论文标题
杨 - 巴克斯特方程的有限能设置理论解决方案
Finite idempotent set-theoretic solutions of the Yang--Baxter equation
论文作者
论文摘要
事实证明，有限的左左左非降级设置理论解决方案$（x，r）$ yang-baxter方程在一组$ x $上取决于$ x $上的左sips simple semogroup结构（尤其是一个有限的同构副本）和某些maps $ q $ q $ q $和$ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x。当关联的结构单体被取消时，这种结构正是恰恰是一个组，并且所有地图$φ_x$等于该组的自动形态。同等地，结构代数$ k [m（x，r）] $是正确的noetherian，或者在特征零中必须是半弹药。结构代数始终是Gelfand-kirillov dimension One的左no代表代数。为了证明这些结果，结果表明，结构Semigroup $ s（x，r）$在有限的许多取消semigroups $ s_u $ the digonal索引中，每个$ s_u $都有一组商$ g_u $，这些$ g_u $是有限的by-by-by-（infinite cyclit），这些组的结构是一个简单的结构。 $ x $等于对角线的情况被$ x $上的单个置换完全描述。
It is proven that finite idempotent left non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation on a set $X$ are determined by a left simple semigroup structure on $X$ (in particular, a finite union of isomorphic copies of a group) and some maps $q$ and $φ_x$ on $X$, for $x\in X$. This structure turns out to be a group precisely when the associated structure monoid is cancellative and all the maps $φ_x$ are equal to an automorphism of this group. Equivalently, the structure algebra $K[M(X,r)]$ is right Noetherian, or in characteristic zero it has to be semiprime. The structure algebra always is a left Noetherian representable algebra of Gelfand--Kirillov dimension one. To prove these results it is shown that the structure semigroup $S(X,r)$ has a decomposition in finitely many cancellative semigroups $S_u$ indexed by the diagonal, each $S_u$ has a group of quotients $G_u$ that is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case that $X$ equals the diagonal is fully described by a single permutation on $X$.
