Algebras defined by Lyndon words and Artin-Schelter regularity

Abstract

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

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由Lyndon词和Artin-Schelter正则定义的代数

设X= {X 1, X 2，⋯，X n} X= \{x_1, x_2， \cdots, x_n\}是一个有限字母，设K K是一个域。我们研究分级K K -代数A = K⟨X⟩/ I A = K\langle X\rangle / I的类C (X, W) \mathfrak {C}(X, W)，由X X生成并具有固定的障碍物W W。首先，我们没有对ww施加限制，并研究了C (X, W) \mathfrak {C} (X, W)中的代数具有多项式增长和有限全局维数d d的情况。接下来我们考虑一类代数C (X, W) \mathfrak {C} (X, W)，它们的障碍集合W W是林登词的反链。核心问题是“当一类C (X, W) \mathfrak {C} (X, W)包含Artin-Schelter正则代数时?”每个类C (X, W) \mathfrak {C} (X, W)定义了一个Lyndon对(N,W) (N,W)，如果N N是有限的，它唯一地决定了全局维数g g dim a g g \，dimA和Gelfand-Kirillov维数g K dim a g K dimA，对于每个A∈C (X, W) A \in \mathfrak {C}(X, W)。我们用< ml:mat找到了一个组合条件

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