Dynamical Skew-Braces

• Dynamical skew-braces are algebraic structures combining a fixed group operation with a family of parameterized quasigroup operations governed by a transition map.
• They yield set-theoretic dynamical Yang–Baxter solutions and offer reformulations in quiver-theoretic and braided groupoid frameworks.
• The theory extends ordinary skew braces by incorporating a parameter space, enabling parallelization of braided groupoids and the derivation of combinatorial invariants.

Dynamical skew-braces are brace-like structures in which the second operation varies over a parameter set and the parameter itself is updated by a transition map. In the formulation recalled by Anai and developed in the groupoid–quiver setting, a dynamical skew brace is a datum

$$(A,\Lambda,\phi,\cdot,\{\bullet_\lambda\}_{\lambda\in\Lambda})$$

where $(A,\phi)$ is a dynamical set over $\Lambda$, $(A,\cdot)$ is a group, and each $(A,\bullet_\lambda)$ is a left quasigroup satisfying the axioms $(\sc da)$ and $(\sc bc)$. These objects produce solutions of the set-theoretic dynamical Yang–Baxter equation, admit a quiver-theoretic reformulation, and, under zero-symmetry, yield skew bracoids or braided groupoids; conversely, every connected braided groupoid can be parallelised into a zero-symmetric dynamical skew brace (Ferri, 2024).

1. Definition and basic algebraic framework

A dynamical set over a nonempty set $\Lambda$ is a pair $(X,\phi)$ with $X$ a set and

$(A,\phi)$0

the transition map. A morphism $(A,\phi)$1 is a family $(A,\phi)$2 such that

$(A,\phi)$3

The category $(A,\phi)$4 is monoidal; for dynamical sets $(A,\phi)$5, $(A,\phi)$6, the tensor product has underlying set $(A,\phi)$7 and transition map

$(A,\phi)$8

A dynamical Yang–Baxter map on $(A,\phi)$9 is a morphism $\Lambda$0 in $\Lambda$1 satisfying the braid relation in that monoidal category (Ferri, 2024).

A dynamical skew brace is then a datum

$\Lambda$2

consisting of a dynamical set $\Lambda$3, a group $\Lambda$4, and a family of left quasigroup structures $\Lambda$5 such that, for all $\Lambda$6 and $\Lambda$7, \begin{align} a\bullet_{\lambda}(b\bullet_{\phi(\lambda,a)} c) &= (a\bullet_{\lambda} b)\bullet_{\lambda} c, \tag{\sc da}\ a\bullet_\lambda(b\cdot c) &= (a\bullet_\lambda b)\cdot a^{-1}\cdot( a\bullet_\lambda c). \tag{\sc bc} \end{align} A morphism $\Lambda$8 is a morphism of dynamical sets $\Lambda$9 such that

$(A,\cdot)$0

If $(A,\cdot)$1 is abelian, one obtains a dynamical brace. The notation $(A,\cdot)$2 denotes left division for $(A,\cdot)$3, and the standard brace computations recorded in the theory are

$(A,\cdot)$4

Compared with an ordinary skew brace, where one has a single second group law satisfying

$(A,\cdot)$5

the dynamical version replaces the fixed operation by a $(A,\cdot)$6-indexed family and couples it to parameter transport through $(A,\cdot)$7 (Guarnieri et al., 2015).

2. Holomorph description and dynamical subgroups

A group $(A,\cdot)$8 equipped with a dynamical set structure $(A,\cdot)$9 is called a dynamical group of bundle type if

$(A,\bullet_\lambda)$0

for all $(A,\bullet_\lambda)$1 and $(A,\bullet_\lambda)$2. This condition is the compatibility needed to place the group law inside the monoidal structure of $(A,\bullet_\lambda)$3 (Ferri, 2024).

Let $(A,\bullet_\lambda)$4 be the holomorph. A dynamical subgroup of $(A,\bullet_\lambda)$5 is a family

$(A,\bullet_\lambda)$6

such that for every $(A,\bullet_\lambda)$7,

$(A,\bullet_\lambda)$8

It is called unital if $(A,\bullet_\lambda)$9 for all $(\sc da)$0, and regular if each projection $(\sc da)$1 is bijective.

The Matsumoto–Anai correspondence identifies dynamical skew brace structures on $(\sc da)$2 with regular dynamical subgroups of $(\sc da)$3. Given a dynamical skew brace, one defines

$(\sc da)$4

and then

$(\sc da)$5

Conversely, if $(\sc da)$6 is regular, the unique $(\sc da)$7 such that $(\sc da)$8 determines the brace law by

$(\sc da)$9

This correspondence is the dynamical analogue of the static holomorph philosophy for ordinary skew braces: the brace structure is encoded by regular subgroup data, but now fiberwise over $(\sc bc)$0 rather than on a single set (Ferri, 2024).

3. Quiver-theoretic reformulation and Yang–Baxter braidings

The Matsumoto–Shimizu functor

$(\sc bc)$1

sends a dynamical set $(\sc bc)$2 to the quiver $(\sc bc)$3 with

$(\sc bc)$4

The notation

$(\sc bc)$5

is used for the arrow with source $(\sc bc)$6 and label $(\sc bc)$7, and

$(\sc bc)$8

for paths (Ferri, 2024).

From a dynamical skew brace

$(\sc bc)$9

one obtains a left unital associative semiloopoid $\Lambda$0 by defining

$\Lambda$1

For each vertex $\Lambda$2, the outgoing star $\Lambda$3 carries a group law

$\Lambda$4

This makes $\Lambda$5 into a quiver-theoretic skew brace, with compatibility

$\Lambda$6

The associated braiding is defined by

$\Lambda$7

and

$\Lambda$8

Theorem 2 of the quiver-theoretic theory states that this $\Lambda$9 is a left non-degenerate braiding on $(X,\phi)$0, and it is involutive precisely when the structure is of abelian type. In arrow notation the braiding produced by a dynamical skew brace is

$(X,\phi)$1

The paper also formulates an equivalent language of post-semiloopoids, and quiver-theoretic skew braces are shown to be equivalent to post-semiloopoids. This places dynamical skew-braces inside a broader quiver/groupoid version of brace theory rather than treating them solely as parameter-dependent set-theoretic objects (Ferri, 2024).

4. Zero-symmetry, skew bracoids, and parallelisation of braided groupoids

The decisive technical hypothesis is zero-symmetry. In a dynamical skew brace it appears as

$(X,\phi)$2

for all $(X,\phi)$3 and $(X,\phi)$4. The same condition is equivalent to unitality of the associated dynamical subgroup $(X,\phi)$5, to the existence of loops at every vertex, and to the statement that the associated quiver is a groupoid rather than merely a left unital associative semiloopoid (Ferri, 2024).

Under zero-symmetry, the quiver-theoretic skew brace becomes a skew bracoid. Since skew bracoids are equivalent to braided groupoids, the associated dynamical Yang–Baxter solution is then a braided groupoid in the sense of the quiver/groupoid theory. The braided groupoid axioms are encoded by a quiver morphism

$(X,\phi)$6

satisfying the hexagon relations and

$(X,\phi)$7

The converse theorem is the structural core of the modern theory: every connected braided groupoid can be parallelised into a zero-symmetric dynamical skew brace. The key device is a Matsumoto–Shimizu labelling, namely a family of bijections

$(X,\phi)$8

such that the transported law

$(X,\phi)$9

is independent of $X$0. The proof proceeds by choosing a base vertex, selecting a maximal Schurian subgroupoid $X$1 that is a groupoid of pairs, and transporting labels along preferred arrows via the $X$2-action. Connectedness is essential here, because it forces all stars $X$3 to have the same cardinality and permits a global identification with one set $X$4.

The resulting conclusion is twofold. First, every connected braided groupoid is obtained from a zero-symmetric dynamical skew brace. Second, every quiver-theoretic skew brace is isomorphic to a disconnected union of dynamical skew braces. For connected objects, the paper states an equivalence

$X$5

The labelling is not canonical, so the equivalence is structural rather than unique. The limits of zero-symmetry are also explicit: Example 5.3 for $X$6 shows a maximal dynamical skew brace with initial vertices, so not every dynamical skew brace produces a groupoid (Ferri, 2024).

5. Combinatorics, maximal families, and explicit invariants

For any group $X$7, the maximal zero-symmetric dynamical skew brace is built from

$X$8

Its cardinality is

$X$9

The associated quiver is homogeneous of weight $(A,\phi)$00: every connected component is a complete quiver of some degree $(A,\phi)$01, and if a component has $(A,\phi)$02 vertices then

$(A,\phi)$03

This leads to integers $(A,\phi)$04, where $(A,\phi)$05 counts the connected components with $(A,\phi)$06 vertices. They satisfy

$(A,\phi)$07

A particularly important case is $(A,\phi)$08: it counts skew brace operations on the fixed group $(A,\phi)$09, since one-point components are exactly ordinary skew brace structures on $(A,\phi)$10 (Ferri, 2024).

For the full maximal dynamical skew brace $(A,\phi)$11, not necessarily zero-symmetric, the initial-vertex counts are also explicit. If a connected component of $(A,\phi)$12 has size $(A,\phi)$13, then the number of initial vertices feeding into it is

$(A,\phi)$14

The paper also defines, for a vertex $(A,\phi)$15, a partition invariant $(A,\phi)$16 obtained by counting how often each automorphism $(A,\phi)$17 occurs in pairs $(A,\phi)$18. This partition depends only on the connected component, although it is not a complete invariant.

Some explicit values are known.

Group $(A,\phi)$19	Nonzero $(A,\phi)$20	Remarks
$(A,\phi)$21	$(A,\phi)$22	One-point component is $(A,\phi)$23
$(A,\phi)$24	$(A,\phi)$25	Used to illustrate noncanonical labellings
$(A,\phi)$26	$(A,\phi)$27	$(A,\phi)$28 prime
$(A,\phi)$29	$(A,\phi)$30	Computed in GAP

These examples show that the combinatorics of dynamical skew-braces is already nontrivial for very small groups. They also show that the zero-symmetric theory is richer than the enumeration of ordinary skew braces alone, because higher $(A,\phi)$31 record genuinely dynamical connected components beyond the isolated one-vertex cases (Ferri, 2024).

6. Relation to ordinary skew braces and adjacent “dynamical” generalisations

Ordinary skew braces provide the static starting point. A skew left brace is a set with two group structures $(A,\phi)$32 and $(A,\phi)$33 satisfying

$(A,\phi)$34

or equivalently a homomorphism

$(A,\phi)$35

and they produce non-degenerate set-theoretic Yang–Baxter solutions (Guarnieri et al., 2015). Dynamical skew braces retain the brace philosophy but alter its form: the multiplicative group $(A,\phi)$36 remains global, while the second operation becomes a family $(A,\phi)$37 indexed by $(A,\phi)$38, and the parameter evolves through $(A,\phi)$39 (Ferri, 2024).

Several nearby theories are dynamic only in a weaker or different sense. Skew brace actions involve a parameter space $(A,\phi)$40 and a family $(A,\phi)$41, yielding reflection-equation solutions, but this is an action theory rather than a theory of dynamical skew braces (Commer, 2018). The family

$(A,\phi)$42

attached to a fixed $(A,\phi)$43 in a left skew brace gives a parameter-dependent family of Yang–Baxter solutions, and in the two-sided case every $(A,\phi)$44 is admissible, but the paper does not define a dynamical skew brace or a parameter-update law (Doikou et al., 2022). Di-skew braces enlarge the additive side from a group to a generalized digroup and decompose their solutions over an idempotent fiber $(A,\phi)$45, which is strongly “dynamical in spirit,” yet that theory also explicitly does not define a dynamical skew brace or study the dynamical Yang–Baxter equation (Albano et al., 21 May 2025).

The closest neighboring enlargement is the theory of skew bracoids. A skew bracoid is a groupoid equipped with local group structures on outgoing stars satisfying the bracoid compatibility law, and earlier work treated them as action-based enlargements of skew braces (Martin-Lyons et al., 2023). Later work showed that certain skew bracoids containing a skew brace correspond to left cancellative semibraces (Colazzo et al., 2024). The distinctive contribution of the dynamical theory is to identify connected braided groupoids, hence connected skew bracoids, with zero-symmetric dynamical skew braces via parallelisation (Ferri, 2024). This sharpens the terminology: “dynamical” can refer either to genuine parameter-dependent brace laws on dynamical sets, or more loosely to internal fibers, actions, or parameter families in adjacent brace-like structures. Only the former is dynamical skew-brace theory in the strict sense.

The resulting picture is a three-way dictionary. Dynamical skew braces live in $(A,\phi)$46, quiver-theoretic skew braces and semiloopoids live in $(A,\phi)$47, and zero-symmetric connected cases coincide with braided groupoids or skew bracoids. This dictionary is the main structural achievement of the current theory and marks dynamical skew-braces as a distinct branch of brace theory rather than a merely suggestive metaphor (Ferri, 2024).