论文标题
$ \ Mathcal r $ operators的立方块中的自相似性
Self-similarity in cubic blocks of $\mathcal R$-operators
论文作者
论文摘要
从线性运算符$ \ Mathcal r $组装的张张量产品中，研究了立方体块。该操作员与有限特征$ p $的字段$ f $在矢量空间中的线性转换$ a $相关联，例如Hietarinta研究的“置换型”运算符。一个很小的区别是，我们不需要$ a $，因此，$ \ Mathcal r $是可逆的；更重要的是，$ \ Mathcal r $的关系不需要Yang-Baxter或其更高类似物的类型。 结果表明，在$ d = 3 $尺寸中，一个$ p^n \ times p^n \ times p^n $ block分解为类似于初始$ \ Mathcal r $的操作员的张量。对此的一种概括涉及$ f $的交换代数，并允许获得有关由四维$ \ Mathcal r $确定的旋转配置的结果。另一个概括涉及引入旋转配置的Boltzmann权重。事实证明，也存在涉及鲍尔茨曼权重的非平凡的自相似性。
Cubic blocks are studied assembled from linear operators $\mathcal R$ acting in the tensor product of $d$ linear "spin" spaces. Such operator is associated with a linear transformation $A$ in a vector space over a field $F$ of a finite characteristic $p$, like "permutation-type" operators studied by Hietarinta. One small difference is that we do not require $A$ and, consequently, $\mathcal R$ to be invertible; more importantly, no relations on $\mathcal R$ are required of the type of Yang--Baxter or its higher analogues. It is shown that, in $d=3$ dimensions, a $p^n\times p^n\times p^n$ block decomposes into the tensor product of operators similar to the initial $\mathcal R$. One generalization of this involves commutative algebras over $F$ and allows to obtain, in particular, results about spin configurations determined by a four-dimensional $\mathcal R$. Another generalization deals with introducing Boltzmann weights for spin configurations; it turns out that there exists a non-trivial self-similarity involving Boltzmann weights as well.
