向簇超代数迈进的一步
A step towards cluster superalgebras
摘要：
引入了一类交换超代数，推广了簇代数。一个簇超代数是由一个超图定义的，这个超图被称为“扩展图”，变换被称为突变。我们证明了“劳伦特现象”的超级模拟，即，证明了簇超代数的所有元素在初始变量下都是Laurent多项式，并找到了不变的前辛形式。簇超代数的例子由Coxeter的frieze模式的超类似物提供。利用Laurent现象构造了一个新的整数序列，它是Somos- $4$序列的推广.
英文摘要：
We introduce a class of commutative superalgebras generalizing cluster algebras. A cluster superalgebra is defined by a hypergraph called an "extended quiver", and transformations called mutations. We prove the super analog of the "Laurent phenomenon", i.e., that all elements of a given cluster superalgebra are Laurent polynomials in the initial variables, and find an invariant presymplectic form. Examples of cluster superalgebras are provided by superanalogs of Coxeter's frieze patterns. We apply the Laurent phenomenon to construct a new integer sequence extending the Somos-$4$ sequence.
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