基于Grassmann扩展的非交换杨-巴克斯特映射及其格子Boussinesq系统的3D一致性研究

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"Grassmann扩展格Boussinesq系统的3个一致性" 在这篇论文中，我们讨论了Grassmann扩展格Boussinesq系统的三个一致性。我们首先基于[15]中提出的思想，制定了一个“Grassmann扩展”方案，用于构造Yang-Baxter映射的非交换（Grassmann）扩展及其相关的PΔEs系统。一、Grassmann扩展的基本概念 Grassmann扩展是一种数学工具，用于构造非交换的扩展系统。这种扩展方法可以用于构造杨-巴特尔（Yang-Baxter）映射的非交换扩展。在这篇论文中，我们使用Grassmann扩展方案构造了杨-巴特尔映射的Grassmann扩展，然后将其应用于构造格子Boussinesq系统。二、Yang-Baxter映射的Grassmann扩展 Yang-Baxter映射是一种数学工具，用于构造可积分系统。在这篇论文中，我们使用Grassmann扩展方案构造了杨-巴特尔映射的Grassmann扩展。这种扩展可以将杨-巴特尔映射压缩为一个新的、可积分的Grassmann格子Boussinesq系统。三、Grassmann格子Boussinesq系统的3D一致性 Grassmann格子Boussinesq系统是一种可积分系统，在这篇论文中，我们证明了某些系统在其Grassmann扩展中保留了3D一致性属性。这种一致性属性是指系统在三维空间中的对称性和不变性。四、结论在这篇论文中，我们制定了一个“Grassmann扩展”方案，用于构造杨-巴特尔映射的非交换扩展及其相关的PΔEs系统。我们证明了某些系统在其Grassmann扩展中保留了3D一致性属性。这种方法可以用于构造其他可积分系统，并且有可能应用于物理、工程和计算机科学等领域。五、相关概念 * Grassmann扩展：一种数学工具，用于构造非交换的扩展系统。 * 杨-巴特尔映射：一种数学工具，用于构造可积分系统。 * 格子Boussinesq系统：一种可积分系统，具有三维空间中的对称性和不变性。 * 3D一致性：一种系统在三维空间中的对称性和不变性。六、应用前景这篇论文的结果可以应用于物理、工程和计算机科学等领域，例如： * 构造新的可积分系统 * 分析复杂系统的行为 * 开发新型的计算机算法这篇论文提出了一种新的Grassmann扩展方案，用于构造杨-巴特尔映射的非交换扩展及其相关的PΔEs系统，并证明了某些系统在其Grassmann扩展中保留了3D一致性属性。这项研究可以为物理、工程和计算机科学等领域的发展提供新的思路和方法。

S. Konstantinou-Rizos / Nuclear Physics B 951 (2020) 114878 5

Fig. 2. Initial values on the (a) vertices, (b) edges.

(z; c)). The geometric interpretation of these maps, can be understood in a similar way as for

quad-graph equations, but with the values being considered on the edges of the quad, as in

Fig. 1-(b).

Similarly to quad-graph systems, we can also interpret Y

ang-Baxter maps on the square, but

considering the values on the edges instead of the vertices (see Fig. 2-(b)).

2.2.2. Lax r

epresentations & integrability

Equation (2)

admits quad-Lax representation, if there is a (Lax) matrix L

= L

(f, f

; λ) ≡

(f, f

λ ∈ C, such that

(

T f, T f

)

(

f, f

)

= L

(

Sf, Sf

)

(

f, f

)

where S and T are shift operators, as deﬁned in section 2.1.1.

Similarly, for Y

ang-Baxter maps, Lax matrix is a matrix L = L(x, a; λ) ≡ L

(x) that satisﬁes

the following matrix refactorisation problem [28]

(u)L

(v) = L

(y)L

(x). (5)

If equation (5) deﬁnes a map (3), then it is called Lax representation of the map. An alternative

way to verify that a map satisﬁes the Yang-Baxter equation is to consider the following matrix

trifactorisation problem

(u)L

(v)L

(w) = L

(x)L

(y)L

(z),

where L

(x) is the same matrix satisfying (5). In particular, if the above trifactorisation prob-

lem implies that u = x, v = y and w = z, then map (3) deﬁned by (5) satisﬁes the parametric

Yang-Baxter equation (4)[17,30].

In the case of quad-graph equations or systems as (2), the possession of Lax representation is

usually used as wo

rking deﬁnition of integrability. However, a stronger integrability criterion

is that of 3D-consistency [4,22] which implies integrability in the sense of Lax representa-

tion.

From the analysis-point-of-vie

w, 3D-consistency is the property of equation (2)to be consis-

tently generalisable in three dimensions, by “adding” a third discrete variable k in the ﬁeld f ,

namely considering f = f(n, m, k). Geometrically, it means that a quad-graph system–as inter-

preted in Fig. 1-(a)–can be generalised and “written” in a consistent wa

y on the faces of the cube

of Fig. 1-(b). That is, we ﬁrst rewrite our system (2)on the bottom, front and left side of the cube,

respectively, as follows:

We usually skip writing explicitly the dependence on the spectral parameter λ.

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基于Grassmann扩展的非交换杨-巴克斯特映射及其格子Boussinesq系统的3D一致性研究

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