Hom-Lie Color Algebras Overview

• Hom-Lie Color Algebras are graded generalizations of Hom-Lie algebras defined by a twist map and a bicharacter that governs the grading structure.
• They support advanced representation theory, cohomology, and formal deformation analysis, extending classical Lie theory to graded and twisted contexts.
• Constructions like omni-Hom-Lie algebras and quadratic extensions illustrate practical applications in structure theory and the classification of regular, deformed algebras.

A Hom-Lie color algebra is a graded generalization of the Hom-Lie algebra concept, enriching the structure with a bicharacter that governs commutation rules according to an abelian grading group. The framework unifies classical color Lie superalgebras, Hom-Lie superalgebras, and associative color algebras under a deformation principle with a twist map, supporting cohomology, representation theory, and deformation analysis. The omni-Hom-Lie construction further organizes these objects through Leibniz-type extensions and allows the classification of regular structures via isotropic subspaces, establishing a structure theory that parallels classical Lie theory but adapts to graded, twisted contexts (Armakan et al., 2020).

1. Formal Structure and Definition

Let $L = \bigoplus_{g\in G} L_g$ be a $G$-graded vector space over a field $\mathbb{k}$ of characteristic 0, and $\varepsilon: G \times G \to \mathbb{k}^\times$ a bicharacter satisfying

$$\varepsilon(\alpha + \beta, \gamma) = \varepsilon(\alpha, \gamma)\varepsilon(\beta, \gamma),\quad \varepsilon(\alpha, \beta + \gamma)=\varepsilon(\alpha, \beta)\varepsilon(\alpha, \gamma),\quad \varepsilon(\alpha, \beta)\varepsilon(\beta, \alpha) = 1$$

for all $\alpha,\beta,\gamma \in G$. A color Hom-Lie algebra is a quadruple $(L, [\cdot,\cdot], \alpha, \varepsilon)$ where $[\cdot,\cdot]:L \otimes L \to L$ is grade-preserving and bilinear, and $\alpha:L\to L$ is even. For homogeneous $x,y,z$,

• $\varepsilon$-skew-symmetry:

$[x, y] = -\varepsilon(|x|, |y|)\, [y, x]$

• Hom-Jacobi identity (cyclic sum):

$$\sum_{\mathrm{cyc}} \varepsilon(|z|, |x|)\, [\alpha(x), [y, z]] = 0$$

If $\alpha$ is invertible and satisfies multiplicativity ($\alpha([x, y]) = [\alpha(x), \alpha(y)]$), the algebra is called regular. When $\alpha = \operatorname{id}$, this recovers an ordinary color Lie algebra; when $\varepsilon$ is trivial, a Hom-Lie algebra structure is recovered (Yuan, 2010, Armakan et al., 2020).

2. Representation Theory and Cohomology

A representation of $(L, [\cdot,\cdot], \alpha, \varepsilon)$ is a triple $(\rho, V, \beta)$ where $V=\bigoplus_{g \in G} V_g$ is $G$-graded, $\beta:V\to V$ is even, and $\rho:L \to \operatorname{End}(V)$ is an even linear map such that: $$\rho([x, y]) \circ \beta = \rho(\alpha(x))\circ \rho(y) - \varepsilon(|x|, |y|)\rho(\alpha(y))\circ \rho(x)$$ The adjoint (ad$\,x(y) = [x, y]$) yields the canonical module structure.

Cohomology is defined on the space of $\varepsilon$-skew, homogeneous $n$-linear maps $C^n(L; V)$: $$(\delta^n f)(x_1, \ldots, x_{n+1}) = \sum_{i=1}^{n+1} (-1)^{i+1} \varepsilon_i\,\rho(\alpha^n(x_i))\,f(\ldots) + \sum_{1\le i<j \le n+1} (-1)^{i+j} \varepsilon_{ij}\, f([x_i, x_j], \ldots)$$ where hats denote omission, and $\varepsilon_i, \varepsilon_{ij}$ are Koszul signs. The operator satisfies $\delta^{n+1}\circ \delta^n=0$, so $$H^n(L; V) = \ker\delta^n / \operatorname{im}\delta^{n-1}$$ (Armakan et al., 2020, Ammar et al., 2012, Abdaoui et al., 2013).

3. Examples, Twisting Procedures, and Universal Enveloping Algebras

Principal examples arise via twisting:

• If $(g,[\cdot, \cdot],\varepsilon)$ is an ordinary color Lie algebra and $\alpha:g\to g$ is an even endomorphism, define $[x, y]_\alpha = \alpha([x, y])$; then $(g, [\cdot,\cdot]_α, \alpha, \varepsilon)$ is Hom-Lie.
• If $(A,\cdot,\alpha)$ is a Hom-associative color algebra, the commutator $[x,y] = x\cdot y - \varepsilon(|x|,|y|)y\cdot x$ yields a Hom-Lie color algebra.
• Iterative twisting by commuting algebra endomorphisms affords families of Hom-Lie color algebras (Yuan, 2010, Armakan et al., 2020).

Universal enveloping algebras of involutive Hom-Lie color algebras are constructed via free tensor algebras and quotienting by Hom-ideals built from twisted commutators. A Poincaré–Birkhoff–Witt theorem holds: for a well-ordered homogeneous basis, strictly descending monomials form a basis for the enveloping algebra (Armakan et al., 2017).

4. Quadratic Structures and Extensions

A quadratic color Hom-Lie algebra is a quintuple $(g,[\cdot,\cdot],\alpha,\varepsilon,B)$, with $B:g\times g\to\mathbb{k}$ nondegenerate, $\varepsilon$-symmetric ($B(x,y)=\varepsilon(|x|,|y|)B(y,x)$), invariant ($B([x,y],z)=B(x,[y,z])$), and $\alpha$-symmetric ($B(\alpha(x),y)=B(x,\alpha(y))$). Constructions include:

• Twisting quadratic color Lie algebras via $\varepsilon$-symmetric automorphisms,
• Passing to commutative Hom-associative algebras,
• Tensor products yielding new quadratic structures.

Extensions:

• Central extensions by $M$ require 2-cocycle conditions; equivalence governed by $H^2(g, \mathbb{k})$.
• $T^*$-extensions with coadjoint representations and 2-cocycles into $g^*$, yielding quadratic extensions with split-signature bilinear forms.
• Double extensions and Faulkner-type constructions generalize further (Ammar et al., 2012, Armakan et al., 2017).

5. Split Regular Algebras and Structure Theory

In the split regular case, maximal abelian graded subalgebras $H$ are used to define roots as nonzero linear forms on $H_0$ whose associated root-spaces are nontrivial. The root system $\Lambda$ yields a decomposition

$$L = U \oplus \bigoplus_{[\gamma] \in \Lambda/\sim} I_{[\gamma]}$$

where $I_{[\gamma]}$ are ideals with centralizer properties. Simplicity criteria depend on root-multiplicativity, maximal length, and connectedness of the root system, generalizing the structure theory for split Lie (super)algebras to the Hom and color context (Cao et al., 2015).

6. Formal Deformation Theory

Formal deformations of a color Hom-Lie algebra $(L, [\cdot,\cdot], \alpha, \varepsilon)$ employ power series brackets preserving $\varepsilon$-skew and the Hom-Jacobi identity order-by-order. Infinitesimal deformations correspond to 2-cocycles, and higher-order obstructions reside in higher cohomology:

• Equivalent deformations are related by formal automorphisms intertwining the twisted bracket and $\alpha$.
• Rigidity theorems and classification up to equivalence are governed by vanishing $H^2(L, L)$ (Abdaoui et al., 2013, Armakan et al., 2020).

7. Color Omni-Hom-Lie and Hom-Leibniz Algebras

The omni-Hom-Lie algebra associated to $(V, \beta)$ is defined on $E = \operatorname{gl}(V) \oplus V$ with twist $$\theta(A + v) = \operatorname{Ad}_\beta(A) + \beta(v)$$ and bracket $\{A + u, B + v\} = [A, B]_\beta + A(v)$: $$[A, B]_\beta = \beta A \beta^{-1} B \beta^{-1} - \varepsilon(|A|,|B|)\beta B \beta^{-1} A \beta^{-1}$$ The structure is that of a color Hom-Leibniz algebra with a nondegenerate $\theta$-invariant pairing. Regular color Hom-Lie structures on $V$ correspond bijectively to $\theta$-stable, maximally isotropic subspaces of $E$ closed under the bracket, amplifying the role of omni structures in classification (Armakan et al., 2020).

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In summary, Hom-Lie color algebras are a broad, technically rich generalization accommodating graded symmetries, twist deformations, advanced extension and cohomology theory, and possess a structure theory analogous to Lie superalgebras but with maximal generality in the graded-twisted category. All major aspects—representation, cohomology, rigidification, extension, omni-constructions, and enveloping algebras—are present and fully characterized in the literature cited above.