On the representability of actions of unital algebras

Technical Summary: On the Representability of Actions of Unital Algebras

Problem Statement
The paper investigates the representability of internal actions within the context of unital non-associative algebras over a field. In categorical algebra, a category is action representable if the functor of split extensions (or internal actions) on an object $X$ is naturally isomorphic to the hom-functor $\text{Hom}(-, [X])$, where $[X]$ is an object called the "actor." While this property holds for groups (where $[X] = \text{Aut}(X)$) and Lie algebras, it is known to fail for many varieties of non-associative algebras, such as commutative associative algebras.

Previous work established that in the setting of semi-abelian categories (which include non-unital algebras), action representability is a highly restrictive property, essentially characterizing only abelian and Lie algebras. The authors address the question of whether this restriction persists when restricting attention to unital algebras. Specifically, they examine the ideally exact categories of unital algebras (which are generally not pointed, unlike semi-abelian categories) to determine if the presence of a multiplicative unit restores action representability.

Methodology
The authors employ a combination of categorical algebra and universal algebra techniques:

1. Ideally Exact Categories: They work within the framework of ideally exact categories (introduced by G. Janelidze), which generalize semi-abelian categories to include non-pointed settings like the category of unital rings ($\text{Ring}$) and unital commutative rings ($\text{CRing}$). Action representability in this context is defined relative to the semi-abelian slice category $(\mathcal{U} \downarrow 0)$.
2. External Weak Actors: For operadic, action accessible varieties of non-associative algebras, the authors utilize the construction of the external weak actor $E(X)$. This is a partial algebra constructed from pairs of endomorphisms satisfying specific identities (the $\lambda/\mu$-rules) derived from the variety's defining equations.
3. The Canonical Map: A central tool is the canonical partial algebra homomorphism $\text{Inn}: X \to E(X)$, defined by mapping an element $x$ to its left and right multiplication operators $(L_x, R_x)$.
4. Unit-Closed Varieties: The analysis focuses on unit-closed varieties, where adjoining an external unit to an algebra preserves membership in the variety. This allows the authors to relate properties of unital algebras back to the underlying non-unital variety.

Key Contributions and Results

• Failure of Accessibility in General Unital Categories: The paper demonstrates that, in general, categories of unital algebras are not even action accessible (a weaker condition than representability). Specifically, Proposition 4.11 proves that the categories of unital commutative algebras ($\text{Com}_1$) and unital non-associative algebras ($\text{Alg}_1$) fail to be action accessible. This is shown by constructing a specific 6-dimensional unital commutative algebra where the split extensions do not admit morphisms into faithful objects.

• Characterization of Unitality via the External Weak Actor: For an operadic, action accessible, and unit-closed variety $\mathcal{V}$, the authors prove a crucial equivalence (Theorem 5.6): For any algebra $X$ in $\mathcal{V}$, the canonical map $\text{Inn}: X \to E(X)$ is an isomorphism if and only if $X$ is a unital algebra.

  • If $X$ is unital, the kernel of $\text{Inn}$ (the annihilator) is trivial, and the existence of a unit allows every element of the external weak actor to be uniquely determined by its action on the unit, ensuring surjectivity.
  • If $\text{Inn}$ is an isomorphism, the inverse image of the identity pair $(\text{id}_X, \text{id}_X)$ provides a multiplicative unit for $X$.

• Main Theorem on Action Representability: The central result (Theorem 5.7) states that if $\mathcal{V}$ is an operadic, action accessible, and unit-closed variety, then the ideally exact category $\mathcal{V}_1$ of unital algebras in $\mathcal{V}$ is action representable. Furthermore, the actor of any unital algebra $X$ is $X$ itself.

  • This implies that for such varieties, the functor of split extensions of a unital algebra $X$ is representable by $X$.
  • This result applies to unital associative algebras, unital commutative associative algebras, unital alternative algebras, and unital Jacobi-Jordan algebras (in characteristic 3).

• Extension to Poisson Algebras: The authors extend these findings to unital Poisson algebras ($\text{Pois}_1$) and unital commutative Poisson algebras ($\text{CPois}_1$). By utilizing the explicit construction of the universal strict general actor (USGA) for Orzech categories of interest, they prove (Theorems 6.7 and 6.8) that these categories are action representable, with the actor of a unital Poisson algebra being the algebra itself.

Significance and Claims
The paper claims to resolve a significant contrast between the non-unital and unital settings in non-associative algebra. While action representability in the semi-abelian (non-unital) setting is rare (limited essentially to Lie algebras), the introduction of a multiplicative unit in operadic, action accessible, unit-closed varieties restores this property universally.

The authors highlight that for unital algebras in these specific varieties, the "actor" is not a complex external object (like a bimultiplier algebra or derivation algebra) but the algebra itself. This simplifies the categorical structure of actions for unital algebras, suggesting that the presence of a unit fundamentally alters the behavior of split extensions and internal actions, making them representable in a direct and intrinsic manner. The work also provides a concrete method for verifying action representability in new varieties by checking the unit-closed property and the existence of $\lambda/\mu$-rules.