论文标题
Veronese子代理和Veronese形态，用于一类Yang-Baxter代数
Veronese subalgebras and Veronese morphisms for a class of Yang-Baxter algebras
论文作者
论文摘要
我们研究Yang-Baxter代数的$ d $ - veronese sibalgebras $ a^{（d）} $ a _x = a_x = a（k，x，r）$与有限的无涉及无涉及的设置理论解决方案$（x，x，r）$ k $ k $ k $ k $ as a field和$ d $ d $ d \ geq 2 $ geq 2 $ geq 2 $ s ande。我们在一代和二次关系方面找到了$ d $ veronese $ a^{（d）} $的明确表示。我们介绍了与$（x，r）$相关的$ d $ veronese解决方案$（y，r_y）$的概念，并使用其杨 - 巴克斯特代数$ a_y = a（k，k，y，r_y）$来定义Veronese morphism $ v_ $ v_ {我们证明，$ v_ {n，d} $的图像是$ d $ - 弗朗西斯subalgebra $ a^{（d）} $，并明确地找到了其内核的最小生成器集。结果在交换案件中与他们的经典类似物一致。我们表明，Yang-Baxter代数$ A（K，X，R）$是PBW代数，并且仅当$（X，R）$是无方形的解决方案时。在这种情况下，$ d $ - 弗朗内斯$ a^{（d）} $也是PBW代数。
We study $d$-Veronese subalgebras $A^{(d)}$ of Yang-Baxter algebras $A_X= A(K, X, r)$ related to finite nondegenerate involutive set-theoretic solutions $(X, r)$ of the Yang-Baxter equation, where $K$ is a field and $d\geq 2$ is an integer. We find an explicit presentation of the $d$-Veronese $A^{(d)}$ in terms of one-generators and quadratic relations. We introduce the notion of a $d$-Veronese solution $(Y, r_Y)$, canonically associated to $(X,r)$ and use its Yang-Baxter algebra $A_Y= A(K, Y, r_Y)$ to define a Veronese morphism $v_{n,d}:A_Y \rightarrow A_X $. We prove that the image of $v_{n,d}$ is the $d$-Veronese subalgebra $A^{(d)}$, and find explicitly a minimal set of generators for its kernel. The results agree with their classical analogues in the commutative case. We show that the Yang-Baxter algebra $A(K, X, r)$ is a PBW algebra if and only if $(X,r)$ is a square-free solution. In this case the $d$-Veronese $A^{(d)}$ is also a PBW algebra.
