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
(
N
,
W
)
(N,W)
, so that the class
C
(
X
,
W
)
\mathfrak {C}(X, W)
contains the enveloping algebra
U
g
U\mathfrak {g}
, of a Lie algebra
g
\mathfrak {g}
. We introduce monomial Lie algebras defined by Lyndon words, and prove results on Gröbner-Shirshov bases of Lie ideals generated by Lyndon-Lie monomials. Finally we classify all two-generated Artin-Schelter regular algebras of global dimension
6
6
and
7
7
occurring as enveloping
U
=
U
g
U = U\mathfrak {g}
of standard monomial Lie algebras. The classification is made in terms of their Lyndon pairs
(
N
,
W
)
(N, W)
, each of which determines also the explicit relations of
U
U
.