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Algebras defined by Lyndon words and Artin-Schelter regularity

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Gateva-Ivanova,  Tatiana
Max Planck Institute for Mathematics, Max Planck Society;

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Gateva-Ivanova, T. (2022). Algebras defined by Lyndon words and Artin-Schelter regularity. Transactions of the American Mathematical Society. Series B, 9, 648-699. doi:10.1090/btran/89.


Cite as: https://hdl.handle.net/21.11116/0000-000A-AC11-1
Abstract
Let $X= \{x_1, x_2, \cdots, x_n\}$ be a finite alphabet, and let $K$ be a
field. We study classes $\mathfrak{C}(X, W)$ of graded $K$-algebras $A =
K\langle X\rangle / I$, generated by $X$ and with a fixed set of obstructions
$W$. Initially we do not impose restrictions on $W$ and investigate the case
when all algebras in $\mathfrak{C} (X, W)$ have polynomial growth and finite
global dimension $d$. Next we consider classes $\mathfrak{C} (X, W)$ of
algebras whose sets of obstructions $W$ are antichains of Lyndon words. The
central question is "when a class $\mathfrak{C} (X, W)$ contains Artin-Schelter
regular algebras?" Each class $\mathfrak{C} (X, W)$ defines a Lyndon pair
$(N,W)$ which determines uniquely the global dimension, $gl\dim A$, and the
Gelfand-Kirillov dimension, $GK\dim A$, for every $A \in \mathfrak{C}(X, W)$.
We find a combinatorial condition in terms of $(N,W)$, so that the class
$\mathfrak{C}(X, W)$ contains the enveloping algebra $U\mathfrak{g}$ of a Lie
algebra $\mathfrak{g}$. We introduce monomial Lie algebras defined by Lyndon
words, and prove results on Groebner-Shirshov bases of Lie ideals generated by
Lyndon-Lie monomials. Finally we classify all two-generated Artin-Schelter
regular algebras of global dimensions $6$ and $7$ occurring as enveloping $U =
U\mathfrak{g}$ of standard monomial Lie algebras. The classification is made in
terms of their Lyndon pairs $(N, W)$, each of which determines also the
explicit relations of $U$.