Globalization of Partial Group Actions on Not Necessarily Associative Algebras and Covariant Representations

1. Problem Statement

The paper addresses the globalization problem for partial group actions in the context of non-associative algebras.

• Context: Partial group actions generalize global actions by allowing a group $G$ to act only on specific sub-objects (pieces) of a mathematical structure. A central question in this field is whether a given partial action can be realized as the restriction of a global action on a larger structure (a "globalization").
• Gap: While globalization has been extensively studied for associative algebras, topological spaces, and modules, there was no systematic framework for non-associative algebras (such as Lie, Jordan, or Malcev algebras) belonging to an arbitrary variety $\mathcal{V}(I)$.
• Specific Challenges:
  1. Extending the concept of partial actions to algebras where the product is not necessarily associative.
  2. Distinguishing between ideal partial actions (where the domains of action are ideals) and generalized partial actions (where domains are merely subalgebras).
  3. Developing a theory of covariant representations (pairs of a module and a group representation compatible with the algebra action) for these partial structures and establishing their functorial properties.

2. Methodology

The authors employ a construction they term the $\Lambda$-construction, originally developed for modules in previous work [31], and adapt it to the algebraic setting of varieties.

• Varieties of Algebras ($\mathcal{V}(I)$): The work is set within a variety of non-associative algebras defined by a set of identities $I$. They utilize free algebras $F_I(M)$ generated by $K$-modules to ensure the resulting structures remain within the specific variety.
• The $\Lambda$-Construction:
  • For a $K$-module $M$ with a partial action $\alpha$, $\Lambda(M)$ is defined as the quotient of the tensor product $K G \otimes M$ by a submodule generated by relations enforcing the partial action consistency: $\{g \otimes \alpha_h(m) - gh \otimes m\}$.
  • The authors extend this to algebras $A \in \mathcal{V}(I)$ by defining a multiplication on $\Lambda(A)$ and proving it satisfies the identities of the variety.
• Two-Pronged Approach:
  1. Ideal Case: When the partial action arises from an ideal partial representation (images are ideals), they construct a global algebra $\Lambda(A)$ where the partial action extends to a global action.
  2. Generalized Case: When the images are only subalgebras (not necessarily ideals), they construct a universal globalization $\Omega(A)$ by taking the free algebra $F_I(\Lambda(A))$ and quotienting by an ideal that enforces the algebraic structure of the original partial action.
• Categorical Framework: The authors define categories of partial covariant representations and "dilated" representations. They utilize adjoint functors to relate the category of partial representations to the category of global (dilated) representations.

3. Key Contributions and Results

A. Globalization of Partial Actions

• Ideal Partial Representations (Theorem 2.6):
  • If $\pi: G \to \text{End}_0(A)$ is an ideal partial representation on an algebra $A \in \mathcal{V}(I)$, the authors construct $\Lambda(A)$.
  • Result: $\Lambda(A)$ is an algebra in $\mathcal{V}(I)$ equipped with a global action $\Theta$ such that $(\Theta, \iota)$ is a universal globalization of the induced partial action.
  • Significance: This generalizes the classical result for unital associative algebras [20] to arbitrary non-associative varieties (e.g., Lie algebras).
• Generalized Partial Actions (Theorem 2.14):
  • For generalized partial actions (where domains are subalgebras), the authors construct $\Omega(A) = F_I(\Lambda(A))/J$.
  • Result: $(\theta, \nu)$ is a universal global action for the generalized partial action. If the action is globalizable, this construction yields a globalization.
  • Example 2.17: Demonstrates the interplay between the globalization of a Lie algebra $L$ and its universal enveloping algebra $U(L)$, showing how the constructions commute.

B. Covariant Representations and Adjoint Functors

• Definition: The paper defines covariant representations for partial representations $\lambda: G \to \text{End}_0(A)$, consisting of a module $M$ and a partial representation $\pi$ satisfying compatibility conditions (e.g., $\pi_g(xm) = \lambda_g(x)\pi_g(m)$).
• The Functor $\Lambda$: The $\Lambda$-construction lifts a partial covariant representation $(\pi, M)$ to a global covariant representation $(\Lambda(\pi), \Lambda(M))$ of the globalized algebra $\Lambda(A)$.
• Dilated Representations: The authors introduce a category of dilated covariant representations, which are global representations equipped with an idempotent operator $T$ that recovers the partial structure.
• Adjointness (Theorem 3.9):
  • They prove that the functor $\Lambda: \text{CovRep}_\lambda(A) \to \text{Dil}_\lambda(A)$ is left adjoint to the restriction functor $F: \text{Dil}_\lambda(A) \to \text{CovRep}_\lambda(A)$.
  • This establishes the universal property of the globalization at the level of representations, clarifying that globalizations are the "most efficient" way to extend partial representations to global ones.

C. Semidirect Products of Lie Algebras

• Commutativity with Semidirect Products (Theorem 4.2):
  • The authors verify that the $\Lambda$-construction commutes with the formation of semidirect products of Lie algebras.
  • Result: If $L$ acts by derivations on $M$, and $\lambda, \pi$ form a partial covariant representation, then:
    $$\Lambda(L \ltimes M) \cong \Lambda(L) \ltimes \Lambda(M)$$
  • The induced global actions on the semidirect products are naturally isomorphic. This is crucial for applications in Lie theory where structures are often built via extensions.

4. Significance

• Unification: The paper provides a unified framework for partial actions across a vast class of algebraic structures (associative, Lie, Jordan, alternative, etc.) by working within the abstract setting of varieties $\mathcal{V}(I)$.
• Structural Integrity: It proves that the globalization process preserves the specific algebraic identities of the variety, ensuring the globalized object remains within the same class of algebras.
• Representation Theory: By establishing the adjoint pair of functors, the paper bridges the gap between partial and global representation theories, offering a systematic method to "dilate" partial symmetries into global ones.
• Applications: The results on semidirect products and the specific handling of Lie algebras open avenues for studying partial symmetries in differential geometry, quantum groups, and non-associative physics models where global symmetries are broken or restricted.

In summary, the paper successfully extends the theory of partial group actions from associative algebras and modules to the broader and more complex realm of non-associative algebras, providing rigorous existence theorems for globalizations and a deep categorical understanding of their representations.