Hom-Braces: Hom-Algebraic Extensions

• Hom-Braces are Hom-type analogues of classical braces, combining an additive group with a Hom-group multiplicative structure and twisting maps that modify distributive laws.
• They are categorized in three types, with type (0) retaining classical additive behavior and types (1) and (2) introducing dual twisting maps for deeper algebraic flexibility.
• Their correspondence with Hom-trusses and links to Yang–Baxter theory make them a promising structure in non-associative geometry and categorical algebra.

Searching arXiv for Hom-braces and related brace literature to ground the article in current papers. Searching arXiv for “Hom-brace”, “Hom-braces”, and closely related brace/truss papers. Hom-braces are Hom-type analogues of classical braces and skew braces, developed as part of a broader Hom-algebraic extension of heap and truss theory. In the framework introduced in "On Hom-Analogues of Heaps and Trusses" (Anowar et al., 1 Sep 2025), they are defined by equipping a set with an “additive” structure, a “multiplicative” Hom-group structure, and twisting maps satisfying Hom-compatible distributive identities. The paper formulates three variants—type $(0)$, type $(1)$, and type $(2)$—and establishes their correspondence with Hom-trusses, thereby placing Hom-braces within a systematic algebraic framework connected to Yang--Baxter theory, non-associative geometry, and categorical algebra (Anowar et al., 1 Sep 2025).

1. Definition and basic variants

The foundational source distinguishes three notions. A Hom–skew left brace of type $(0)$ is a triple $(B,\star,\bullet,\alpha)$ such that $(B,\star)$ is a group, $(B,\bullet,\alpha)$ is a Hom-group with bijective twisting map $\alpha$, the map $\alpha$ is multiplicative for $\star$, and the twisted left distributive law

$(1)$0

holds for all $(1)$1, where $(1)$2 denotes the inverse of $(1)$3 in $(1)$4 (Anowar et al., 1 Sep 2025).

A Hom–brace of type $(1)$5 is obtained by strengthening the additive side: $(1)$6 must be abelian and the analogous right distributive law

$(1)$7

must also hold (Anowar et al., 1 Sep 2025). The construction is explicitly arranged so that setting $(1)$8 recovers the classical skew brace or brace, depending on whether the additive group is nonabelian or abelian (Anowar et al., 1 Sep 2025).

The higher variants introduce two twisting maps. A Hom–brace of type $(1)$9 is a quintuple $(2)$0 in which $(2)$1 is an abelian Hom-group, $(2)$2 is a Hom-group, $(2)$3 is an abelian Hom-group automorphism of $(2)$4, $(2)$5 is multiplicative for $(2)$6, and two twisted distributive identities hold: $(2)$7

$(2)$8

A Hom–brace of type $(2)$9 uses the same ambient data but replaces these by more symmetric “level-2” identities involving $(0)$0 and $(0)$1 iterates (Anowar et al., 1 Sep 2025).

The three notions are not arbitrary variations. They are designed to capture different levels of Hom-twisting while retaining the brace-style interaction between an additive and a multiplicative structure. The data and axioms also make clear that the additive and multiplicative parts are no longer treated symmetrically: in type $(0)$2, the additive side is classical while the multiplicative side is Hom; in types $(0)$3 and $(0)$4, both sides carry Hom-structure, but with potentially distinct twisting maps (Anowar et al., 1 Sep 2025).

2. Hom-group background and internal structure

The multiplicative part of a Hom-brace is a Hom-group in the sense used in (Anowar et al., 1 Sep 2025). Thus $(0)$5 satisfies Hom-associativity and multiplicativity,

$(0)$6

together with a unit $(0)$7 satisfying

$(0)$8

and twisted inverses in the usual sense (Anowar et al., 1 Sep 2025). For type $(0)$9 and type $(B,\star,\bullet,\alpha)$0, the additive structure is also a Hom-group, and in that case it is required to be abelian (Anowar et al., 1 Sep 2025).

A basic structural remark established in the paper is that twisting commutes with taking inverses in the additive part: $(B,\star,\bullet,\alpha)$1 This compatibility is elementary but important: it ensures that the distributive identities are stable under the twisting maps and that the usual brace-style manipulations remain meaningful in the Hom setting (Anowar et al., 1 Sep 2025).

The paper also proves an equivalence theorem for the higher variants. When $(B,\star,\bullet,\alpha)$2, the type $(B,\star,\bullet,\alpha)$3 and type $(B,\star,\bullet,\alpha)$4 definitions coincide; equivalently, $(B,\star,\bullet,\alpha)$5-Hom-braces of type $(B,\star,\bullet,\alpha)$6 and type $(B,\star,\bullet,\alpha)$7 are the same structures. Moreover, setting $(B,\star,\bullet,\alpha)$8 yields exactly Hom-braces of type $(B,\star,\bullet,\alpha)$9 (Anowar et al., 1 Sep 2025). This result shows that the distinction between level-1 and level-2 formulations is largely one of presentation when the two twisting maps are identified.

A plausible implication is that the two-twist formalism is most meaningful when $(B,\star)$0 and $(B,\star)$1 are genuinely distinct. In the one-twist case, the theory collapses to a single coherent Hom-brace notion, which is structurally closer to classical brace theory.

3. Relation to classical braces and skew braces

Hom-braces are introduced as Hom-analogues of classical braces and skew braces. The recovery statement is explicit: setting the twisting maps equal to the identity recovers the classical structures (Anowar et al., 1 Sep 2025). In that sense, Hom-braces are not a parallel theory but a deformation or twisting of the established brace formalism.

Classical skew braces are recalled in the literature as sets endowed with two group laws $(B,\star)$2 satisfying the brace identity

$(B,\star)$3

with the classical brace obtained when $(B,\star)$4 is abelian (Bardakov et al., 2020). The compatibility in type $(B,\star)$5 Hom-braces is formally analogous, but one of the group structures is replaced by a Hom-group and the interaction is expressed through twisted distributive laws rather than the untwisted classical identity (Anowar et al., 1 Sep 2025).

The relation is especially clear in the type $(B,\star)$6 definition. There, the additive structure $(B,\star)$7 remains an ordinary group, while $(B,\star)$8 is a Hom-group. The left distributive identity

$(B,\star)$9

is the direct Hom-analogue of the usual skew-brace law, with the multiplicative side carrying the Hom deformation (Anowar et al., 1 Sep 2025). When $(B,\bullet,\alpha)$0 is abelian and the right-hand analogue is added, the result matches the classical left-and-right brace pattern in a twisted setting.

A separate branch of the literature studies $(B,\bullet,\alpha)$1-homomorphic braces, sometimes also labeled “Hom-Brace” in the sense that the map

$(B,\bullet,\alpha)$2

is a group homomorphism (Nasybullov et al., 2024). This terminology is distinct from the Hom-algebraic use of “Hom-brace” in (Anowar et al., 1 Sep 2025). In the $(B,\bullet,\alpha)$3-homomorphic setting, one remains within classical brace theory and imposes an additional homomorphism condition on the $(B,\bullet,\alpha)$4-map; in the Hom-algebraic setting, the primary deformation lies in the replacement of associativity by Hom-associativity and the introduction of twisting maps (Bardakov et al., 2020, Nasybullov et al., 2024). The terminological overlap is therefore a source of possible confusion.

4. Correspondence with Hom-trusses

A central theorem of the theory is that Hom-braces and Hom-trusses are “two sides of the same coin” (Anowar et al., 1 Sep 2025). This correspondence is formulated in several versions.

For a unital Hom-truss of type $(B,\bullet,\alpha)$5,

$(B,\bullet,\alpha)$6

with multiplicative unit $(B,\bullet,\alpha)$7, the paper defines an additive operation by

$(B,\bullet,\alpha)$8

With this operation, $(B,\bullet,\alpha)$9 becomes a Hom-brace of type $\alpha$0 (Anowar et al., 1 Sep 2025). Conversely, any Hom-brace of type $\alpha$1 gives rise to a unital Hom-truss of type $\alpha$2 through

$\alpha$3

with the same $\alpha$4 and $\alpha$5 (Anowar et al., 1 Sep 2025).

The same pattern extends to higher types. For an idempotent Hom-truss of type $\alpha$6 or type $\alpha$7 with unit $\alpha$8, the additive operation is again

$\alpha$9

and the twisting maps are unchanged. The Hom-heap axioms together with Hom-distributivity imply the relevant brace axioms. Conversely, a Hom-brace of type $\alpha$0 or type $\alpha$1, idempotent in the sense that $\alpha$2, yields a unital Hom-truss of the corresponding kind by setting

$\alpha$3

with $\alpha$4 retained (Anowar et al., 1 Sep 2025).

This correspondence is structurally significant because it places Hom-braces within the larger heap–truss–brace web. In the classical setting, trusses generalize rings by replacing the additive group with a heap; the Hom version suggests a similar generalization under twisting. A plausible implication is that many constructions may be transferred between the truss and brace sides with little loss of information, provided the required unital or idempotent hypotheses are present.

5. Examples and constructions

The foundational paper does not provide a fully worked-out nontrivial Hom-brace from scratch, but it explains that the brace–truss correspondence allows one to recycle nontrivial Hom-truss examples (Anowar et al., 1 Sep 2025). The basic example is derived from the classical odd-integers truss: $\alpha$5 This is a unital Hom-truss of type $\alpha$6, indeed an ordinary truss, with unit $\alpha$7 for $\alpha$8 (Anowar et al., 1 Sep 2025).

Defining

$\alpha$9

one obtains

$\star$0

and $\star$1 is a Hom-group, in fact a group (Anowar et al., 1 Sep 2025). The left- and right-distributive laws become

$\star$2

which reduce to the familiar identity $\star$3. Hence

$\star$4

is a Hom-brace of type $\star$5 (Anowar et al., 1 Sep 2025).

The paper adds that one may similarly twist by any truss-automorphism $\star$6 to produce a non-trivial Hom-brace with genuine $\star$7 (Anowar et al., 1 Sep 2025). This suggests that examples are expected to arise naturally by transporting known truss constructions through automorphism twisting rather than by ad hoc direct construction.

For context, classical and $\star$8-homomorphic brace theory provides many explicit constructions. Bardakov, Neshchadim, and Yadav construct $\star$9-homomorphic skew braces when the additive group is either a free group or a free abelian group (Bardakov et al., 2020), and Nasybullov and Novikov give a complete classification of $(1)$00-homomorphic braces on $(1)$01 via commuting matrices $(1)$02 (Nasybullov et al., 2024). These results are not Hom-braces in the Hom-algebraic sense of (Anowar et al., 1 Sep 2025), but they delineate a nearby constructional landscape from which future Hom-analogues may plausibly emerge.

6. Modules, categories, and research directions

The paper does not develop a module theory for Hom-braces directly. It states instead that, by the brace–truss correspondence, any module over the underlying Hom-truss—left, right, or bimodule in the sense of Section 5 of the paper—can be transported to a module over the associated Hom-brace (Anowar et al., 1 Sep 2025). This is a transfer principle rather than a standalone intrinsic module theory.

Categorically, Hom-braces of a fixed type form a category whose forgetful functor to Hom-groups, via the multiplicative part, is faithful. The paper further states that the left adjoint, when it exists, can be described by a “free brace on a Hom-group” construction using the corresponding free truss (Anowar et al., 1 Sep 2025). This positions Hom-braces within categorical algebra not merely as isolated algebraic gadgets but as objects participating in adjunction-based free constructions.

The applications highlighted are prospective rather than fully developed. The paper emphasizes three motivations. First, in Yang--Baxter theory, classical skew braces classify nondegenerate involutive set-theoretic solutions of the Yang--Baxter equation, and Hom-braces should similarly classify twisted solutions when one allows a self-map $(1)$03 (Anowar et al., 1 Sep 2025). Second, in non-associative geometry, Hom-braces may serve as multiplicative analogues controlling the “group-like” symmetries related to Hom-Lie affgebras (Anowar et al., 1 Sep 2025). Third, in categorical algebra, Hom-braces sit at the crossroads of Hom-groups, Hom-racks, and Hom-rings, suggesting new monoidal categories and Hom-operad structures (Anowar et al., 1 Sep 2025).

These statements define the current frontier of the subject. The theory in (Anowar et al., 1 Sep 2025) is explicitly foundational: it establishes definitions, equivalences, and brace–truss correspondences, but leaves the systematic development of examples, representation theory, homological invariants, and Yang--Baxter applications largely open.

7. Terminological scope and adjacent meanings of “brace”

Within algebra, “brace” terminology is not unique. The Hom-braces of (Anowar et al., 1 Sep 2025) belong to the lineage of classical braces, skew braces, heaps, and trusses. By contrast, "The Homotopy Braces Formality Morphism" (Willwacher, 2011) concerns Braces algebras, the operad $(1)$04, and homotopy–braces or $(1)$05-morphisms in deformation quantization. There, a braces algebra is described via multilinear operations $(1)$06 and $(1)$07, the operad $(1)$08, and its minimal resolution $(1)$09 (Willwacher, 2011).

The two usages are mathematically distinct. In (Willwacher, 2011), “braces” refers to an operadic structure related to Gerstenhaber theory, Hochschild cochains, and Kontsevich–Tamarkin formality. In (Anowar et al., 1 Sep 2025), “Hom-brace” refers to a brace-like algebraic object built from group or Hom-group data with distributive identities. The shared word does not indicate a common definition.

This distinction matters in bibliographic and conceptual orientation. Hom-braces in the sense of Hom-algebra are part of the brace–truss–ring–Yang--Baxter web, whereas homotopy–braces belong to the operadic and deformation-quantization setting (Anowar et al., 1 Sep 2025, Willwacher, 2011). Keeping these meanings separate avoids conflating two unrelated research programs that happen to use the same noun.