# Item

ITEM ACTIONSEXPORT

Released

Journal Article

#### Algebras defined by Lyndon words and Artin-Schelter regularity

##### External Resource

https://doi.org/10.1090/btran/89

(Publisher version)

https://doi.org/10.48550/arXiv.1905.11281

(Preprint)

##### Fulltext (restricted access)

There are currently no full texts shared for your IP range.

##### Fulltext (public)

Gateva-Ivanova_Algebras defined by Lyndon words and Artin-Schelter regularity_2022.pdf

(Publisher version), 604KB

MPIM Preprint series 2019-32.pdf

(Preprint), 331KB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

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 $U$.

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 $U$.