Algebras defined by Lyndon words and Artin-Schelter regularity
Algebras defined by Lyndon words and Artin-Schelter regularity
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DOI：
10.1090/btran/89
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发表时间：
2019-05
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通讯作者：
T. Gateva-Ivanova
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作者：
T. Gateva-Ivanova
设X = {x1，x2，cdots，xn} X= \{x_1，x_2，\cdots，xn\}是有限字母表，K是域.研究了分次K -代数A = K <$X <$/ I A = K\langle X\rangle / I的类C（X，W）\mathfrak {C}（X，W），它由X X生成，且有一个固定的障碍集W W .最初，我们不施加限制W W和调查的情况下，代数在C（X，W）\mathfrak {C}（X，W）有多项式增长和有限的全球尺寸d d。接下来，我们考虑代数类C（X，W）\mathfrak {C}（X，W），其障碍集W W是林登词的反链。中心问题是“当一个类C（X，W）\mathfrak {C}（X，W）包含Artin-Schelter正则代数？”每个类C（X，W）\mathfrak {C}（X，W）定义一个Lyndon对（N，W）（N，W），如果N N是有限的，则它唯一地确定全局维数g l d i m A gl\，dimA和Gelfand-Kirillov维数G K d i m A GK dimA，对于每个A ∈ C（X，W）A \in \mathfrak {C}（X，W）。我们找到了一个组合条件，
Let X = { x 1 , x 2 , ⋯ , x n } X= \{x_1, x_2, \cdots , x_n\} be a finite alphabet, and let K K be a field. We study classes C ( X , W ) \mathfrak {C}(X, W) of graded K K -algebras A = K ⟨ X ⟩ / I A = K\langle X\rangle / I , generated by X X and with a fixed set of obstructions W W . Initially we do not impose restrictions on W W and investigate the case when the algebras in C ( X , W ) \mathfrak {C} (X, W) have polynomial growth and finite global dimension d d . Next we consider classes C ( X , W ) \mathfrak {C} (X, W) of algebras whose sets of obstructions W W are antichains of Lyndon words. The central question is “when a class C ( X , W ) \mathfrak {C} (X, W) contains Artin-Schelter regular algebras?” Each class C ( X , W ) \mathfrak {C} (X, W) defines a Lyndon pair ( N , W ) (N,W) , which, if N N is finite, determines uniquely the global dimension, g l d i m A gl\,dimA , and the Gelfand-Kirillov dimension, G K d i m A GK dimA , for every A ∈ C ( X , W ) A \in \mathfrak {C}(X, W) . We find a combinatorial condition in terms of <mml:mat