrow{\mu} G_0), $$ with $G_0$ and $G_1$ Lie groups, $\mu: G_1 \to G_0$ a Lie group homomorphism, and an action of $G_0$ on $G_1$ by automorphisms: $$ g_0 \cdot g_1, $$ satisfying $$ \mu(g_0 \cdot g_1) = g_0 \mu(g_1) g_0^{-1}, \quad \mu(\gamma) \gamma' \mu(\gamma)^{-1} = \gamma \gamma' \gamma^{-1} $$ for all $g_0 \in G_0$, $\gamma, \gamma' \in G_1$. There exists a one-to-one correspondence between strict Lie 2-groups and crossed modules, which forms the basis for their classification and study [1006.1262, 2602.02041]. ## 2. Crossed Homomorphisms: Definition and Coherence Given Lie 2-groups $\mathcal{P}$ and $\mathcal{Q}$, and an action $(\phi, \phi_0)$ of $\mathcal{P}$ on $\mathcal{Q}$ (i.e., $\phi: P_1 \to \mathrm{Aut}(Q_1)$ and $\phi_0: P_0 \to \mathrm{Aut}(Q_0)$), a crossed homomorphism is a pair of smooth maps $$ \Delta : P_1 \to Q_1, \quad \Delta_0: P_0 \to Q_0, $$ satisfying: 1. **Crossed homomorphism property:** $$ \Delta(p p') = \Delta(p) \, \phi(p)(\Delta(p')), \qquad \forall\, p, p' \in P_1 $$ and analogously for $\Delta_0$ on $P_0$. 2. **Lie groupoid morphism property:** $$ s_{Q}(\Delta(p)) = \Delta_0(s_P(p)), $$ $$ t_{Q}(\Delta(p)) = \Delta_0(t_P(p)), $$ $$ \Delta(p * p') = \Delta(p) * \Delta(p') $$ for composable $p, p'$. Equivalently, the graph $$ \operatorname{Gr}(\Delta) = \{\,(\Delta(p), p) \mid p \in P_1\,\} $$ defines a Lie 2-subgroup of the semidirect product $\mathcal{Q} \rtimes \mathcal{P}$, projecting isomorphically to $\mathcal{P}$ [2602.02041]. In the crossed module formalism, crossed homomorphisms coincide with smooth maps $\alpha: G \to H'$ between group objects (for appropriate morphisms $(f_H, f_G)$), satisfying the twisted cocycle and boundary compatibility conditions: $$ \alpha(gg') = \alpha(g)\, (f_G(g) \cdot \alpha(g')), \quad \partial'(\alpha(g)) = f_G(g) $$ [1006.1262]. ## 3. Crossed Homomorphisms and Rota–Baxter Operators A crucial recent result is the formal relationship between crossed homomorphisms and relative Rota–Baxter operators on Lie 2-groups. In the semidirect product context, a relative Rota–Baxter operator is a Lie groupoid morphism characterized by compatibility with the group actions. The key identification is that: - The formal inverse of a relative Rota–Baxter operator is a crossed homomorphism, and vice versa. - For a crossed homomorphism $\Delta$, the induced operator $$ \widehat{\Delta}(q, p) = (\phi(p^{-1})(q^{-1} \cdot \Delta(p)), e_P) $$ satisfies the axioms of a relative Rota–Baxter operator [2602.02041, Theorem 5.4]. This relationship enables factorization theorems at the 2-group level and the construction of categorical Yang–Baxter solutions, generalizing classical group-theoretic results. ## 4. Structural Characterizations and Graph Theoretic Classification Crossed homomorphisms on Lie 2-groups admit comprehensive structural characterizations: - **Graph Classification:** Crossed homomorphisms correspond bijectively to Lie 2-subgroups of the semidirect product $\mathcal{Q} \rtimes \mathcal{P}$ with graphs that project isomorphically to $\mathcal{P}$ [2602.02041]. - **Derived Actions:** Every crossed homomorphism $\Delta$ induces a "derived" action: $$ \widetilde{\phi}(p)(q) = \Delta(p) \, \phi(p)(q) \, \Delta(p)^{-1} $$ for which $\Delta$ is a genuine 1-cocycle. - **Infinitesimalization:** The differential of a crossed homomorphism between Lie 2-groups provides a crossed homomorphism between the corresponding Lie 2-algebras [2602.02041, Theorem 5.5]. These characterizations underpin the role of crossed homomorphisms in higher gauge theory, stacky Lie groups, and the structure theory of 2-group extensions [1006.1262]. ## 5. Exemplary Constructions ### Central Extensions and the String 2-Group A paradigmatic example is furnished by U(1)–central extensions of the based loop group $\Omega G$ of a compact, semisimple, and simply connected Lie group $G$ [2303.13176]: $$ 1 \longrightarrow U(1) \longrightarrow \widetilde{\Omega G} \xrightarrow{\pi} \Omega G \longrightarrow 1. $$ Restricting to loops supported on $(0,\pi)$, $H = \widetilde{\Omega_{(0,\pi)} G}$, with $G = P_e G$ the path group. The crossed module $$ 1 \longrightarrow U(1) \longrightarrow H \xrightarrow{\partial} G \longrightarrow 1 $$ is defined with $\partial(\Phi) = \pi(\Phi)|_{[0,\pi]}$ and the $G$-action on $H$ by conjugation. The Peiffer identity is enforced strictly by the "disjoint commutativity" of central extensions. If $G$ is semisimple and simply connected, all such extensions are disjoint commutative; hence the resulting 2-group is strict. ### Crossed Homomorphisms into Trivial 2-Groups For the crossed module $(U(1) \to \{1\})$, any character $\chi: G \to U(1)$ is a crossed homomorphism, encoding weak 2-group morphisms into the trivial 2-group [1006.1262]. ### Abelian Semidirect Example For trivial 2-groups on $(\mathbb{R} \rightrightarrows \mathbb{R})$, setting $\Delta(p) = e^{a p}$ for $a \neq 0$ and the conjugation action by $e^{a p}$ realizes a nontrivial crossed homomorphism [2602.02041]. ## 6. Cohomological and Categorical Context Crossed homomorphisms constitute the 1-cocycles in nonabelian cohomology $H^1(\mathcal{P};\mathcal{Q})$ for Lie 2-groups. Correspondingly, - They classify 2-group extensions and gauge equivalence classes in higher homotopy theory and topological quantum field theories. - Crossed homomorphisms yield categorical solutions to the Yang–Baxter equation on the 2-groupoid, extending set-theoretic solutions in the group case. - They implement factorization theorems analogously to how Rota–Baxter operators factor Lie groups, yielding decompositions of Lie 2-groups into matched pairs or cross modules [2602.02041]. ## 7. Strictness, Disjoint Commutativity, and Morphism Weakening Disjoint commutativity in central extensions of loop groups is pivotal for obtaining strict Lie 2-group structures from crossed modules, as shown explicitly for the string 2-group in [2303.13176]. In the absence of strictness, crossed homomorphisms compensate the lack of exactness in morphisms by providing coherent higher transformations, captured categorically by Peiffer liftings and monoidal coherence maps. The structure of crossed homomorphisms thus interpolates between strict and weak morphisms of higher group objects, with implications for the classification, deformation, and homotopical invariance of 2-groupoids. Source: https://www.emergentmind.com/topics/crossed-homomorphisms-on-lie-2-groups