Finitely presented quadratic algebras of intermediate growth
In this article, we give two examples of finitely presented quadratic algebras (algebras presented by quadratic relations) of intermediate growth.
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Koçak, D. 2019-06-15T19:49:12Z 2019-06-15T19:49:12Z 2015 Finitely presented quadratic algebras of intermediate growth / D. Koçak // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 69-88. — Бібліогр.: 18 назв. — англ. 1726-3255 2010 MSC:16P90, 16S37, 16S30, 17B70. https://nasplib.isofts.kiev.ua/handle/123456789/154743 In this article, we give two examples of finitely presented quadratic algebras (algebras presented by quadratic relations) of intermediate growth. I wish to thank Inna Capdeboscq for calling my attention to subalgebras of Kac-Moody algebras and Efim Zelmanov for the idea of considering Veronese subalgebras to get quadratic algebras. I also thank my advisor Rostislav Grigorchuk for his assistance in writing this paper. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Finitely presented quadratic algebras of intermediate growth Article published earlier |
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Finitely presented quadratic algebras of intermediate growth |
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Finitely presented quadratic algebras of intermediate growth Koçak, D. |
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Finitely presented quadratic algebras of intermediate growth |
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Finitely presented quadratic algebras of intermediate growth |
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Finitely presented quadratic algebras of intermediate growth |
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Finitely presented quadratic algebras of intermediate growth |
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finitely presented quadratic algebras of intermediate growth |
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Koçak, D. |
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Algebra and Discrete Mathematics |
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Інститут прикладної математики і механіки НАН України |
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In this article, we give two examples of finitely presented
quadratic algebras (algebras presented by quadratic relations) of intermediate growth.
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Finitely presented quadratic algebras of intermediate growth / D. Koçak // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 69-88. — Бібліогр.: 18 назв. — англ. |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 20 (2015). Number 1, pp. 69–88
© Journal “Algebra and Discrete Mathematics”
Finitely presented quadratic algebras
of intermediate growth
Dilber Koçak∗
Communicated by R. I. Grigorchuk
Abstract. In this article, we give two examples of finitely
presented quadratic algebras (algebras presented by quadratic rela-
tions) of intermediate growth.
1. Introduction
Let A be a finitely generated algebra over a field k with generating set
S = {x1, . . . , xm}. We denote by An the subspace of elements of degree
at most n, then A =
⋃∞
n=0 An. The growth function γS
A of A with respect
to S is defined as the dimension of the vector space An over k,
γS
A(n) = dimk(An)
The function γS
A depends on the generating set S. This dependence can be
removed by introducing an equivalence relation: Let f and g be eventually
monotone increasing and positive valued functions on N. Set f � g if and
only if there exist N > 0, C > 0, such that f(n) 6 g(Cn), for n > N , and
f ∼ g if and only if f � g and g � f . The equivalence class of f is called
the growth rate of f . Simple verification shows that growth functions of
an algebra with respect to different generating sets are equivalent. The
growth rate is a useful invariant for finitely generated algebraic structures
∗The author was partially supported by NSF grant DMS-1207699.
2010 MSC: 16P90, 16S37, 16S30, 17B70.
Key words and phrases: Finitely presented algebras, growth of algebras,
quadratic relations.
70 Finitely presented quadratic algebras
such as groups, semigroups and algebras. The notion of growth function for
groups was introduced by Schwarz [Šva55] and independently by Milnor
[Mil68]. The description of groups of polynomial growth was obtained
by Gromov in his celebrated work [Gro81]. He proved that every finitely
generated group of polynomial growth contains a nilpotent subgroup of
finite index.
The study of growth of algebras dates back to the papers by Gelfand
and Kirillov, [GK66a,GK66b]. In this paper we are mainly interested in
finitely presented algebras whose growth functions behave in intermediate
way i.e., they grow faster than any polynomial function but slower than
any exponential function. Govorov gave the first examples of finitely
generated semigroups and associative algebras of intermediate growth
in [Gov72]. Examples of algebras of intermediate growth can also be
found in [Ste75, Smi76, She80, Ufn80, KKM83]. The first examples of
finitely generated groups of intermediate growth were constructed by
Grigorchuk [Gri83,Gri84]. It is still an open problem whether there exists
a finitely presented group of intermediate growth. In contrast, there are
examples of finitely presented algebras of intermediate growth. The first
example is the universal enveloping algebra of a Lie algebra W with basis
{w−1, w0, w1, w2, . . . } and brackets defined by [wi, wj ] = (i − j)wi+j . W
is a subalgebra of the generalized Witt algebra WZ (see [AS74, p.206] for
definitions). It was proven in [Ste75] that W has a finite presentation
with two generators and six relations. It is also a graded algebra with
generators of degree −1 and 2. Since W has linear growth, its universal
enveloping algebra is an example of finitely presented associative algebra
of intermediate growth.
The main goal of this paper is to present examples of finitely presented
quadratic algebras (algebras defined by quadratic relations) of interme-
diate growth. The class of quadratic algebras contains a class of finitely
presented algebras, called Koszul algebras. They play an important role
in many studies. In [PP05], it is conjectured that the Hilbert series of a
Koszul algebra A is a rational function and in particular, the growth of
A is either polynomial or exponential.
In order to construct our first example of a finitely presented quadratic
algebra of intermediate growth, we consider the Kac-Moody algebra for
the generalized Cartan matrix A =
(
2 −2
−2 2
)
. This is a graded Lie algebra
of polynomial growth whose generators are of degree 1. Next, we consider
a suitable subalgebra and its universal enveloping algebra.
D. Koçak 71
Theorem 1. Let U be the associative algebra with generators x, y and
relations x3y − 3x2yx + 3xyx2 − yx3 = 0, y3x − 3y2xy + 3yxy2 − xy3 = 0.
Then
(i) It is the universal enveloping algebra of a subalgebra of the the
Kac-Moody algebra for the generalized Cartan matrix A =
(
2 −2
−2 2
)
.
(ii) U is a graded algebra with generators of degree 1.
(iii) It has intermediate growth of type e
√
n.
(iv) The Veronese subalgebra V4(U) of U is a quadratic algebra given
by 14 generators and 96 quadratic relations and it has the same
growth type with U .
The Kac-Moody algebra for the generalized Cartan matrix A =
(
2 −2
−2 2
)
is the affine Lie algebra A
(1)
1 . (For the definition of Kac-Moody
algebras and classification of affine Lie algebras see [Kac85]). It has a
subalgebra which is isomorphic to the Lie subalgebra L of sl2(C[t]) which
consists of all matrices with entries on and under the diagonal divisible
by t. That is,
L =
{
a = (aij)2×2 | aij ∈ C[t], tr(a) = 0
and for (i, j) 6= (1, 2), t divides aij
}
with the usual Lie bracket [a, b] = ab−ba. It follows from [Kac85, Theorem
9.11] that L is finitely presented. In this paper we will prove this by
using the axioms of Lie bracket without mentioning the theory of Kac-
Moody algebras. In Section 2 we show that L is a finitely presented
graded Lie algebra whose generators are all of degree 1 and L has linear
growth. In Section 3 we explain the relation between the growth of a Lie
algebra and its universal enveloping algebra. In Section 4 we consider
the Veronese subalgebra of U to obtain a finitely presented quadratic
algebra of intermediate growth and in Section 5 we complete the proof of
Theorem 1. In Section 6 we give another example of finitely presented
associative algebra A of intermediate growth related to the example of
the monoid in [Kob95]. A has the following presentation:
A = 〈a, b, c | b2a = ab2, b2c = aca, acc = 0,
aba = 0, abc = 0, cba = 0, cbc = 0〉
We show that A has intermediate growth of type e
√
n and its Veronese
subalgebra V3(A) is an example of finitely presented quadratic algebra of
72 Finitely presented quadratic algebras
intermediate growth. In Section 7 , we give an explicit presentation of
the Veronese subalgebra V4(U) of the first construction U as an example
of a finitely presented quadratic algebra of intermediate growth.
2. An example of a finitely presented Lie Algebra of linear
growth
The following example is a subalgebra of the Kac-Moody Algebra for
the generalized Cartan matrix A =
(
2 −2
−2 2
)
[Kac85].
Consider the subalgebra L of Sl2(C[t]) over C (i.e., matrices of trace 0
with entries in C[t])) which consists of matrices whose entries on and
under the diagonal are divisible by t. That is,
L =
{
a = (aij)2x2| aij ∈ C[t], tr(a) = 0
and for (i, j) 6= (1, 2), t divides aij
}
with the usual Lie bracket [a, b] = ab − ba.
Proposition 1. Let L be the Lie algebra described above. Then it has
the following properties.
(i) L is finitely presented with generators
x :=
(
0 1
0 0
)
and y :=
(
0 0
t 0
)
and the defining relations [x, [x, [x, y]]] = 0 and [y, [y, [y, x]]] = 0.
(ii) L =
⊕
k>1
Lk is graded and generated by L1.
(iii) L has linear growth.
Proof. Take
x1 := x =
(
0 1
0 0
)
, y1 := y =
(
0 0
t 0
)
, and let z1 :=
(
t 0
0 −t
)
.
In fact, define
xi :=
(
0 ti−1
0 0
)
, yi :=
(
0 0
ti 0
)
, and let zi :=
(
ti 0
0 −ti
)
for i > 1.
D. Koçak 73
An arbitrary element w ∈ L is of the form:
w =
(
∑n
i=1 mit
i
∑n
i=1 kit
i−1
∑n
i=1 lit
i
∑n
i=1 −mit
i
)
=
n
∑
i=1
kixi +
n
∑
i=1
liyi +
n
∑
i=1
mizi.
So, any element of L can be written as a linear combination of xi, yi, zi
for i > 1 and {xi, yi, zi}
∞
i=1 forms a linearly independent set over C.
Algebra L has the following relations
[xi, yj ] = zi+j−1, (1)
[xi, zj ] = −2xi+j , (2)
[yi, zj ] = 2yi+j , (3)
[xi, xj ] = 0, (4)
[yi, yj ] = 0, (5)
[zi, zj ] = 0. (6)
for i, j > 1. In particular,
xi+1 = −
1
2
[xi, z1], yi+1 =
1
2
[yi, z1], zi = [xi, y1].
It follows that L is generated by x1 and y1. In order to show that all the
relations (1)–(6) can be derived from the relations [x1, [x1, [x1, y1]]] = 0
and [y1, [y1, [y1, x1]]] = 0, we apply induction on i + j = n. If i + j = 2,
the relations (1)–(6) hold trivially. If i + j = 3,
[x1, y2] = [x1,
[y1, z1]
2
]
= −
1
2
([z1, [x1, y1]] + [y1, [z1, x1]])
= [x2, y1]
= z2,
[x1, z2] = [x1, [x2, y1]]
= −[y1, [x1, x2]] + [x2, [y1, x1]](since [x1, x2] = 0)
= [x2, [x1, y1]]
= [x2, z1]
= −2x3,
[y1, z2] = [y1, [x1, y2]]
= −([y2, [y1, x1]] + [x1, [y2, y1]]) (since [y1, y2] = 0)
= [y2, z1]
= 2y3.
74 Finitely presented quadratic algebras
The relations (4)-(5) for n = 3 correspond to relations of L0. Observe the
following three equations for [z2, z1],
[z2, z1] = [[x2, y1], z1]
= −([[z1, x2], y1] + [[y1, z1], x2])
= [[x2, z1], y1] + [x2, [y1, z1]]
= −2[x3, y1] + 2[x2, y2]
= k,
[z2, z1] = [[x1, y2], z1]
= −([[z1, x1], y2] + [[y2, z1], x1])
= [[x1, z1], y2] + [x1, [y2, z1]]
= −2[x2, y2] + 2[x1, y3]
= l,
[z2, z1] = [z2, [x1, y1]
= −([y1, [z2, x1]] + [x1, [y1, z2]])
= 2[x3, y1] − 2[x1, y3]
= m.
3 · [z2, z1] = k + l + m = 0. So, (1)–(6) hold for n = 3. Now, suppose that
(1)–(6) hold for i + j 6 n for some n > 3. For 1 6 i 6 n − 1,
[xi, yj+1] =
1
2
[xi, [yj , z1]]
= −
1
2
([z1, [xi, yj ]] + [yj , [z1, xi]])
= [xi+1, yj ],
−2xn+1 = [xn, z1]
= −
1
2
[[x1, zn−1], z1]
=
1
2
([[z1, x1], zn−1] + [[zn−1, z1], x1])
= [x2, zn−1],
and
[xi, zj+1] = [xi, [x1, yj+1]]
= −([yj+1, [xi, x1]] + [x1, [yj+1, xi]])
= [x1, zi+j ].
D. Koçak 75
Similarly, it can be shown that
2yn+1 = [yi, zj+1]
for any i, j > 1 such that i + j = n. So (1)–(3) hold for i + j = n + 1.
[x1, xn] = −
1
2
[x1, [xi, zj ]]
=
1
2
([zj , [x1, xi]] + [xi, [zj , x1]])
= −
1
2
[xi, [x1, zj ]]
= [xi, xj ]
This equality implies [xi, xj ] = [xj , xi]. Similarly, one checks that [yi, yj ] =
[yj , yi]. Hence, (4)–(5) hold for i + j = n + 1.
Finally, we need check that (6) holds for i + j = n + 1.
[z1, zn] = [z1, [xn, y1]] = 2[xn+1, y1] − 2[xn, y2]
= [z1, [xn−1, y2]] = 2[xn, y2] − 2[xn−1, y3]
...
= [z1, [x1, yn]] = 2[x2, yn] − 2[x1, yn+1]
implies that n · [z1, zn] = 2[xn+1, y1] − 2[x1, yn+1] and,
2[x1, yn+1] = [x1, [y1, zn]] = −[zn, [x1, y1]] − [y1, [zn, x1]]
= [z1, zn] + 2[xn+1, y1].
So [z1, zn] = 0. Now, consider [zi, zj ] for i ∈ {1, . . . , n − 1},
[zi, zj ] = [zi, [xj , y1]] = −([y1, [zi, xj ]] + [xj , [y1, zi]])
= 2[xi+j , y1] − 2[xj , yi+1],
and
[xj , yi+1] =
1
2
[xj , [yi, z1] = −
1
2
([z1, [xj , yi]] + [yi, [z1, xj ]])
= −
1
2
([z1, zn] + [yi, 2xj+1])
= [xj+1, yi]
By applying this i times we get [xj , yi+1] = [xn, y1] , so that
[zi, zj ] = 0 for i + j = n + 1
76 Finitely presented quadratic algebras
i.e., (6) holds for i + j = n + 1. By (1) - (3), the set {xi, yi, zi}
∞
i=1 forms
a basis for L as a vector space. It can be observed that L =
⊕
k>1
Lk where
L2k−1 = 〈xk〉 ⊕ 〈yk〉 and Lk = 〈zk〉 for k > 1. Since
[L2k−1, L2m−1] ⊆ L2(k+m−1), [L2k, L2m] = 0,
[L2k−1, L2m] ⊆ L2(k+m)−1,
L admits an N-gradation given by the sum of occurrences of x and y in
each commutator i.e., L =
⊕
k>1 Lk is a graded Lie algebra generated by
two elements of degree 1 (deg(a) = min{n|a ∈
⊕n
k=1 Lk)}) and L has
linear growth (dim Li ∈ {1, 2} for i > 1 ).
Remark 1. We notice that L also admits a Z-gradation. It is a 3-graded
Lie algebra (in the sense of [dO03]) over C generated by elements x of
degree 1 and y of degree −1 .
3. The relation between the growth of a Lie algebra and
its universal enveloping algebra
Let L be any Lie algebra over a field k and U(L) be its universal en-
veloping algebra. For an ordered basis u1, u2, . . . of L, monomials ui1
. . . uir
with i1 6 i2 6 · · · 6 ir form a basis for U(L) (Poincaré-Birkhoff-Witt
Theorem ([Ber78])). If L =
⊕
Ln is a graded Lie algebra such that all
the components are finite dimensional, then
∞
∑
n=0
bntn =
∞
∏
n=1
(1 − tn)−an (7)
where an := dim(Ln) and bn:=number of monomials of length n in U(L)
([Smi76]). The proof of the following proposition can be found in various
papers ([Ber83], [Pet93], [BG00]).
Proposition 2. If an and bn are related by (7) and an ∼ nd, then
bn ∼ en
d+1
d+2 .
Corollary 1. If a Lie algebra L grows polynomially then its universal
enveloping algebra U(L) has intermediate growth. In particular, if L has
linear growth, then U(L) has growth of type e
√
n.
D. Koçak 77
4. Veronese subalgebra of an associative graded algebra
Let A = k〈x1, . . . , xm〉 be a free associative algebra over a field k with
generating set {x1, . . . , xm}. Each element u of A can be written uniquely
as
u = u0 + u1 + · · · + ul,
where A0 = k, ui ∈ Ai and Ai is the vector space over k spanned by mi
monomials of length i. Let R = {f1, f2, . . . , fs} be a finite set of non-zero
homogeneous polynomials and I be the ideal generated by R. Since I is
generated by homogeneous polynomials, the factor algebra à = A/I is
graded:
à = Ã0 ⊕ Ã1 ⊕ · · · ⊕ Ãn ⊕ . . .
where Ãi = (Ai + I)/I ∼= Ai/(Ai ∩ I). For d > 1, a Veronese subalgebra
of à is defined as
Vd(Ã) := k ⊕ Ãd ⊕ Ã2d ⊕ . . .
It is straightforward to see that,
growth of à ∼ growth of Vd(Ã)
Proposition 3. [BF85] For sufficiently large d, Vd(Ã) is quadratic.
Proof. Let d1, . . . , ds be the degrees of f1, f2, . . . , fs respectively and
d > max{di, 1 6 i 6 s}. For any two words v′, v′′ such that
deg(v′) + di + deg(v′′) = d
consider the element v′fiv
′′ ∈ Ad, and for any two words w′, w′′ such that
deg(w′) + di + deg(w′′) = 2d
consider the element w′fiw
′′ ∈ A2d. Let R∗ = {v′fiv
′′, w′fiw
′′} for i ∈
{1, . . . , s} and a be a homogeneous element from A(n) ∩ I. Say a =
∑
αvfiw, where α ∈ k, v and w are words. If we choose a summand and
represent v = v1v2, deg(v1) is a multiple of d, 0 6 deg(v2) < d. Similarly,
w = w2w1, deg(w1) is a multiple of d, 0 6 deg(w2) < d. Then we will get
deg(v2fiw2) = d or 2d. Hence v2fiw2 ∈ R∗. It shows that Vd(A) ∩ I is an
ideal generated by the elements of R∗ and an element v′fiv
′′ is a linear
combination of free generators of A(n) whereas w′fiw
′′ is a quadratic
element in these generators. So Vd(Ã) = Vd(A)/(Vd(A) ∩ I) is a quadratic
algebra.
78 Finitely presented quadratic algebras
5. Proof of Theorem 1
Let L = 〈x1, . . . , xm | f1 = 0, . . . , fr = 0〉 where each of fi is a linear
combination of the commutators (elements of the form [xi1
, . . . , xik
] with
an arbitrary distribution of parentheses inside). Then the universal en-
veloping algebra U(L) of L is an associative algebra with the identical
set of generators and relations, where the commutators are thought of
as in the ordinary associative sense: [x, y] = xy − yx [Bou89, Proposi-
tion 2, p.14]. The universal enveloping algebra U(L) of L = 〈x1, y1 |
[x1, [x1, [x1, y1]]] = 0, [y1, [y1, [y1, x1]]] = 0〉 has the following presenta-
tion:
U(L) = 〈x1, y1 | x3
1y1 − 3x2
1y1x1 + 3x1y1x2
1 − y1x3
1 = 0,
y3
1x1 − 3y2
1x1y1 + 3y1x1y2
1 − x1y3
1 = 0〉.
So, the associative algebra U in Theorem 1 is the universal enveloping
algebra U(L) of L. By Proposition 2, since L has linear growth, the growth
rate of U(L) is intermediate of type e
√
n . In order to obtain a quadratic
algebra of intermediate growth we consider a Veronese subalgebra of V4(U)
as explained in the previous section and conclude that for a given finitely
presented graded algebra with all generators of degree 1, one can construct
a finitely presented graded algebra with all relations of degree 2. V4(U)
is an example of a finitely presented graded algebra with intermediate
growth. It has 14 generators and 96 relations. In the next section we
compute all these relations.
6. A construction based on Kobayashi’s example
In this section we construct another example of a finitely presented
associative algebra with quadratic relations whose growth function is
intermediate. For this, we consider the following example of a monoid
with 0 that appears in the paper of Kobayashi [Kob95].
M = 〈a, b, c | ba = ab, bc = aca, acc = 0〉
where w(a) = w(c) = 1, w(b) = 2, w is a positive weight function on M .
Kobayashi shows that M is a finitely presented monoid with solvable word
problem which cannot be presented by a regular complete system. In order
to prove that it cannot be presented by a regular complete system, he
proves that M has intermediate growth. Now, we consider the semigroup
D. Koçak 79
algebra k[M ] over a field k. k[M ] has the same presentation and growth
function with M . So k[M ] is an example of finitely presented associative
graded algebra of intermediate growth. But the generators of k[M ] have
degrees deg(a) = deg(c) = 1 and deg(b) = 2. To construct a quadratic
algebra with these properties, we need to consider an algebra whose
generators are all of degree 1. Thus we consider the following monoid:
M̃ = 〈a, b, c | b2a = ab2, b2c = aca, acc = 0,
aba = 0, abc = 0, cba = 0, cbc = 0〉
where w(a) = w(b) = w(c) = 1.
Now, we have the monoid algebra A := k[M̃ ] over a field k:
A = 〈a, b, c | b2a = ab2, b2c = aca, acc = 0,
aba = 0, abc = 0, cba = 0, cbc = 0〉
where deg(a) = deg(b) = deg(c) = 1. To show that A has intermediate
growth, we first find a complete rewriting system for A. Let ≺ be the
shortlex order on 〈X〉 based on the order a ≺ b ≺ c i.e.,
w1 ≺ w2 implies |w1| < |w2| or |w1| = |w2| & w1 ≺lex w2.
Then A has the rewriting system R consisting of the following relations
b2a → ab2
b2c → aca
acc → 0
aba → 0
abc → 0
cba → 0
cbc → 0
It is easily seen that R is Noetherian. By applying the Knuth-Bendix algo-
rithm, we obtain the following complete rewriting system R∞ equivalent
to R:
R∞ = {b2a → ab2, b2c → aca, aba → 0, abc → 0, cba → 0, cbc → 0}
∪
∞
⋃
n=1
{ancan−1c → 0}.
80 Finitely presented quadratic algebras
A monomial (word) m is called irreducible with respect to the rewriting
system R if all the rewriting rules act trivially on m. The set of all
irreducible words with respect to R is denoted by Irr(R). Since R∞ is
a complete rewriting system, Irr(R∞) is the set of words which do not
contain u as a subword for any u → v ∈ R∞. By Bergman’s Diamond
Lemma [Ber78], Irr(R∞), forms a basis for A. Words in Irr(R∞) are of
the following form
bsam1cam2c . . . amr calbk
where s ∈ {0, 1}, l, k ∈ N∪{0} and 0 6 m1 6 m2 6 · · · 6 mr, mi ∈ N∪{0}
for i ∈ {1, . . . r}. So, the number of words in Irr(R∞) of length n is equal
to
n
∑
j=0
(2j+1) · |{(m1, . . . , mr) | 06m16 . . .6mr, m1+. . .+mr =n−j−r}|
=
n
∑
j=0
(2j + 1) · p(n − j)
where p(n) is the number of partitions of n. Hence
γA(n) ∼ p(n) ∼ e
√
n.
A is an example of finitely presented graded algebra with generators of
degree 1 and intermediate growth function and its Veronese subalgebra
V3(A) can be presented by finitely many quadratic relations (to be precise
with 21 generators and 280 relations).
7. Appendix: Presentation of the Veronese
subalgebra V4(U) of U
As we noted in the Section 5, U(L) is an associative algebra with
generators x, y and the set of relations
R = {x3y−3x2yx+3xyx2−yx3 = 0, y3x−3y2xy+3yxy2−xy3 = 0}.
Since R is a set of two homogeneous polynomials, U is a graded algebra.
Let V4(U) be the Veronese subalgebra of U . It was proven in Section 4
that V4(U) is a graded algebra generated by the set S of monomials of
length 4 over {x, y} and the set of relations R∗ = {fi = 0, vfiw = 0}
where v, w are monomials such that l(v) + l(w) = 4 and, f1 = x3y −
3x2yx + 3xyx2 − yx3, f2 = y3x − 3y2xy + 3yxy2 − xy3. Basically, R∗
D. Koçak 81
is the set of homogeneous polynomials of degree 4 or 8 generated by
R = {f1 = 0, f2 = 0} in k[x, y]. Since there are 48 different pairs (v, w)
of monomials, R∗ consists of 2 homogeneous polynomials of degree 4:
(i) yx3 = x3y − 3x2yx + 3xyx2, (ii) y3x = xy3 − 3yxy2 + 3y2xy
and 96 homogeneous polynomials of degree 8:
1) xyx2x4 = x4yx3 − 3x3yx4 + 3x2yxx4,
2) x3yx4 = x4x2yx − 3x4xyx2 + 3x4yx3,
3) x2y2yx3 = x3yy2x2 − 3x2yxy2x2 + 3x2y2xyx2,
4) xyx2x3y = x4yx2y − 3x3yx3y + 3x2yxx3y,
5) x3yx3y = x4x2y2 − 3x4xyxy + 3x4yx2y,
6) x2y2yx2y = x3yy2xy − 3x2yxy2xy + 3x2y2xyxy,
7) xyx2x2yx = x4yxyx − 3x3yx2yx + 3x2yxx2yx,
8) x2y2x4 = x2yxx2yx − 3x2yxxyx2 + 3x2yxyx3,
9) x2y2yxyx = x3yy3x − 3x2yxy3x + 3x2y2xy2x,
10) xyx2x2y2 = x4yxy2 − 3x3yx2y2 + 3x2yxx2y2,
11) x2y2x3y = x2yxx2y2 − 3x2yxxyxy + 3x2yxyx2y,
12) x2y2yxy2 = x3yy4 − 3x2yxy4 + 3x2y2xy3,
13) xyx2xyx2 = x4y2x2 − 3x3yxyx2 + 3x2yxxyx2,
14) xyxyx4 = xyx2x2yx − 3xyx2xyx2 + 3xyx2yx3,
15) xy3yx3 = xyxyy2x2 − 3xy2xy2x2 + 3xy3xyx2,
16) xyx2xy2x = x4y3x − 3x3yxy2x + 3x2yxxy2x,
17) xy3x4 = xy2xx2yx − 3xy2xxyx2 + 3xy2xyx3,
18) xy3yxyx = xyxyy3x − 3xy2xy3x + 3xy3xy2x,
19) xyx2xyxy = x4y2xy − 3x3yxyxy + 3x2yxxyxy,
20) xyxyx3y = xyx2x2y2 − 3xyx2xyxy + 3xyx2yx2y,
21) xy3yx2y = xyxyy2xy − 3xy2xy2xy + 3xy3xyxy,
22) xyx2xy3 = x4y4 − 3x3yxy3 + 3x2yxxy3,
23) xy3x3y = xy2xx2y2 − 3xy2xxyxy + 3xy2xyx2y,
24) xy3yxy2 = xyxyy4 − 3xy2xy4 + 3xy3xy3,
25) y2x2x4 = yx3yx3 − 3yx2yx4 + 3yxyxx4,
26) yx2yx4 = yx3x2yx − 3yx3xyx2 + 3yx3yx3,
27) yxy2yx3 = yx2yy2x2 − 3yxyxy2x2 + 3yxy2xyx2,
28) x2y2x2yx = yx3yxyx − 3yx2yx2yx + 3yxyxy2xy,
29) yxy2x4 = yxyxx2yx − 3yxyxxyx2 + 3yxyxyx3,
30) yxy2yxyx = yx2yy3x − 3yxyxy3x + 3yxy2xy2x,
31) y2x2x2y2 = yx3yxy2 − 3yx2yx2y2 + 3yxyxx2y2,
82 Finitely presented quadratic algebras
32) yxy2x3y = yxyxx2y2 − 3yxyxxyxy + 3yxyxyx2y,
33) yxy2yxy2 = yx2yy4 − 3yxyxy4 + 3yxy2xy3,
34) y2x2x3y = yx3yx2y − 3yx2yx3y + 3yxyxx3y,
35) yx2yx3y = yx3x2y2 − 3yx3xyxy + 3yx3yx2y,
36) yxy2yx2y = yx2yy2xy − 3yxyxy2xy + 3yxy2xyxy,
37) y2x2xyx2 = yx3y2x2 − 3yx2yxyx2 + 3yxyxxyx2,
38) y2xyx4 = y2x2x2yx − 3y2x2xyx2 + 3y2x2yx3,
39) y4yx3 = y2xyy2x2 − 3y3xy2x2 + 3y4xyx2,
40) y2x2xyxy = yx3y2xy − 3yx2yxyxy + 3yxyxxyxy,
41) y2xyx3y = y2x2x2y2 − 3y2x2xyxy + 3y2x2yx2y,
42) y4yx2y = y2xyy2xy − 3y3xy2xy + 3y4xyxy,
43) y2x2xy2x = yx3y3x − 3yx2yxy2x + 3yxyxxy2x,
44) y4x4 = y3xx2yx − 3y3xxyx2 + 3y3xyx3,
45) y4yxyx = y2xyy3x − 3y3xy3x + 3y4xy2x,
46) y2x2xy3 = yx3y4 − 3yx2yxy3 + 3yxyxxy3,
47) y4x3y = y3xx2y2 − 3y3xxyxy + 3y3xyx2y,
48) y4yxy2 = y2xyy4 − 3y3xy4 + 3y4xy3,
49) x2yxx4 = x4xyx2 − 3x4yx3 + 3x3yx4,
50) xy3x4 = x2y2yx3 − 3xyxyyx3 + 3xy2xyx3,
51) x3yy2x2 = x4y3x − 3x3yxy2x + 3x3yyxyx,
52) x2yxx3y = x4xyxy − 3x4yx2y + 3x3yx3y,
53) xy3x3y = x2y2yx2y − 3xyxyyx2y + 3xy2xyx2y,
54) x3yy2xy = x4y4 − 3x3yxy3 + 3x3yyxy2,
55) x2yxx2yx = x4xy2x − 3x4yxyx + 3x3yx2yx,
56) xy3x2yx = x2y2yxyx − 3xyxyyxyx + 3xy2xyxyx,
57) x2y2y2x2 = x2yxy3x − 3x2y2xy2x + 3x2y2yxyx,
58) x2yxx2y2 = x4xy3 − 3x4yxy2 + 3x3yx2y2,
59) xy3x2y2 = x2y2yxy2 − 3xyxyyxy2 + 3xy2xyxy2,
60) x2y2y2xy = x2yxy4 − 3x2y2xy3 + 3x2y2yxy2,
61) xy2xx4 = xyx2xyx2 − 3xyx2yx3 + 3xyxyx4,
62) xy3xyx2 = x2y2y2x2 − 3xyxyy2x2 + 3xy2xy2x2,
63) xyxyy2x2 = xyx2y3x − 3xyxyxy2x + 3xyxyyxyx,
64) xy2xx2yx = xyx2xy2x − 3xyx2yxyx + 3xyxyx2yx,
65) xy3xy2x = x2y2y3x − 3xyxyy3x + 3xy2xy3x,
66) xy3y2x2 = xy2xy3x − 3xy3xy2x + 3xy3yxyx,
67) xy2xx3y = xyx2xyxy − 3xyx2yx2y + 3xyxyx3y,
68) xy3xyxy = x2y2y2xy − 3xyxyy2xy + 3xy2xy2xy,
D. Koçak 83
69) xyxyy2xy = xyx2y4 − 3xyxyxy3 + 3xyxyyxy2,
70) xy2xx2y2 = xyx2xy3 − 3xyx2yxy2 + 3xyxyx2y2,
71) xy3xy3 = x2y2y4 − 3xyxyy4 + 3xy2xy4,
72) xy3y2xy = xy2xy4 − 3xy3xy3 + 3xy3yxy2,
73) yxyxx4 = yx3xyx2 − 3yx3yx3 + 3yx2yx4,
74) y4x4 = yxy2yx3 − 3y2xyyx3 + 3y3xyx3,
75) yx2yy2x2 = yx3y3x − 3yx2yxy2x + 3yx2yyxyx,
76) yxyxx2yx = yx3xy2x − 3yx3yxyx + 3yx2yx2yx,
77) y4x2yx = yxy2yxyx − 3y2xyyxyx + 3y3xyxyx,
78) yxy2y2x2 = yxyxy3x − 3yxy2xy2x + 3yxy2yxyx,
79) yxyxx2y2 = yx3xy3 − 3yx3yxy2 + 3yx2yx2y2,
80) y4x2y2 = yxy2yxy2 − 3y2xyyxy2 + 3y3xyxy2,
81) yxy2y2xy = yxyxy4 − 3yxy2xy3 + 3yxy2yxy2,
82) yxyxx3y = yx3xyxy − 3yx3yx2y + 3yx2yx3y,
83) y4x3y = yxy2yx2y − 3y2xyyx2y + 3y3xyx2y,
84) yx2yy2xy = yx3y4 − 3yx2yxy3 + 3yx2yyxy2,
85) y3xx4 = y2x2xyx2 − 3y2x2yx3 − 3y2xyx4,
86) y4xyx2 = yxy2y2x2 − 3y2xyy2x2 + 3y3xy2x2,
87) y2xyy2x2 = y2x2y3x − 3y2xyxy2x + 3y2xyyxyx,
88) y3xx3y = y2x2xyxy − 3y2x2yx2y + 3y2xyx3y,
89) y4xyxy = yxy2y2xy − 3y2xyy2xy + 3y3xy2xy,
90) y2xyy2xy = y2x2y4 − 3y2xyxy3 + 3y2xyyxy2,
91) y3xx2yx = y2x2yx2x − 3y2x2yxyx + 3y2xyx2yx,
92) y4xy2x = yxy2y3x − 3y2xyy3x + 3y3xy3x,
93) y4y2x2 = y3xy3x − 3y4xy2x + 3y4yxyx,
94) y3xx2y2 = y2x2xy3 − 3y2x2yxy2 + 3y2xyx2y2,
95) y4xy3 = yxy2y4 − 3y2xyy4 − 3y2xyy4 + 3y3xy4,
96) y4y2xy = y3xy4 − 3y4xy3 + 3y4yxy2.
We can rename the generators as follows:
y4 = Y1, y3x = Y2, y2xy = Y3, y2x2 = Y4,
yxy2 = Y5, yxyx = Y6, yx2y = Y7, yx3 = Y8,
xy3 = X1, xy2x = X2, xyxy = X3, xyx2 = X4,
x2y2 = X5, x2yx = X6, x3y = X7, x4 = X8.
So the relations will be
(i) Y8 = X7 − 3X6 + 3X4, (ii) Y2 = X1 − 3Y5 + 3Y3
84 Finitely presented quadratic algebras
1) X4X8 = X8Y8 − 3X7X8 + 3X6X8,
2) X7X8 = X8X6 − 3X8X4 + 3X8Y8,
3) X5Y8 = X7Y4 − 3X6Y4 + 3X5X4,
4) X4X7 = X8Y7 − 3X7X7 + 3X6X7,
5) X7X7 = X8X5 − 3X8X3 + 3X8Y7,
6) X5Y7 = X7Y3 − 3X6Y3 + 3X5X3,
7) X4X6 = X8Y6 − 3X7X6 + 3X6X6,
8) X5X8 = X6X6 − 3X6X4 + 3X6Y8,
9) X5Y6 = X7Y2 − 3X6Y2 + 3X5X2,
10) X4X5 = X8Y5 − 3X7X5 + 3X6X5,
11) X5X7 = X6X5 − 3X6X3 + 3X6Y7,
12) X5Y5 = X7Y1 − 3X6Y1 + 3X5X1,
13) X4X4 = X8Y4 − 3X7X4 + 3X6X4,
14) X3X8 = X4X6 − 3X4X4 + 3X4Y8,
15) X1Y8 = X3Y4 − 3X2Y4 + 3X1X4,
16) X4X2 = X8Y2 − 3X7X2 + 3X6X2,
17) X1X8 = X2X6 − 3X2X4 + 3X2Y8,
18) X1Y6 = X3Y2 − 3X2Y2 + 3X1X2,
19) X4X3 = X8Y3 − 3X7X3 + 3X6X3,
20) X3X7 = X4X5 − 3X4X3 + 3X4Y7,
21) X1Y7 = X3Y3 − 3X2Y3 + 3X1X3,
22) X4X1 = X8Y1 − 3X7X1 + 3X6X1,
23) X1X7 = X2X5 − 3X2X3 + 3X2Y7,
24) X1Y5 = X3Y1 − 3X2Y1 + 3X1X1,
25) Y4X8 = Y8Y8 − 3Y7X8 + 3Y6X8,
26) Y7X8 = Y8X6 − 3Y8X4 + 3Y8Y8,
27) Y5Y8 = Y7Y4 − 3Y6Y4 + 3Y5X4,
28) Y4X6 = Y8Y6 − 3Y7X6 + 3Y6X6,
29) Y5X8 = Y6X6 − 3Y6X4 + 3Y6Y8,
30) Y5Y6 = Y7Y2 − 3Y6Y2 + 3Y5X2,
31) Y4X5 = Y8Y5 − 3Y7X5 + 3Y6X5,
32) Y5X7 = Y6X5 − 3Y6X3 + 3Y6Y7,
33) Y5Y5 = Y7Y1 − 3Y6Y1 + 3Y5X1,
34) Y4X7 = Y8Y7 − 3Y7X7 + 3Y6X7,
35) Y7X7 = Y8X5 − 3Y8X3 + 3Y8Y7,
36) Y5Y7 = Y7Y3 − 3Y6Y3 + 3Y5X3,
37) Y4X4 = Y8Y4 − 3Y7X4 + 3Y6X4,
D. Koçak 85
38) Y3X8 = Y4X6 − 3Y4X4 + 3Y4Y8,
39) Y1Y8 = Y3Y4 − 3Y2Y4 + 3Y1X4,
40) Y4X3 = Y8Y3 − 3Y7X3 + 3Y6X3,
41) Y3X7 = Y4X5 − 3Y4X3 + 3Y4Y7,
42) Y1Y7 = Y3Y3 − 3Y2Y3 + 3Y1X3,
43) Y4X2 = Y8Y2 − 3Y7X2 + 3Y6X2,
44) Y1X8 = Y2X6 − 3Y2X4 + 3Y2Y8,
45) Y1Y6 = Y3Y2 − 3Y2Y2 + 3Y1X2,
46) Y4X1 = Y8Y1 − 3Y7X1 + 3Y6X1,
47) Y1X7 = Y2X5 − 3Y2X3 + 3Y2Y7,
48) Y1Y5 = Y3Y1 − 3Y2Y1 + 3Y1X1,
49) X6X8 = X8X4 − 3X8Y8 + 3X7X8,
50) X1X8 = X5Y8 − 3X3Y8 + 3X2Y8,
51) X7Y4 = X8Y2 − 3X7X2 + 3X7Y6,
52) X6X7 = X8X3 − 3X8Y7 + 3X7X7,
53) X1X7 = X5Y7 − 3X3Y7 + 3X2Y7,
54) X7Y3 = X8Y1 − 3X7X1 + 3X7Y5,
55) X6X6 = X8X2 − 3X8Y6 + 3X7X6,
56) X1X6 = X5Y6 − 3X3Y6 + 3X2Y6,
57) X5Y4 = X6Y2 − 3X5X2 + 3X5Y6,
58) X6X5 = X8X1 − 3X8Y5 + 3X7X5,
59) X1X5 = X5Y5 − 3X3Y5 + 3X2Y5,
60) X5Y3 = X6Y1 − 3X5X1 + 3X5Y5,
61) X2X8 = X4X4 − 3X4Y8 + 3X3X8,
62) X1X4 = X5Y4 − 3X3Y4 + 3X2Y4,
63) X3Y4 = X4Y2 − 3X3X2 + 3X3Y6,
64) X2X6 = X4X2 − 3X4Y6 + 3X3X6,
65) X1X2 = X5Y2 − 3X3Y2 + 3X2Y2,
66) X1Y4 = X2Y2 − 3X1X2 + 3X1Y6,
67) X2X7 = X4X3 − 3X4Y7 + 3X3X7,
68) X1X3 = X5Y3 − 3X3Y3 + 3X2Y2,
69) X3Y3 = X4Y1 − 3X3X1 + 3X3Y5,
70) X2X5 = X4X1 − 3X4Y5 + 3X3X5,
71) X1X1 = X5Y1 − 3X3Y1 + 3X2Y1,
72) X1Y3 = X2Y1 − 3X1X1 + 3X1Y5,
73) Y6X8 = Y8X4 − 3Y8Y8 + 3Y7X8,
74) Y1X8 = Y5Y8 − 3Y3Y8 + 3Y2Y8,
86 Finitely presented quadratic algebras
75) Y7Y4 = Y8Y2 − 3Y7X2 + 3Y7Y6,
76) Y6X6 = Y8X2 − 3Y8Y6 + 3Y7X6,
77) Y1X6 = Y5Y6 − 3Y3Y6 + 3Y2Y6,
78) Y5Y4 = Y6Y2 − 3Y5X2 + 3Y5Y6,
79) Y6X5 = Y8X1 − 3Y8Y5 + 3Y7X5,
80) Y1X5 = Y5Y5 − 3Y3Y5 + 3Y2Y5,
81) Y5Y3 = Y6Y1 − 3Y5X1 + 3Y5Y5,
82) Y6X7 = Y8X3 − 3Y8Y7 + 3Y7X7,
83) Y1X7 = Y5Y7 − 3Y3Y7 + 3Y2Y7,
84) Y7Y3 = Y8Y1 − 3Y7X1 + 3Y7Y5,
85) Y2X8 = Y4X4 − 3Y4Y8 + 3Y3X8,
86) Y1X4 = Y5Y4 − 3Y3Y4 + 3Y2Y4,
87) Y3Y4 = Y4Y2 − 3Y3X2 + 3Y3Y6,
88) Y2X7 = Y4X3 − 3Y4Y7 + 3Y3X7,
89) Y1X3 = Y5Y3 − 3Y3Y3 + 3Y2Y3,
90) Y3Y3 = Y4Y1 − 3Y3X1 + 3Y3Y5,
91) Y2X6 = Y4X2 − 3Y4Y6 + 3Y3X6,
92) Y1X2 = X5Y2 − 3Y3Y2 + 3Y2Y2,
93) Y1Y4 = Y2Y2 − 3Y1X2 + 3Y1Y6,
94) Y2X5 = Y4X1 − 3Y4Y5 + 3Y3X5,
95) Y1X1 = Y5Y1 − 3Y3Y1 + 3Y2Y1,
96) Y1Y3 = Y2Y1 − 3Y1X1 + 3Y1Y5.
We see that V4(U) is a quadratic algebra with generators X1, . . . , X8,
Y1, . . . Y8 and relations (i), (ii), 1)–96). This may not be the simplest
presentation of V4(U). Observe that the generators Y8 and Y2 are linear
combinations of other generators by (i) and (ii), so they can be removed
from the generating set.
Acknowledgements
I wish to thank Inna Capdeboscq for calling my attention to subalge-
bras of Kac-Moody algebras and Efim Zelmanov for the idea of considering
Veronese subalgebras to get quadratic algebras. I also thank my advisor
Rostislav Grigorchuk for his assistance in writing this paper.
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Contact information
D. Koçak Department of Mathematics, Texas A&M Uni-
versity, College Station, Texas 77840
E-Mail(s): dkocak@math.tamu.edu
Received by the editors: 06.03.2015
and in final form 09.07.2015.
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