Isotopes of Biracks and Zhang Twists of Algebras

Abstract

In this paper, we introduce the notion of an ${\mathbb{N}}^p$-graded birack and construct its isotope. Every involutive ${\mathbb{N}}^p$-graded birack gives rise to an ${\mathbb{N}}^p$-graded Yang-Baxter algebra. We study the relation between isotopes of involutive ${\mathbb{N}}^p$-graded biracks and Zhang twists of ${\mathbb{N}}^p$-graded Yang-Baxter algebras. As an example, Yang-Baxter algebras determined by distributive solutions are proved to be Zhang twists of polynomial algebras.

1. Introduction

The Yang-Baxter equation is a cornerstone of mathematical physics that arises in the study of integrable systems, quantum groups, and quantum information theory. Its origins trace back to foundational papers by Yang [1] and Baxter [2]. Let V be a vector space. A solution of the Yang-Baxter equation is a linear mapping $R:V\otimes V\to V\otimes V$ such that the equality

$$(id\otimes R)(R\otimes id)(id\otimes R)=(R\otimes id)(id\otimes R)(R\otimes id)$$

of operators holds in the space $End(V\otimes V\otimes V)$. The problem of describing all possible solutions is highly complex, so some simplifications have been introduced.

A particularly nice class of solutions is provided by set-theoretic solutions proposed by Drinfel’d [3]. Let X be a set and $r:X\times X\to X\times X$ a bijective mapping. r is called a set-theoretic solution of the Yang-Baxter equation if the following braid relation holds:

$$(id\times r)(r\times id)(id\times r)=(r\times id)(id\times r)(r\times id).$$

It is clear that a set-theoretic solution extends to a linear one, but, more importantly, set-theoretic solutions lead to their own remarkable algebraic and combinatorial structures. Set-theoretic solutions of the Yang-Baxter equation have found applications in several areas of theoretical and mathematical physics. For example, in the theory of integrable systems, they describe how multi-particle scattering factorizes into pairwise interactions satisfying the Yang-Baxter equation. More recently, involutive non-degenerate set-theoretic solutions have been applied in quantum information theory, where they are related to quantum entanglement and the construction of quantum gates via unitary representations of the braid group. Set-theoretic solutions of the Yang-Baxter equation also have interesting applications in several areas of mathematics, such as non-commutative algebra, knot theory, low-dimensional topology, and the theory of tensor categories. During the last three decades, the study of set-theoretic solutions and related structures has notably intensified; see, for instance, [4,5,6,7,8,9,10,11,12,13].

Among the various algebraic structures, skew braces have emerged as a particularly noteworthy concept. Skew braces were introduced in [12], inspired by the earlier works [14,15]. Every skew brace yields a non-degenerate set-theoretic solution of the Yang-Baxter equation (not necessarily involutive) [12]. Moreover, the theory of skew braces is closely connected to the study of indecomposable solutions and multipermutation solutions [8,16,17]. On the algebra side, each set-theoretic solution $(X,r)$ associates a quadratic Yang-Baxter algebra $A(X,r)$ as proposed in [10], Section 6. If X is finite and r is non-degenerate and involutive, then $A(X,r)$ has remarkable algebraic, homological, and combinatorial properties [11,18]. A natural question that arises is how to use the theory of Yang-Baxter algebras to study solutions of the Yang-Baxter equation. The motivation of this paper is to find more ways to study properties of Yang-Baxter algebras.

Deformation theory is very useful in the study of algebraic structures. In light of the deep connections between set-theoretic solutions of the Yang-Baxter equation and various algebraic structures, it is meaningful to investigate the relationships between the deformations of these algebraic structures.

It is known that there is a one-to-one correspondence between non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation and involutive biracks. A birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is an algebra such that $(X,\circ ,{\setminus }_{\circ })$ is a left quasigroup, $(X,•,{/}_{•})$ is a right quasigroup, and some additional identities are satisfied. Such a correspondence allows us to study solutions of the Yang-Baxter equations via involutive biracks.

In quasigroup theory (see, e.g., [19], Section II.2), isotopy is a standard method for transforming one quasigroup into another. In their work on involutive set-theoretic solutions of the Yang-Baxter equation of multipermutation level 2, Jedlička, Pilitowska, and Zamojska-Dzienio used a special form of isotope to construct non-distributive solutions from distributive solutions [20]. Based on their work, in Section 3, we introduce the notions of an ${\mathbb{N}}^p$-graded birack and its isotope and then discuss their properties. Our main theorem is presented in Section 4. Each involutive ${\mathbb{N}}^p$-graded birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ determines an ${\mathbb{N}}^p$-graded Yang-Baxter algebra $A(X,{\mathbf{R}}^{\circ })$. We find that, under some conditions, the algebra $A^{\ast }(X,{\mathbf{R}}^{\ast })$ determined by an isotope $(X,\ast ,{\setminus }_{\ast },\diamond ,{/}_{\diamond })$ of the birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is isomorphic to a Zhang twist of the algebra $A(X,{\mathbf{R}}^{\circ })$ (Theorem 1). The Zhang twist is a twist of graded algebras introduced by J.J. Zhang in order to describe graded algebras whose graded module categories are equivalent [21]. Many important properties of graded algebras, including the Gelfand–Kirillov dimension, and the Noetherian property, being a domain, global dimension, and Artin–Schelter regularity, are preserved under Zhang twist. In fact, our work is inspired by the observation that, in a certain sense, both Zhang twists and isotopes can be viewed as deformations that modify “left translations”.

We provide some examples. Especially, we show that the Yang-Baxter algebras arising from distributive solutions are Zhang twists of polynomial algebras. This class of algebras is believed to share properties similar to those of polynomial algebras. We will discuss them in detail in a subsequent paper.

2. Preliminaries

Throughout, $\mathbb{k}$ is a base field of characteristic zero. All vector spaces and algebras are over $\mathbb{k}$.

2.1. Quasigroups and Biracks

A left quasigroup is an algebra $(X,\circ ,{\setminus }_{\circ })$ with two binary operations ∘ and ${\setminus }_{\circ }$ such that, for every $x,y\in X$, the following conditions hold:

$$x\circ \left(x{\setminus }_{\circ }y\right){=y=x\setminus }_{\circ }(x\circ y).$$

(1)

A right quasigroup is defined analogously as an algebra $(X,•,{/}_{•})$ with two binary operations • and ${/}_{•}$ such that, for every $x,y\in X$, the following conditions hold:

$$\left(y{/}_{•}x\right)•x=y=(y•x){/}_{•}x.$$

(2)

Let $(X,\circ ,{\setminus }_{\circ })$ be a left quasigroup. For each $x\in X$, the left translation by x is denoted by ${\mathcal{L}}_x$, i.e., ${\mathcal{L}}_x\left(a\right)=x\circ a$ for any $a\in X$. Condition (1) guarantees that each ${\mathcal{L}}_x$ is bijective, with ${\mathcal{L}}_x^{-1}{\left(a\right)=x\setminus }_{\circ }a$. Similarly, Condition (2) guarantees that each right translation ${\mathcal{R}}_x$ by x is bijective with ${\mathcal{R}}_x^{-1}\left(a\right)=a{/}_{•}x$ for any $a\in X$.

For a left quasigroup $(X,\circ ,{\setminus }_{\circ })$, a bijection $f:X\to X$ is called an automorphism of $(X,\circ ,{\setminus }_{\circ })$ if

$$f(x\circ y)=f\left(x\right)\circ f\left(y\right)$$

for every $x,y\in X$. It is easy to see that a bijection $f:X\to X$ is an automorphism of $(X,\circ ,{\setminus }_{\circ })$ if and only if

$$f{\mathcal{L}}_x{f}^{-1}={\mathcal{L}}_{f\left(x\right)}$$

for every $x\in X$.

An automorphism of a right quasigroup can be defined analogously. A bijection $f:X\to X$ is an automorphism of $(X,•,{/}_{•})$ if and only if $f{\mathcal{R}}_x{f}^{-1}={\mathcal{R}}_{f\left(x\right)}$ for every $x\in X$.

A left quasigroup $(X,\circ ,{\setminus }_{\circ })$ is called

• non-degenerate if the mapping ${x\mapsto x\setminus }_{\circ }x$ is a bijection;
• right cyclic if, for every $x,y,z\in X$,

  $$\left({x\setminus }_{\circ }y\right){{\setminus }_{\circ }\left({x\setminus }_{\circ }z\right)=\left({y\setminus }_{\circ }x\right)\setminus }_{\circ }\left({y\setminus }_{\circ }z\right),$$

  or, equivalently, ${\mathcal{L}}_x{\mathcal{L}}_{{x\setminus }_{\circ }y}={\mathcal{L}}_y{\mathcal{L}}_{{y\setminus }_{\circ }x}$.

An algebra $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ with four binary operations is called a birack if $(X,\circ ,{\setminus }_{\circ })$ is a left quasigroup, $(X,•,{/}_{•})$ is a right quasigroup, and the following holds for any $x,y,z\in X$:

$$\begin{array}{cc}\hfill x\circ (y\circ z)& =(x\circ y)\circ \left(\right(x•y)\circ z),\hfill \\ \hfill (x\circ y)•\left(\right(x•y)\circ z)& =(x•(y\circ z\left)\right)\circ (y•z),\hfill \\ \hfill (x•y)•z& =(x•(y\circ z\left)\right)•(y•z).\hfill \end{array}$$

If the set X is finite, we call the birack finite.

Example 1.

Let X be a non-empty set. An algebra $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ such that, for every $x,y\in X$,

$${x\circ y=y,x\setminus }_{\circ }y=y,$$

$$x•y=x,x{/}_{•}y=x$$

is certainly a birack. It is called a projection birack.

Let $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ be a birack.

• It is called involutive if it additionally satisfies

  $$(x\circ y)\circ (x•y)=xand(x\circ y)•(x•y)=y$$

  for any $x,y\in X$.
• It is said to satisfy Condition lri provided that

  $$(x\circ y)•x=y=x\circ (y•x),$$

  for any $x,y\in X$. That is, ${\mathcal{R}}_x={\mathcal{L}}_x^{-1}$ for all $x\in X$.

Remark 1.

An involutive birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ satisfies Condition lri if and only if

$${(x\circ y)\setminus }_{\circ }{x=y\setminus }_{\circ }x,$$

(3)

for any $x,y\in X$. In fact, in an involutive birack, ${x•y=(x\circ y)\setminus }_{\circ }x$. Hence,

$${\mathcal{R}}_y\left(x\right)={\mathcal{L}}_y^{-1}\left(x\right)\iff (x\circ y){{\setminus }_{\circ }x=y\setminus }_{\circ }x.$$

The following one-to-one correspondence between involutive biracks and non-degenerate right cyclic left quasigroups is well-known (see, e.g., [22], Proposition 1.5).

Lemma 1.

(1) 

Let $\left({X,\circ ,\setminus }_{\circ },•,{/}_{•}\right)$ be an involutive birack. Then, $(X,\circ ,{\setminus }_{\circ })$ is a non-degenerate right cyclic left quasigroup.

(2) 

Conversely, let $(X,\circ ,{\setminus }_{\circ })$ be a non-degenerate right cyclic left quasigroup. For every $x,y\in X$, define

$$x•y={\mathcal{R}}_y{\left(x\right)=(x\circ y)\setminus }_{\circ }x\mathit{and}x{/}_{•}y={\mathcal{R}}_y^{-1}\left(x\right).$$

Then, the algebra $\left({X,\circ ,\setminus }_{\circ },•,{/}_{•}\right)$ is an involutive birack.

The operations ${\setminus }_{\circ }$ and ${/}_{•}$ are connected by

$$\left({x\setminus }_{\circ }x\right){/}_{•}\left({x\setminus }_{\circ }x\right)=x\mathrm{and}\left(x{/}_{•}x\right){\setminus }_{\circ }\left(x{/}_{•}x\right)=x.$$

2.2. Set-Theoretic Solution of the Yang-Baxter Equation

By a quadratic set, we mean a pair $(X,r)$, where X is a nonempty set and $r:X\times X\to X\times X$ is a bijective mapping. We write the image of $(x,y)$ under r as

$$r(x,y)=\left({}^{x}y,x^y\right).$$

The above equation defines two binary operations on X:

• a left action $\mathcal{L}:X\times X\to X$ given by ${\mathcal{L}}_x\left(y\right)={}^{x}y$;
• a right action $\mathcal{R}:X\times X\to X$ given by ${\mathcal{R}}_y\left(x\right)=x^y$,

• for all $x,y\in X$. In other words, the maps $\mathcal{L}$ and $\mathcal{R}$ are precisely the first and second components of r, respectively. That is,

  $$r(x,y)=\left({\mathcal{L}}_x\left(y\right),{\mathcal{R}}_y\left(x\right)\right).$$

A quadratic set $(X,r)$ is called

• non-degenerate if the mappings ${\mathcal{L}}_x$ and ${\mathcal{R}}_x$ are bijections for all $x\in X$;
• involutive if $r^2={id}_{X^2}$;
• square-free if $r(x,x)=(x,x)$ for every $x\in X$;
• a quantum binomial set if it is non-degenerate, involutive, and square-free.

A set-theoretic solution of the Yang-Baxter equation (YBE) is a quadratic set $(X,r)$ such that the braid relation holds:

$$(id\times r)(r\times id)(id\times r)=(r\times id)(id\times r)(r\times id).$$

In this case, $(X,r)$ is also called a braided set.A solution is called non-degenerate (involutive, or square-free) if it is non-degenerate (involutive, or square-free) as a quadratic set. If X is finite, then $(X,r)$ is called a finite solution.

• Convention In the following, by a solution (of the YBE), we mean a set-theoretic, non-degenerate, and involutive solution of the YBE.

Each quadratic set $(X,r)$ determines a set of quadratic defining relations $\mathbf{R}\left(r\right)$ defined by

$$xy-y^{\prime }{x}^{\prime }\in \mathbf{R}\left(r\right),\mathrm{whenever}r(x,y)=(y^{\prime },x^{\prime }).$$

The unital semigroup $S(X,r)=\langle X;\mathbf{R}\left(r\right)\rangle$, with generators X and relations $\mathbf{R}\left(r\right)$, is called the semigroup associated with $(X,r)$. The group $G(X,r)$ associated with $(X,r)$ is defined analogously. The algebra associated with $(X,r)$ is defined as $A(X,r)=\mathbb{k}\langle X\rangle /\left(\mathbf{R}\right(r\left)\right)$. When $(X,r)$ is a solution of the YBE, $S(X,r)$, resp. $G(X,r)$, $A(X,r)$ is called the Yang-Baxter (YB) semigroup, resp. the YB group, the YB algebra.

Let $(X,r)$ be a non-degenerate quadratic set. It is well-known that $(X,r)$ is a set-theoretic solution of the YBE if and only if the following conditions hold for all $x,y,z\in X$ (e.g., [9] (Lemma 2.5)):

$$\mathbf{l}\mathbf{1}:{}^x\left({}^{y}z\right)={}^{{}^{x}y}\left({}^{x^y}z\right),\mathbf{r}\mathbf{1}:{\left(x^y\right)}^z={\left(x^{{}^{y}z}\right)}^{y^z},\mathbf{lr}\mathbf{3}:{\left({}^{x}y\right)}^{\left({}^{x^y}\left(z\right)\right)}={}^{\left(x^{{}^{y}z}\right)}\left(y^z\right).$$

For every non-degenerate solution of the YBE, Condition l1 implies that the mapping $x\mapsto {\mathcal{L}}_x$ extends canonically to a group homomorphism

$$\mathcal{L}:G(X,r)⟶Sym\left(X\right),$$

where $Sym\left(X\right)$ is the symmetric group on X. The homomorphism $\mathcal{L}$ defines the canonical left action of $G(X,r)$ on the set X. Similarly, r1 provides the canonical right action of $G(X,r)$ on the set X.

The YB permutation group $\mathcal{G}(X,r)$ of a solution $(X,r)$ is the subgroup of the symmetric group $Sym\left(X\right)$ on X generated by $\{{\mathcal{L}}_x:x\in X\}$.

Example 2.

Let X be a set with $\left|X\right|⩾2$, and define the map τ by $\tau (x,y)=(y,x)$ for all $x,y\in X$. Then, the pair $(X,\tau )$ is called the trivial solution.

One can see that $(X,r)$ is the trivial solution if and only if

$${}^{x}y=y\mathit{and}{x}^y=x,$$

for all $x,y\in X$, or, equivalently,

$${\mathcal{L}}_x={id}_X={\mathcal{R}}_x,$$

for all $x\in X$.

In this case, the YB algebra $A(X,r)$ is just the commutative polynomial algebra $\mathbb{k}\left[X\right]$, and $\mathcal{G}(X,r)={id}_X$ is the trivial group.

Example 3

([3]). Let X be a set with $\left|X\right|⩾2$ and f a bijection on X. Define the map $r:X\times X⟶X\times X$ as

$$r(x,y)=(f\left(y\right),f^{-1}\left(x\right)),$$

for all $x,y\in X$. Then, $(X,r)$ is a non-degenerate involutive solution of the YBE. Clearly, ${\mathcal{L}}_x=f$ and ${\mathcal{R}}_x=f^{-1}$ for all $x\in X$. Moreover, $(X,r)$ is a trivial solution if and only if $f={\mathrm{id}}_X$.

It is known (see, e.g., [22], Lemma 1.2) that there is a one-to-one correspondence between solutions of the YBE and involutive biracks.

Lemma 2.

(1) 

If $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is an involutive birack, then defining

$$r(x,y)=(x\circ y,x•y)$$

yields a solution $(X,r)$ of the YBE.

(2) 

Let $(X,r)$ be a solution of the YBE with $r(x,y)=({\mathcal{L}}_x\left(y\right),{\mathcal{R}}_y\left(x\right))$. Defining

$$x\circ y={\mathcal{L}}_x{\left(y\right),x\setminus }_{\circ }y={\mathcal{L}}_x^{-1}\left(y\right),$$

$$x•y={\mathcal{R}}_y\left(x\right),x{/}_{•}y={\mathcal{R}}_y^{-1}\left(x\right),$$

we obtain an involutive birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$.

Such a correspondence allows us to treat each solution as an involutive birack.

3. Isotopes of Quasigroups

This section serves as preparation for Section 4. We introduce the notion of an ${\mathbb{N}}^p$-graded birack and generalize the isotope construction of [20] to this graded setting.

We say a left quasigroup $(X,\circ ,{\setminus }_{\circ })$ is ${\mathbb{N}}^p$-graded if there is a partition on X:

$$X=X_1\cup X_2\cup \cdots \cup X_p$$

and, for any $x\in X,y\in X_i$,

$$x\circ y\in X_i.$$

(4)

If $x\in X_i$, then the degree vector is assigned as

$$\left|x\right|=(0,\dots ,0,1,0,\dots ,0)\in {\mathbb{N}}^p,$$

where the 1 is in the i-th position.

A map $f:X\to X$ is called graded if $\left|f\right(x\left)\right|=\left|x\right|$ for any $x\in X$, or equivalently $f\left(X_i\right)\subseteq X_i$ for $1⩽i⩽p$. If, in addition, f is an automorphism of $(X,\circ ,{\setminus }_{\circ })$, then f is called a graded automorphism.

Condition (4) means that all left translation is graded.

Let $\varphi =\{{\varphi }_s\mid s=1,\dots ,p\}$ be a sequence of commuting graded bijections of X. For a degree vector $\alpha =(a_1,\dots ,a_p)$, set ${\varphi }^{\alpha }={\varphi }_1^{a_1}\dots {\varphi }_p^{a_p}$. Define on the set X new binary operations:

$$x\ast y:=x\circ {\varphi }^{\left|x\right|}\left(y\right)={\mathcal{L}}_x{\varphi }^{\left|x\right|}\left(y\right),$$

$${x\setminus }_{\ast }y:={\varphi }^{-\left|x\right|}\left({x\setminus }_{\circ }y\right)={\varphi }^{-\left|x\right|}{\mathcal{L}}_x^{-1}\left(y\right).$$

The algebra $(X,\ast ,{\setminus }_{\ast })$ is called the ϕ-isotope of $(X,\circ ,{\setminus }_{\circ })$.

It is easy to see that

$$x\ast \left({x\setminus }_{\ast }y\right)={\mathcal{L}}_x{\varphi }^{\left|x\right|}{\varphi }^{-\left|x\right|}{\mathcal{L}}_x^{-1}\left(y\right)=y$$

and

$${x\setminus }_{\ast }(x\ast y)={\varphi }^{-\left|x\right|}{\mathcal{L}}_x^{-1}{\mathcal{L}}_x{\varphi }^{\left|x\right|}\left(y\right)=y.$$

So, the $\varphi$-isotope $(X,\ast ,{\setminus }_{\ast })$ is also a left quasigroup. Moreover, the grading of $(X,\circ ,{\setminus }_{\circ })$ induces an ${\mathbb{N}}^p$-grading on $(X,\ast ,{\setminus }_{\ast })$.

Lemma 3.

Let $(X,\circ ,{\setminus }_{\circ })$ be a non-degenerate ${\mathbb{N}}^p$-graded left quasigroup and let $\varphi =\{{\varphi }_s\mid s=1,\dots ,p\}$ be a sequence of commuting graded bijections of X. Then, the ϕ-isotope $(X,\ast ,{\setminus }_{\ast })$ is also non-degenerate.

Proof. 

The left quasigroup $(X,\circ ,{\setminus }_{\circ })$ is ${\mathbb{N}}^p$-graded, say

$$X=X_1\cup X_2\cup \cdots \cup X_p.$$

(5)

All left translations are graded. By the non-degeneracy of the left quasigroup, for each subset $X_s$, the mapping

$$\begin{array}{ccc}\hfill X_s& ⟶& X_s\hfill \\ \hfill x& ⟼& {\mathcal{L}}_x^{-1}{\left(x\right)=x\setminus }_{\circ }x\hfill \end{array}$$

is bijective. Each ${\varphi }_s$ is bijective, so the mapping

$$\begin{array}{ccc}\hfill X_s& ⟶& X_s\hfill \\ \hfill x& ⟼& {\varphi }_s{\mathcal{L}}_x^{-1}{\left(x\right)=x\setminus }_{*}x\hfill \end{array}$$

is also bijective. Further, (5) is a partition of X. Therefore, the mapping

$$\begin{array}{ccc}\hfill T_{\varphi }:X& ⟶& X\hfill \\ \hfill x& ⟼& {\varphi }_s{\mathcal{L}}_x^{-1}{\left(x\right)=x\setminus }_{*}x\hfill \end{array}$$

is bijective. The $\varphi$-isotope $(X,*,{\setminus }_{*})$ is also non-degenerate. □

Lemma 4.

Let $(X,\circ ,{\setminus }_{\circ })$ be an ${\mathbb{N}}^p$-graded right cyclic left quasigroup, and $\varphi =\{{\varphi }_s\mid s=1,\dots ,p\}$ be a sequence of commuting ${\mathbb{N}}^p$-graded automorphisms of $(X,\circ ,{\setminus }_{\circ })$ such that

$${\mathcal{L}}_{{\varphi }_s\left(x\right)}={\mathcal{L}}_x,$$

(6)

for every $1⩽s⩽p$ and any $x\in X$. Then, the ϕ-isotope $(X,\ast ,{\setminus }_{\ast })$ is also right cyclic.

Proof. 

We need to prove

$$\left(x{\setminus }_{\ast }y\right){{\setminus }_{\ast }\left(x{\setminus }_{\ast }z\right)=\left(y{\setminus }_{\ast }x\right)\setminus }_{\ast }\left(y{\setminus }_{\ast }z\right),$$

(7)

for every $x,y,z\in X$. Note that $\left\{{\varphi }_s\mid s=1,\cdots ,p\right\}$ is a sequence of automorphisms of $(X,\circ ,{\setminus }_{\circ })$. Then, ${\mathcal{L}}_{{\varphi }_s\left(x\right)}={\varphi }_s{\mathcal{L}}_x{\varphi }_s^{-1}$ for every $x\in X$ and $1⩽s⩽p$. In this case, Condition (6) is equivalent to

$${\varphi }_s{\mathcal{L}}_x={\mathcal{L}}_x{\varphi }_s.$$

(8)

That is, every ${\varphi }_s$ commutes with all left translations.

By the definition of the $\varphi$-isotope, for the left-hand side of (7), we have that

$$\begin{array}{cc}& \left(x{\setminus }_{\ast }y\right){\setminus }_{\ast }\left(x{\setminus }_{\ast }z\right)\hfill \\ \hfill =& \left({\varphi }^{-\left|x\right|}{\mathcal{L}}_x^{-1}\left(y\right)\right){\setminus }_{\ast }\left({\varphi }^{-\left|x\right|}{\mathcal{L}}_x^{-1}\left(z\right)\right)\hfill \\ \hfill =& {\varphi }^{-|{\varphi }^{-\left|x\right|}\left({\mathcal{L}}_x^{-1}\left(y\right)\right)|}{\mathcal{L}}_{{\varphi }^{-\left|x\right|}{\mathcal{L}}_x^{-1}\left(y\right)}^{-1}{\varphi }^{-\left|x\right|}{\mathcal{L}}_x^{-1}\left(z\right).\hfill \end{array}$$

Since the quasigroup $(X,\circ ,{\setminus }_{\circ })$ is ${\mathbb{N}}^p$-graded, all the left translation ${\mathcal{L}}_x$ is graded. Moreover, each ${\varphi }_s$, $s=1,\dots ,p$, is graded. Hence,

$${\varphi }^{-|{\varphi }^{-\left|x\right|}\left({\mathcal{L}}_x^{-1}\left(y\right)\right)|}{\mathcal{L}}_{{\varphi }^{-\left|x\right|}{\mathcal{L}}_x^{-1}\left(y\right)}^{-1}={\varphi }^{-\left|y\right|}{\mathcal{L}}_{{\varphi }^{-\left|x\right|}{\mathcal{L}}_x^{-1}\left(y\right)}^{-1}.$$

It follows from Condition (6) that ${\mathcal{L}}_{{\varphi }^{-\left|x\right|}{\mathcal{L}}_x^{-1}\left(y\right)}^{-1}={\mathcal{L}}_{{\mathcal{L}}_x^{-1}\left(y\right)}^{-1}$. Therefore,

$$\begin{array}{cc}& \left(x{\setminus }_{\ast }y\right){\setminus }_{\ast }\left(x{\setminus }_{\ast }z\right)\hfill \\ \hfill =& {\varphi }^{-\left|y\right|}{\mathcal{L}}_{{\mathcal{L}}_x^{-1}\left(y\right)}^{-1}{\varphi }^{-\left|x\right|}{\mathcal{L}}_x^{-1}\left(z\right)\hfill \\ \hfill \stackrel{\left(8\right)}{=}& {\varphi }^{-\left(\right|x|+|y\left|\right)}{\mathcal{L}}_{{\mathcal{L}}_x^{-1}\left(y\right)}^{-1}{\mathcal{L}}_x^{-1}\left(z\right).\hfill \end{array}$$

Similarly, for the right-hand side of (7), we have

$$\begin{array}{cc}& \left(y{\setminus }_{\ast }x\right){\setminus }_{\ast }\left(y{\setminus }_{\ast }z\right)\hfill \\ \hfill =& \left({\varphi }^{-\left|y\right|}{\mathcal{L}}_y^{-1}\left(x\right)\right){\setminus }_{\ast }\left({\varphi }^{-\left|y\right|}{\mathcal{L}}_y^{-1}\left(z\right)\right)\hfill \\ \hfill =& {\varphi }^{-|{\varphi }^{-\left|y\right|}{\mathcal{L}}_y^{-1}\left(x\right)|}{\mathcal{L}}_{{\varphi }^{-\left|y\right|}{\mathcal{L}}_y^{-1}\left(x\right)}^{-1}\left(x\right){\varphi }^{-\left|y\right|}{\mathcal{L}}_y^{-1}\left(z\right)\hfill \\ \hfill =& {\varphi }^{-\left|x\right|}{\mathcal{L}}_{{\varphi }^{-\left|y\right|}{\mathcal{L}}_y^{-1}\left(x\right)}^{-1}\left(x\right){\varphi }^{-\left|y\right|}{\mathcal{L}}_y^{-1}\left(z\right)\hfill \\ \hfill =& {\varphi }^{-\left|x\right|}{\mathcal{L}}_{{\mathcal{L}}_y^{-1}\left(x\right)}^{-1}{\varphi }^{-\left|y\right|}{\mathcal{L}}_y^{-1}\left(z\right)\hfill \\ \hfill =& {\varphi }^{-\left(\right|x|+|y\left|\right)}{\mathcal{L}}_{{\mathcal{L}}_y^{-1}\left(x\right)}^{-1}{\mathcal{L}}_y^{-1}\left(z\right).\hfill \end{array}$$

The quasigroup $(X,\circ ,{\setminus }_{\circ })$ is right cyclic, so ${\mathcal{L}}_x{\mathcal{L}}_{{x\setminus }_{\circ }y}={\mathcal{L}}_y{\mathcal{L}}_{{y\setminus }_{\circ }x}$ for any $x,y\in X$. Taking inverses yields ${\mathcal{L}}_{{\mathcal{L}}_x^{-1}\left(y\right)}^{-1}{\mathcal{L}}_x^{-1}={\mathcal{L}}_{{\mathcal{L}}_y^{-1}\left(x\right)}^{-1}{\mathcal{L}}_y^{-1}$ for any $x,y\in X$.

In conclusion, we obtain that (7) holds that, for every $x,y,z\in X$, the $\varphi$-isotope $(X,\ast ,{\setminus }_{\ast })$ is also right cyclic. □

We know that involutive biracks are in one-to-one correspondence with non-degenerate right cyclic left quasigroups. We now define the isotope of an ${\mathbb{N}}^p$-graded birack.

A birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is called ${\mathbb{N}}^p$-graded if there is a partition

$$X=X_1\cup \cdots \cup X_p$$

such that, for any $x\in X,y\in X_i$,

$$x\circ y\in X_i\mathrm{and}y•x\in X_i.$$

Remark 2.

If $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is an involutive birack, then $(X,\circ ,{\setminus }_{\circ })$ is a non-degenerate right cyclic left quasigroup. The birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is an ${\mathbb{N}}^p$-graded birack if and only if $(X,\circ ,{\setminus }_{\circ })$ is an ${\mathbb{N}}^p$-graded left quasigroup.

Let $\left({X,\circ ,\setminus }_{\circ },•,{/}_{•}\right)$ be an ${\mathbb{N}}^p$-graded involutive birack. Then, $(X,\circ ,{\setminus }_{\circ })$ is an ${\mathbb{N}}^p$-graded non-degenerate right cyclic left quasigroup. Let $\varphi =\left\{{\varphi }_s\mid s=1,\cdots ,p\right\}$ be a sequence of commuting graded bijection of the set X such that the $\varphi$-isotope $\left({X,\ast ,\setminus }_{\ast }\right)$ of $(X,\circ {\setminus }_{\circ })$ is right cyclic. Then, the left quasigroup $\left({X,\ast ,\setminus }_{\ast }\right)$ uniquely determines the involutive birack $\left({X,\ast ,\setminus }_{\ast },\diamond ,{/}_{\diamond }\right)$ as follows:

$$\begin{array}{c}x\ast y={\mathcal{L}}_x{\varphi }^{\left|x\right|}\left(y\right),\hfill \\ {x\diamond y=(x\ast y)\setminus }_{\ast }x={\varphi }^{-\left|y\right|}{\mathcal{L}}_{{\mathcal{L}}_x{\varphi }^{\left|x\right|}\left(y\right)}^{-1}\left(x\right)={\varphi }^{-\left|y\right|}(x•{\varphi }^{\left|x\right|}\left(y\right)).\hfill \end{array}$$

(9)

The birack $\left({X,\ast ,\setminus }_{\ast },\diamond ,{/}_{\diamond }\right)$ is called the ϕ-isotope of $\left({X,\circ ,\setminus }_{\circ },•,{/}_{•}\right)$.

All left translations in $\left({X,\circ ,\setminus }_{\circ },•,{/}_{•}\right)$ and ${\varphi }_s,1⩽s⩽p$, are graded, so $\left({X,\ast ,\setminus }_{\ast },\diamond ,{/}_{\diamond }\right)$ is also an ${\mathbb{N}}^p$-graded birack.

Lemma 5.

Let $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ be an ${\mathbb{N}}^p$-graded involutive birack satisfying Condition  lri   and $\varphi =\left\{{\varphi }_s\mid s=1,\cdots ,p\right\}$ be a sequence of commuting graded bijections. Then, the ϕ-isotope $(X,\ast ,{\setminus }_{\ast },\diamond ,{/}_{\diamond })$ satisfies Condition  lri  if and only if in $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$,

$${\mathcal{L}}_{{\varphi }^{\left|y\right|}\left(x\right)}\left(y\right)={\mathcal{L}}_x\left(y\right)$$

(10)

for any $x,y\in X$. In this case,

$${y\diamond x=x\setminus }_{\ast }y={\varphi }^{-\left|x\right|}{\mathcal{L}}_x^{-1}\left(y\right),$$

for any $x,y\in X$.

Proof. 

By the definition of Condition lri, we only need to prove that Condition (6) is equivalent to

$${y\diamond x=x\setminus }_{\ast }y$$

for all $x,y\in X$.

Indeed, $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ satisfies Condition lri. Then, for every $x,y\in X$, we have

$$\begin{array}{ccc}\hfill {y\diamond x=x\setminus }_{\ast }y& \iff & {\varphi }^{-\left|x\right|}{\mathcal{L}}_{{\mathcal{L}}_y{\varphi }^{\left|y\right|}\left(x\right)}^{-1}\left(y\right)={\varphi }^{-\left|x\right|}{\mathcal{L}}_x^{-1}\left(y\right)\hfill \\ & \iff & {\varphi }^{-\left|x\right|}(y•{\varphi }^{\left|y\right|}\left(x\right))={\varphi }^{-\left|x\right|}(y•x)\hfill \\ & \iff & y•{\varphi }^{\left|y\right|}\left(x\right)=y•x\hfill \\ & \iff & {\mathcal{R}}_{{\varphi }^{\left|y\right|}\left(x\right)}\left(y\right)={\mathcal{R}}_x\left(y\right)\hfill \\ & \iff & {\mathcal{L}}_{{\varphi }^{\left|y\right|}\left(x\right)}\left(y\right)={\mathcal{L}}_x\left(y\right).\hfill \end{array}$$

Therefore, $(X,\ast ,{\setminus }_{\ast },\diamond ,{/}_{\diamond })$ satisfies Condition lri if and only if $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ satisfies Condition (10). □

Remark 3.

It is easy to see that in an birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$, Condition (6) implies Condition (10).

Proposition 1.

Let $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ be an ${\mathbb{N}}^p$-graded involutive birack satisfying Condition lri and $\varphi =\left\{{\varphi }_s\mid s=1,\cdots ,p\right\}$ be a sequence of commuting graded automorphisms of $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ satisfying Condition (6). Then, the ϕ-isotope $(X,\ast ,{\setminus }_{\ast },\diamond ,{/}_{\diamond })$ of $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is also an ${\mathbb{N}}^p$-graded involutive birack satisfying Condition lri.

Proof. 

The birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is an ${\mathbb{N}}^p$-graded involutive birack, and then $(X,\circ ,{\setminus }_{\circ })$ is a non-degenerate right cyclic left quasigroup by Lemma 1. By Lemmas 3 and 4, the $\varphi$-isotope $(X,\ast ,{\setminus }_{\ast })$ is also non-degenerate right cyclic. It determines the ${\mathbb{N}}^p$-graded involutive birack $(X,\ast ,{\setminus }_{\ast },\diamond ,{/}_{\diamond })$. By Lemma 5, $(X,\ast ,{\setminus }_{\ast },\diamond ,{/}_{\diamond })$ also satisfies Condition  lri. □

4. Isotopes and Zhang Twists

In this section, we will show the relation between the isotopes of involutive biracks and the Zhang twists of algebras.

Recall that, in this paper, the term solution refers to a set-theoretic, non-degenerate, and involutive solution of the YBE. By Lemma 2, solutions of the YBE are in one-to-one correspondence with involutive biracks; hence, every solution can be realized by an involutive birack.

Let us recall the notions needed in this section. Let $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ be an involutive birack.

• For each $x\in X$, ${\mathcal{L}}_x$ denotes the left translation by x, and ${\mathcal{R}}_x$ denotes the right translation by x. That is,

  $${\mathcal{L}}_x\left(a\right)=x\circ a;$$

  $${\mathcal{R}}_x\left(a\right)=a•x,$$

  for any $a\in X$.
• It determines a solution $(X,r)$ of the YBE, where

  $$r(x,y)=(x\circ y,x•y),$$

  for any $x,y\in X$.
• It determines an algebra $A=A(X,{\mathbf{R}}^{\circ })$, generated by X with quadratic relations ${\mathbf{R}}^{\circ }$ defined by

  $$xy=(x\circ y)(x•y)$$

  for any $x,y\in X$. If the birack is ${\mathbb{N}}^p$-graded, then the grading on X naturally defines a grading on A so that A is an ${\mathbb{N}}^p$-graded algebra.

Remark 4.

(1) 

For an (${\mathbb{N}}^p$-graded) involutive birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$, the (${\mathbb{N}}^p$-graded) algebra $A=A(X,{\mathbf{R}}^{\circ })$ is precisely the YB algebra of the solution corresponding to the involutive birack. In some cases, we simply call $A=A(X,{\mathbf{R}}^{\circ })$ the algebra associated with the birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$.

(2) 

Under the correspondence between a solution of the YBE and an involutive birack, the left action coincides with the left translation, and both are denoted by $\mathcal{L}$. Similarly, the right action coincides with the right translation, and both are denoted by $\mathcal{R}$.

Let B be any ${\mathbb{N}}^p$-graded algebra. Let $\varphi =\left\{{\varphi }_s\mid s=1,\cdots ,p\right\}$ be a sequence of commuting ${\mathbb{N}}^p$-graded algebra automorphisms of B. Then, the Zhang twist $B^{\varphi }$ of B by $\varphi$ is defined as follows: $B^{\varphi }=B$ as an ${\mathbb{N}}^p$-graded vector space, and the new multiplication ⋆ of $B^{\varphi }$ is determined by

$$a\star b=a{\varphi }^{\left|a\right|}\left(b\right),$$

where ${\varphi }^{\left|a\right|}={\varphi }_1^{a_1}\cdots {\varphi }_p^{a_p}$ if the degree of a is $\left|a\right|=\left(a_1,\cdots ,a_p\right)$. We refer to [21] for basic properties of twisted algebras. The following lemma can be derived from [21], Proposition 2.5.

Lemma 6.

Let B be an ${\mathbb{N}}^p$-graded algebra. If an ${\mathbb{N}}^p$-graded algebra C is a Zhang twist of B, then B is a Zhang twist of C.

Now we outline the following main theorem.

Theorem 1.

Let $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ be an ${\mathbb{N}}^p$-graded involutive birack satisfying Condition  lri. It determines an ${\mathbb{N}}^p$-graded algebra $A=A(X,{\mathbf{R}}^{\circ })$. Let $\varphi =\left\{{\varphi }_s\mid s=1,\cdots ,p\right\}$ be a sequence of commuting graded automorphisms of $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ satisfying Condition (6).

(1) 

The automorphisms $\left\{{\varphi }_s\mid s=1,\cdots ,p\right\}$ naturally induce a sequence of commuting ${\mathbb{N}}^p$-graded automorphisms on the algebra A.

(2) 

The ϕ-isotope $(X,\ast ,{\setminus }_{\ast },\diamond ,{/}_{\diamond })$ of $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is also an ${\mathbb{N}}^p$-graded involutive birack satisfying Condition  lri. It determines an ${\mathbb{N}}^p$-graded algebra $A^{\ast }=A^{\ast }(X,{\mathbf{R}}^{\ast })$.

(3) 

The algebra $A^{\ast }$ obtained by the isotope $(X,\ast ,{\setminus }_{\ast },\diamond ,{/}_{\diamond })$ is isomorphic to the Zhang twist $A^{\varphi }$ of A by ϕ.

Proof.  

(1) It is direct.

(2) The fact that the $\varphi$-isotope $(X,\ast ,{\setminus }_{\ast },\diamond ,{/}_{\diamond })$ of $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is also an ${\mathbb{N}}^p$-graded involutive birack satisfying Condition lri follows from Proposition 1. Then, it determines an ${\mathbb{N}}^p$-graded algebra $A^{\ast }=A^{\ast }(X,{\mathbf{R}}^{\ast })$.

(3) The isotope $(X,\ast ,{\setminus }_{\ast },\diamond ,{/}_{\diamond })$ satisfies Condition lri. Hence,

$$\begin{array}{ccc}\hfill x\ast y& =& {\mathcal{L}}_x{\varphi }^{\left|x\right|}\left(y\right),\hfill \\ \hfill x\diamond y& =& {y\setminus }_{\ast }x={\varphi }^{-\left|y\right|}{\mathcal{L}}_y^{-1}\left(x\right),\hfill \end{array}$$

for any $x,y\in X$. Then, for the algebra $A^{\ast }=(A^{\ast },X,{\mathbf{R}}^{\ast })$, the relation ${\mathbf{R}}^{\ast }$ is defined by

$$\begin{array}{ccc}\hfill xy& =& (x\ast y)(x\diamond y)=(x\ast y)\left(y{\setminus }_{\ast }x\right)\hfill \\ \hfill & =& {\mathcal{L}}_x\left({\varphi }^{\left|x\right|}\left(y\right)\right){\varphi }^{-\left|y\right|}\left({\mathcal{L}}_y^{-1}\left(x\right)\right).\hfill \end{array}$$

(11)

The multiplication of the Zhang twist $A^{\varphi }$ is determined by

$$x\star y=x{\varphi }^{\left|x\right|}\left(y\right),$$

for any $x,y\in X$. Recall that relation ${\mathbf{R}}^{\circ }$ of A is defined by

$$\begin{array}{ccc}\hfill xy& =& (x\circ y)(x•y)\hfill \\ & =& (x\circ y)\left(y{\setminus }_{\circ }x\right),\hfill \end{array}$$

for any $x,y\in X$. We have

$$\begin{array}{ccc}\hfill x\star y& =& x{\varphi }^{\left|x\right|}\left(y\right)\hfill \\ \hfill & =& (x\circ {\varphi }^{\left|x\right|}\left(y\right))\left({\varphi }^{\left|x\right|}\left(y\right){\setminus }_{\circ }x\right)\hfill \\ \hfill & =& {\mathcal{L}}_x\left({\varphi }^{\left|x\right|}\left(y\right)\right){\mathcal{L}}_{{\varphi }^{\left|x\right|}\left(y\right)}^{-1}\left(x\right)\hfill \\ \hfill & \stackrel{\left(6\right)}{=}& {\mathcal{L}}_x\left({\varphi }^{\left|x\right|}\left(y\right)\right){\mathcal{L}}_y^{-1}\left(x\right)\hfill \\ \hfill & =& {\mathcal{L}}_x\left({\varphi }^{\left|x\right|}\left(y\right)\right)\star {\varphi }^{-|{\mathcal{L}}_x\left({\varphi }^{\left|x\right|}\left(y\right)\right)|}\left({\mathcal{L}}_y^{-1}\left(x\right)\right)\hfill \\ \hfill & =& {\mathcal{L}}_x\left({\varphi }^{\left|x\right|}\left(y\right)\right)\star {\varphi }^{-\left|y\right|}\left({\mathcal{L}}_y^{-1}\left(x\right)\right).\hfill \end{array}$$

(12)

The last equation follows from that $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is an ${\mathbb{N}}^p$-graded birack and each ${\varphi }_s$, $s=1,\cdots ,p$, are graded automorphisms of $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$. So, $|{\mathcal{L}}_x\left(y\right)|=|y|$ and $|{\varphi }^{\left|x\right|}\left(y\right)|=|y|$ for any $x,y\in X$.

With (11) and (12), the map $\mathrm{\Psi }:A^{\ast }\to A^{\varphi }$ defined by $\mathrm{\Psi }\left(x\right)=x$ for any $x\in X$ is an isomorphism of algebras; the Zhang twist $A^{\varphi }$ of A is isomorphic to the algebra $A^{\ast }$. □

The Zhang twist preserves many important properties of graded algebras. Thus, by Theorem 1, the study of a structurally complex YB algebra can be reduced—via isotopy—to that of a simpler one.

5. Examples

In this section, we provide some examples illustrating Theorem 1.

5.1. Distributive Solutions

We first show that YB algebras of distributive solutions are Zhang twists of polynomial algebras.

A birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is called distributive if, for every $x,y,z\in X$,

$$x\circ (y\circ z)=(x\circ y)\circ (x\circ z),$$

$$(y•z)•x=(y•x)•(z•x).$$

If the birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is involutive, then it is distributive if and only if, for every $x,y,z\in X$,

$$x\circ (y\circ z)=(x\circ z)\circ (x\circ z),$$

(13)

i.e., ${\mathcal{L}}_x{\mathcal{L}}_y={\mathcal{L}}_{{\mathcal{L}}_x\left(y\right)}{\mathcal{L}}_x$ [20], Corollary 5.7. Following from [20], Corollary 5.4 and [8], Lemma 7.1, we know that an involutive distributive birack satisfies Condition lri.

A solution of the YBE is called distributive if it corresponds to a distributive involutive birack.

For the rest of this subsection, $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is a finite distributive involutive birack. There is an equivalence ∼ defined on X:

$$x\sim y\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{\mathcal{L}}_x={\mathcal{L}}_y.$$

This equivalence provides a partition on X. That is,

$$X=\bigcup _{x\in X}\left[x\right],$$

(14)

where $\left[x\right]=\{a\in X|x\sim a\}.$ By Lemma 3.1 and Corollary 5.4 in [20], a distributive involutive birack is 2-reductive, i.e.,

$$(x\circ y)\circ z=y\circ z(\iff {\mathcal{L}}_{{\mathcal{L}}_x\left(y\right)}={\mathcal{L}}_y)$$

(15)

for any $x,y,z\in X$. Then, for any $y\in \left[x\right]$ and $x\in X$, ${\mathcal{L}}_x\left(y\right)\in \left[x\right]$, (14) provides a grading on the birack. Assume that X is partitioned into p equivalence classes, numbering them as $X_1,\cdots ,X_p$. The birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ is an ${\mathbb{N}}^p$-graded birack.

In each equivalence class $X_s$, select a representative element $x_s$. We have ${\mathcal{L}}_y={\mathcal{L}}_{x_s}$ for any $y\in X_s$. We denote the inverse of these equal mappings as ${\varphi }_s$; i.e., ${\varphi }_s={\mathcal{L}}_{x_s}^{-1}$. Directly from (13) and (15), we obtain that all left translations are automorphisms of the birack satisfying Condition (6). Moreover, the permutation groups generated by the left transformations are abelian. So, $\varphi =\left\{{\varphi }_s\mid s=1,\cdots ,p\right\}$ is a sequence of commuting graded automorphisms of the birack satisfying Condition (6). In the $\varphi$-isotope $(X,\ast ,{\setminus }_{\ast },\diamond ,{/}_{\diamond })$ of the birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$, for any $x,y\in X$,

$$\begin{array}{c}x\ast y={\mathcal{L}}_x{\varphi }^{\left|x\right|}\left(y\right)={\mathcal{L}}_x{\mathcal{L}}_x^{-1}\left(y\right)=y,\hfill \\ {x\diamond y=y\setminus }_{\ast }x={\varphi }^{-\left|y\right|}{\mathcal{L}}_y^{-1}\left(x\right)=x.\hfill \end{array}$$

All left and right translations are the identity, so the birack $(X,\ast ,{\setminus }_{\ast },\diamond ,{/}_{\diamond })$ is a projection birack. It corresponds to a trivial solution of the YBE, whose YB algebra $A^{\ast }$ is just the polynomial algebra $\mathbb{k}\left[X\right]$. By Theorem 1, the algebra $A^{\ast }$ is isomorphic to a Zhang twist of the algebra A, the algebra associated with the birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$. Symmetrically, the algebra A is a Zhang twist of the algebra $A^{*}$ by Lemma 6.

In conclusion, we obtain the following proposition.

Proposition 2.

The YB algebra of a finite distributive solution $(X,r)$ of the YBE is a Zhang twist of the polynomial algebra over the set X.

We provide a concrete example.

Example 4.

Let $(X=\{x_0,x_1,x_2,x_3\},\circ ,{\setminus }_{\circ },•,{/}_{•})$ be the following involutive distributive birack:

$$\begin{array}{ccccc}\circ & x_0& x_1& x_2& x_3\\ ine{x}_0& x_0& x_1& x_3& x_2\\ x_1& x_0& x_1& x_3& x_2\\ x_2& x_1& x_0& x_2& x_3\\ x_3& x_1& x_0& x_2& x_3\end{array}\begin{array}{ccccc}•& x_0& x_1& x_2& x_3\\ ine{x}_0& x_0& x_1& x_3& x_2\\ x_1& x_0& x_1& x_3& x_2\\ x_2& x_1& x_0& x_2& x_3\\ x_3& x_1& x_0& x_2& x_3\end{array}$$

That is, ${\mathcal{L}}_{x_0}={\mathcal{L}}_{x_1}={\mathcal{R}}_{x_0}={\mathcal{R}}_{x_1}=\left(x_2{x}_3\right)$ and ${\mathcal{L}}_{x_2}={\mathcal{L}}_{x_3}={\mathcal{R}}_{x_2}={\mathcal{R}}_{x_3}=\left(x_0{x}_1\right)$. The partition

$$X=\{x_0,x_1\}\cup \{x_2,x_3\}$$

provides an ${\mathbb{N}}^2$-grading on the birack. That is, $|x_0|=|x_1|=(1,0)$ and $|x_2|=|x_3|=(0,1)$.

The birack determines a solution $(X,r)$, where $r(x,y)=({\mathcal{L}}_x\left(y\right),{\mathcal{R}}_y\left(x\right))$ for $x,y\in \{x_0,x_1,x_2,x_3\}$. The associated YB algebra is the algebra A generated by $x_0,x_1,x_2,x_3$ with relations $\mathbf{R}$ defined by

$$\mathbf{R}=\{x_0{x}_1-x_1{x}_0,x_2{x}_3-x_3{x}_2,x_0{x}_2-x_3{x}_1,x_0{x}_3-x_2{x}_1,x_1{x}_2-x_3{x}_0,x_1{x}_3-x_2{x}_0\}.$$

By defining $|x_0|=|x_1|=(1,0)$ and $|x_2|=|x_3|=(0,1)$, the algebra A is an ${\mathbb{N}}^2$-graded algebra.

Define ${\varphi }_1=\left(x_2{x}_3\right)$ and ${\varphi }_2=\left(x_0{x}_1\right)$. It is direct to check that $\varphi =\{{\varphi }_1,{\varphi }_2\}$ is a sequence of commuting graded automorphisms of the birack satisfying Condition (6). The ϕ-isotope is a projection birack.

On the other hand, the maps $\varphi =\{{\varphi }_1,{\varphi }_2\}$ induce a sequence of commuting graded automorphisms of A by defining

$$\begin{array}{ccc}{\varphi }_1\left(x_2\right)=\left(x_3\right),& {\varphi }_1\left(x_3\right)=\left(x_2\right),& {\varphi }_1\left(x_i\right)=\left(x_i\right),\mathit{for}i=0,1,\\ {\varphi }_2\left(x_0\right)=\left(x_1\right),& {\varphi }_2\left(x_1\right)=\left(x_0\right),& {\varphi }_2\left(x_i\right)=\left(x_i\right),\mathit{for}i=2,3.\end{array}$$

The Zhang twist $A^{\varphi }$ is just the polynomial algebra $\mathbb{k}[x_0,x_1,x_2,x_3]$.

5.2. A Non-Distributive Solution

In this subsection, we present an example of non-distributive solution whose YB algebra is an iterated Zhang twist of a polynomial algebra.

Let $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ be an involutive birack, where the set X is

$$X=\{x_1,x_2,\cdots ,x_8,a,b,c,d\}$$

and the left and right translations are given by the following permutations:

$$\begin{array}{c}{\mathcal{L}}_a={\mathcal{R}}_a=\left(bd\right)\left(x_1{x}_2\right)\left(x_3{x}_4\right)\left(x_5{x}_6\right)\left(x_7{x}_8\right),\hfill \\ {\mathcal{L}}_c={\mathcal{R}}_c=\left(bd\right)\left(x_1{x}_5\right)\left(x_2{x}_6\right)\left(x_3{x}_7\right)\left(x_4{x}_8\right),\hfill \\ {\mathcal{L}}_b={\mathcal{R}}_b=\left(ac\right)\left(x_1{x}_3\right)\left(x_2{x}_4\right)\left(x_5{x}_7\right)\left(x_6{x}_8\right),\hfill \\ {\mathcal{L}}_d={\mathcal{R}}_d=\left(ac\right)\left(x_1{x}_8\right)\left(x_2{x}_7\right)\left(x_3{x}_6\right)\left(x_5{x}_4\right),\hfill \\ {\mathcal{L}}_{x_i}={\mathcal{R}}_{x_i}={id}_X,1⩽i⩽8.\hfill \end{array}$$

It satisfies Condition lri. Direct calculation shows that ${\mathcal{L}}_{{}^{a}b}{\mathcal{L}}_b={\mathcal{L}}_d{\mathcal{L}}_b\ne {\mathcal{L}}_b{\mathcal{L}}_a$. Hence, the birack is not distributive.

The partition $X=X_1\bigcup X_2\bigcup X_3$, where

$$X_1=\{x_i\mid 1⩽i⩽8\},X_2=\{a,c\},X_3=\{b,d\},$$

yields an ${\mathbb{N}}^3$-grading on the birack. Set

$${\varphi }_1={id}_X,{\varphi }_2=\left(bd\right)\left(x_1{x}_6\right)\left(x_2{x}_5\right),{\varphi }_3=\left(ac\right)\left(x_1{x}_6\right)\left(x_3{x}_8\right).$$

Then, $\varphi =\{{\varphi }_1,{\varphi }_2,{\varphi }_3\}$ is a sequence of commuting ${\mathbb{N}}^p$-graded automorphisms of $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$. The $\varphi$-isotope is the birack $(X,\ast ,{\setminus }_{\ast },\diamond ,{/}_{\diamond })$, where the left and right translations are given by

$$\begin{array}{c}{\mathcal{L}}_a^{\prime }={\mathcal{R}}_a^{\prime }=\left(x_1{x}_5\right)\left(x_2{x}_6\right)\left(x_3{x}_4\right)\left(x_7{x}_8\right),\hfill \\ {\mathcal{L}}_c^{\prime }={\mathcal{R}}_c^{\prime }=\left(x_1{x}_8\right)\left(x_2{x}_4\right)\left(x_3{x}_6\right)\left(x_5{x}_7\right),\hfill \\ {\mathcal{L}}_b^{\prime }={\mathcal{R}}_b^{\prime }=\left(x_1{x}_2\right)\left(x_5{x}_6\right)\left(x_4{x}_7\right)\left(x_4{x}_8\right),\hfill \\ {\mathcal{L}}_d^{\prime }={\mathcal{R}}_d^{\prime }=\left(x_1{x}_3\right)\left(x_6{x}_8\right)\left(x_4{x}_5\right)\left(x_2{x}_7\right),\hfill \\ {\mathcal{L}}_{x_i}^{\prime }={\mathcal{R}}_{x_i}^{\prime }={id}_X,1⩽i⩽8.\hfill \end{array}$$

It is a distributive birack. Then, by Theorem 1, Proposition 2, and the symmetry of Zhang twists (Lemma 6), the algebra associated with the birack $(X,\circ ,{\setminus }_{\circ },•,{/}_{•})$ can be obtained from the polynomial algebra $\mathbb{k}\left[X\right]$ by applying two successive Zhang twists.

6. Conclusions

In this paper, we have introduced the notion of an ${\mathbb{N}}^p$-graded birack and developed its isotope theory. By the well-established correspondence between set-theoretic involutive non-degenerate solutions of the YBE and involutive biracks, we extended this correspondence to the graded setting. Each involutive ${\mathbb{N}}^p$-graded birack naturally associates an ${\mathbb{N}}^p$-graded YB algebra. Our main result establishes a precise algebraic link between isotopy and Zhang twisting: under suitable conditions, the YB algebra arising from an isotope of a given birack is isomorphic to a Zhang twist of the original algebra.

This connection not only enriches the interplay between combinatorial structures (biracks and their isotopes) and noncommutative graded algebras but also provides a new perspective on deformation techniques in the context of the YBE. Since Zhang twists preserve many homological and ring-theoretic properties, such as being Noetherian, having a finite Gelfand–Kirillov dimension, or satisfying Artin–Schelter regularity, our result suggests that isotopic biracks give rise to YB algebras with closely related algebraic behaviors, even when their underlying combinatorial structures may differ significantly.

As a concrete illustration, we have shown that YB algebras associated with finite distributive solutions are Zhang twists of commutative polynomial algebras. This places such algebras within a well-understood class of deformations of polynomial rings, offering a pathway to transfer known results from commutative algebra to this noncommutative setting. In future work, we plan to explore the full implications of this viewpoint, particularly with regard to (i) the classification of multipermutation solutions through their associated graded algebras and (ii) the role of higher-grade isotopies in constructing new families of solutions whose algebraic properties are prescribed in advance.