Polynomial identities and asymptotic
methods
Antonio Giambruno
Dipartimento di Matematica ed Applicazioni
Università di Palermo
– p.
F = field of characteristic zero,
A= associative algebra over F .
X = {x1 , x2 , . . .} a countable set and F hXi= the free
associative algebra on X over F.
– p.
F = field of characteristic zero,
A= associative algebra over F .
X = {x1 , x2 , . . .} a countable set and F hXi= the free
associative algebra on X over F.
f = f (x1 , . . . , xn ) ∈ F hXi is a polynomial
identity for A, if f (a1 , . . . , an ) = 0, for all a1 , . . . , an ∈ A.
Definition.
– p.
Introduction
Problem. Describe the identities of a given algebra.
– p.
Introduction
Problem. Describe the identities of a given algebra.
If A = Mn (F ) = the algebra of n × n matrices over F , the description of
the identities is known only for n = 2.
– p.
Introduction
Problem. Describe the identities of a given algebra.
If A = Mn (F ) = the algebra of n × n matrices over F , the description of
the identities is known only for n = 2.
An effective way: combine algebraic and analytical methods.
– p.
Introduction
Problem. Describe the identities of a given algebra.
If A = Mn (F ) = the algebra of n × n matrices over F , the description of
the identities is known only for n = 2.
An effective way: combine algebraic and analytical methods.
The idea of applying numerical methods for investigating the identities
was originally realized in the associative case by Regev.
– p.
Introduction
Problem. Describe the identities of a given algebra.
If A = Mn (F ) = the algebra of n × n matrices over F , the description of
the identities is known only for n = 2.
An effective way: combine algebraic and analytical methods.
The idea of applying numerical methods for investigating the identities
was originally realized in the associative case by Regev. The same
analytical approach was also effectively applied in Lie theory.
– p.
Introduction
Problem. Describe the identities of a given algebra.
If A = Mn (F ) = the algebra of n × n matrices over F , the description of
the identities is known only for n = 2.
An effective way: combine algebraic and analytical methods.
The idea of applying numerical methods for investigating the identities
was originally realized in the associative case by Regev. The same
analytical approach was also effectively applied in Lie theory.
Given an algebra A over F , one associates to A a numerical sequence
cn (A), n = 1, 2, . . . , called the sequence of codimensions of A.
– p.
Introduction
Problem. Describe the identities of a given algebra.
If A = Mn (F ) = the algebra of n × n matrices over F , the description of
the identities is known only for n = 2.
An effective way: combine algebraic and analytical methods.
The idea of applying numerical methods for investigating the identities
was originally realized in the associative case by Regev. The same
analytical approach was also effectively applied in Lie theory.
Given an algebra A over F , one associates to A a numerical sequence
cn (A), n = 1, 2, . . . , called the sequence of codimensions of A.
The sequence cn (A), n = 1, 2 . . . , gives in some way a measure of the
polynomial relations vanishing in the algebra A and in general has
overexponential growth.
– p.
Generalities
Id(A) = {f ∈ F hXi | f ≡ 0 in A} = the T-ideal
of F hXi of polynomial identities of A.
Definition.
– p.
Generalities
Id(A) = {f ∈ F hXi | f ≡ 0 in A} = the T-ideal
of F hXi of polynomial identities of A.
Definition.
A is a PI-algebra if Id(A) 6= 0.
– p.
Generalities
Id(A) = {f ∈ F hXi | f ≡ 0 in A} = the T-ideal
of F hXi of polynomial identities of A.
Definition.
A is a PI-algebra if Id(A) 6= 0.
Kemer (1978). Every proper T-ideal of F hXi is finitely
generated as a T-ideal.
– p.
Generalities
Id(A) = {f ∈ F hXi | f ≡ 0 in A} = the T-ideal
of F hXi of polynomial identities of A.
Definition.
A is a PI-algebra if Id(A) 6= 0.
Kemer (1978). Every proper T-ideal of F hXi is finitely
generated as a T-ideal.
For every n ≥ 1, let
Vn = spanF {xσ(1) · · · xσ(n) | σ ∈ Sn }
be the space of multilinear polynomials in x1 , . . . , xn .
– p.
Generalities
Id(A) = {f ∈ F hXi | f ≡ 0 in A} = the T-ideal
of F hXi of polynomial identities of A.
Definition.
A is a PI-algebra if Id(A) 6= 0.
Kemer (1978). Every proper T-ideal of F hXi is finitely
generated as a T-ideal.
For every n ≥ 1, let
Vn = spanF {xσ(1) · · · xσ(n) | σ ∈ Sn }
be the space of multilinear polynomials in x1 , . . . , xn .
Remark.
Since charF = 0, Id(A) is determined by its multilinear
polynomials (polarization process).
– p.
Definition.
The non-negative integer
Vn
cn (A) = dimF
Vn ∩ Id(A)
is called the nth codimension of A.
– p.
Definition.
The non-negative integer
Vn
cn (A) = dimF
Vn ∩ Id(A)
is called the nth codimension of A.
Remark.
A is a P I-algebra if and only if cn (A) < n!, for some n ≥ 1.
– p.
Definition.
The non-negative integer
Vn
cn (A) = dimF
Vn ∩ Id(A)
is called the nth codimension of A.
Remark.
A is a P I-algebra if and only if cn (A) < n!, for some n ≥ 1.
Regev (1972). If A is a PI-algebra, then there exists d ≥ 1
such that cn (A) ≤ dn , for all n.
– p.
Definition.
The non-negative integer
Vn
cn (A) = dimF
Vn ∩ Id(A)
is called the nth codimension of A.
Remark.
A is a P I-algebra if and only if cn (A) < n!, for some n ≥ 1.
Regev (1972). If A is a PI-algebra, then there exists d ≥ 1
such that cn (A) ≤ dn , for all n.
Notation.
[x1 , x2 ] = x1 x2 − x2 x1
[x1 , x2 , . . . , xn ] = [[[x1 , x2 ], x3 ], . . .], xn ]
hf1 , . . . , ft iT = the T-ideal generated by f1 , . . . , ft
– p.
Examples
A = F [x]
– p.
Examples
A = F [x]
Id(A) = h[x1 , x2 ]iT
– p.
Examples
A = F [x]
Id(A) = h[x1 , x2 ]iT
cn (A) = 1, for all n ≥ 1.
– p.
Examples
A = F [x]
Id(A) = h[x1 , x2 ]iT
cn (A) = 1, for all n ≥ 1.
G = he1 , e2 , . . . | ei ej = −ej ei , for all i, ji is the infinite
dimensional Grassmann algebra over F .
– p.
Examples
A = F [x]
Id(A) = h[x1 , x2 ]iT
cn (A) = 1, for all n ≥ 1.
G = he1 , e2 , . . . | ei ej = −ej ei , for all i, ji is the infinite
dimensional Grassmann algebra over F .
Id(G) = h[x1 , x2 , x3 ]iT
– p.
Examples
A = F [x]
Id(A) = h[x1 , x2 ]iT
cn (A) = 1, for all n ≥ 1.
G = he1 , e2 , . . . | ei ej = −ej ei , for all i, ji is the infinite
dimensional Grassmann algebra over F .
Id(G) = h[x1 , x2 , x3 ]iT
cn (G) = 2n−1 , for all n ≥ 1.
– p.

U T2 = 
F F
0 F

 2 × 2 upper triangular matrices
– p.

U T2 = 
F F
0 F

 2 × 2 upper triangular matrices
Id(U T2 ) = h[x1 , x2 ][x3 , x4 ]iT
– p.

U T2 = 
F F
0 F

 2 × 2 upper triangular matrices
Id(U T2 ) = h[x1 , x2 ][x3 , x4 ]iT
cn (U T2 ) = 2n−1 (n − 2) + 2 for all n ≥ 1.
– p.

U T2 = 
F F
0 F

 2 × 2 upper triangular matrices
Id(U T2 ) = h[x1 , x2 ][x3 , x4 ]iT
cn (U T2 ) = 2n−1 (n − 2) + 2 for all n ≥ 1.


F F
 2 × 2 matrices over F
M2 (F ) = 
F F
– p.

U T2 = 
F F
0 F

 2 × 2 upper triangular matrices
Id(U T2 ) = h[x1 , x2 ][x3 , x4 ]iT
cn (U T2 ) = 2n−1 (n − 2) + 2 for all n ≥ 1.


F F
 2 × 2 matrices over F
M2 (F ) = 
F F
Id(M2 (F )) = h[[x1 , x2 ]2 , x3 ], St4 (x1 , . . . , x4 )iT
n−1
4√
cn (M2 (F )) ≃ n πn ,
n→∞
– p.

U T2 = 
F F
0 F

 2 × 2 upper triangular matrices
Id(U T2 ) = h[x1 , x2 ][x3 , x4 ]iT
cn (U T2 ) = 2n−1 (n − 2) + 2 for all n ≥ 1.


F F
 2 × 2 matrices over F
M2 (F ) = 
F F
Id(M2 (F )) = h[[x1 , x2 ]2 , x3 ], St4 (x1 , . . . , x4 )iT
n−1
4√
cn (M2 (F )) ≃ n πn ,
n→∞
where St4 (x1 , . . . , x4 ) =
P
σ∈S4 (sgn σ)xσ(1)
· · · xσ(4) .
– p.

F F F

A=
 0
0


0 F 

0 0
– p.

F F F

A=
 0
0


0 F 

0 0
Id(A) = h[x1 , x2 ]x3 x4 iT
– p.

F F F

A=
 0
0


0 F 

0 0
Id(A) = h[x1 , x2 ]x3 x4 iT
cn (A) = n(n − 1), for all n ≥ 2.
– p.
Kemer (1978). For a PI-algebra A, cn (A), n = 1, 2, . . . , is
either polynomially bounded or grows exponentially.
– p.
Kemer (1978). For a PI-algebra A, cn (A), n = 1, 2, . . . , is
either polynomially bounded or grows exponentially.
G.-Zaicev (1999). For a PI-algebra A,
p
exp(A) = lim n cn (A)
n→∞
exists and is an integer called the PI-exponent of the
algebra A.
– p.
Kemer (1978). For a PI-algebra A, cn (A), n = 1, 2, . . . , is
either polynomially bounded or grows exponentially.
G.-Zaicev (1999). For a PI-algebra A,
p
exp(A) = lim n cn (A)
n→∞
exists and is an integer called the PI-exponent of the
algebra A.
Examples.
exp(F [x]) = 1,
exp(U T2 ) = exp(G) = 2, exp(M2 (F )) = 4
– p.
Methods
Sn = the symmetric group acts on the left on
Vn = spanF {xσ(1) · · · xσ(n) | σ ∈ Sn }:
– p. 1
Methods
Sn = the symmetric group acts on the left on
Vn = spanF {xσ(1) · · · xσ(n) | σ ∈ Sn }:
if σ ∈ Sn , f (x1 , . . . , xn ) ∈ Vn , then
σf (x1 , . . . , xn ) = f (xσ(1) , . . . , xσ(n) ).
– p. 1
Methods
Sn = the symmetric group acts on the left on
Vn = spanF {xσ(1) · · · xσ(n) | σ ∈ Sn }:
if σ ∈ Sn , f (x1 , . . . , xn ) ∈ Vn , then
σf (x1 , . . . , xn ) = f (xσ(1) , . . . , xσ(n) ).
If Q = Id(A) is a T-ideal, Vn ∩ Q is invariant under this
action. Hence
Vn
Vn (A) =
Vn ∩ Q
has a structure of Sn -module.
– p. 1
Methods
Sn = the symmetric group acts on the left on
Vn = spanF {xσ(1) · · · xσ(n) | σ ∈ Sn }:
if σ ∈ Sn , f (x1 , . . . , xn ) ∈ Vn , then
σf (x1 , . . . , xn ) = f (xσ(1) , . . . , xσ(n) ).
If Q = Id(A) is a T-ideal, Vn ∩ Q is invariant under this
action. Hence
Vn
Vn (A) =
Vn ∩ Q
has a structure of Sn -module.
Let χn (A) be its Sn -character (called the n-th cocharacter
of A).
– p. 1
By complete reducibility we can write
X
χn (A) =
mλ χ λ
λ⊢n
where for a partition λ of n, χλ is the irreducible
Sn -character associated to λ and mλ ≥ 0 is the
corresponding multiplicity.
– p. 1
By complete reducibility we can write
X
χn (A) =
mλ χ λ
λ⊢n
where for a partition λ of n, χλ is the irreducible
Sn -character associated to λ and mλ ≥ 0 is the
corresponding multiplicity.
Berele. For a PI-algebra A, the multiplicities mλ are
polynomially bounded.
– p. 1
Amitsur-Regev. If A is a PI-algebra, there exist integers k, l such that
χn (A) =
X
mλ χλ
λ⊢n
λ∈H(k,l)
where H(k, l) =
S
n≥1 {λ
= (λ1 , λ2 , . . .) ⊢ n | λk+1 ≤ l}.
– p. 1
Amitsur-Regev. If A is a PI-algebra, there exist integers k, l such that
χn (A) =
X
mλ χλ
λ⊢n
λ∈H(k,l)
S
where H(k, l) = n≥1 {λ = (λ1 , λ2 , . . .) ⊢ n | λk+1 ≤ l}.
Thus H(k, l) = the set of all diagrams whose shape lies in the hook
shaped region
6
k
?
l -
– p. 1
G.-Zaicev. If A is a PI-algebra, there exist integers
k ≥ l ≥ 0, m ≥ 0 such that
X
χn (A) =
mλ χ λ
λ⊢n
λ∈SH(k,l,m)
where SH(k, l, m)=
6
k
.
6
?
m
m
?
l-
– p. 1
Key result.
X
χλ (1) ≃ Cnt (k + l)n .
n→∞
λ∈H(k,l)
– p. 1
Exponent, proper exponent and Lie exponent
For every n ≥ 1,
Vn = spanF {xσ(1) · · · xσ(n) | σ ∈ Sn }
is the space of multilinear polynomials in x1 , . . . , xn ,
– p. 1
Exponent, proper exponent and Lie exponent
For every n ≥ 1,
Vn = spanF {xσ(1) · · · xσ(n) | σ ∈ Sn }
is the space of multilinear polynomials in x1 , . . . , xn ,
Pn = spanF {[xσ(1) · · · xσ(i1 ) ] · · · [xσ(it +1) · · · xσ(n) ] | σ ∈ Sn }
is the space of multilinear proper polynomials in x1 , . . . , xn ,
– p. 1
Exponent, proper exponent and Lie exponent
For every n ≥ 1,
Vn = spanF {xσ(1) · · · xσ(n) | σ ∈ Sn }
is the space of multilinear polynomials in x1 , . . . , xn ,
Pn = spanF {[xσ(1) · · · xσ(i1 ) ] · · · [xσ(it +1) · · · xσ(n) ] | σ ∈ Sn }
is the space of multilinear proper polynomials in x1 , . . . , xn ,
Ln = spanF {[xn , xσ(1) · · · xσ(n−1) | σ ∈ Sn−1 }
is the space of multilinear Lie polynomials in x1 , . . . , xn .
– p. 1
Exponent, proper exponent and Lie exponent
For every n ≥ 1,
Vn = spanF {xσ(1) · · · xσ(n) | σ ∈ Sn }
is the space of multilinear polynomials in x1 , . . . , xn ,
Pn = spanF {[xσ(1) · · · xσ(i1 ) ] · · · [xσ(it +1) · · · xσ(n) ] | σ ∈ Sn }
is the space of multilinear proper polynomials in x1 , . . . , xn ,
Ln = spanF {[xn , xσ(1) · · · xσ(n−1) | σ ∈ Sn−1 }
is the space of multilinear Lie polynomials in x1 , . . . , xn .
Any T-ideal of polynomial identities of an algebra
with 1 can be generated by its proper polynomials.
Remark.
– p. 1
If A is an associative algebra, define
Vn
cn (A) = dimF
,
Vn ∩ Id(A)
cLn (A)
cpn (A)
Pn
= dimF
,
Pn ∩ Id(A)
Ln
= dimF
,
Ln ∩ Id(A)
– p. 1
If A is an associative algebra, define
Vn
cn (A) = dimF
,
Vn ∩ Id(A)
cLn (A)
cpn (A)
Pn
= dimF
,
Pn ∩ Id(A)
Ln
= dimF
,
Ln ∩ Id(A)
the n-th codimension, the n-th proper codimension and
the n-th Lie codimension of A, respectively.
– p. 1
If A is an associative algebra, define
Vn
cn (A) = dimF
,
Vn ∩ Id(A)
cLn (A)
cpn (A)
Pn
= dimF
,
Pn ∩ Id(A)
Ln
= dimF
,
Ln ∩ Id(A)
the n-th codimension, the n-th proper codimension and
the n-th Lie codimension of A, respectively.
Also define
p
p
n
L
p
exp (A) = lim
cLn (A), exp (A) = lim n cpn (A),
n→∞
n→∞
the Lie exponent and the proper exponent (provided they
exist).
– p. 1
If A is any algebra, then
cLn (A) ≤ cpn (A) ≤ cn (A),
for all n ≥ 1.
– p. 1
If A is any algebra, then
cLn (A) ≤ cpn (A) ≤ cn (A),
for all n ≥ 1.
Examples
A = F [x]
cn (A) = 1, cpn (A) = 0, cLn (A) = 0, for all n ≥ 1.
– p. 1
If A is any algebra, then
cLn (A) ≤ cpn (A) ≤ cn (A),
for all n ≥ 1.
Examples
A = F [x]
cn (A) = 1, cpn (A) = 0, cLn (A) = 0, for all n ≥ 1.
exp(M2 (F )) = 4, expL (M2 (F )) = 3, hence
expp (M2 (F )) = 3.
– p. 1

A=
G G
0
0

.
exp(A) = expL (A) = 2 and, so, expp (A) = 2.
– p. 1

A=
G G
0
0

.
exp(A) = expL (A) = 2 and, so, expp (A) = 2.
Let A be the algebra of (k + 1) × (k + 1) matrices with
zero last row:


a11 . . . a1,k+1
 .

.
 ..
.. 


A = {
 | aij ∈ F }.
ak1 . . . ak,k+1 


0 ...
0
expL (A) = exp(A) = k 2 . Hence also expp (A) = k 2 .
– p. 1
G.-Zaicev (2007). Let A be a finitely generated
PI-algebra.
Then
1) expL (A) and expp (A) exist and are integers;
2) expL (A) = expp (A) = exp(A) or exp(A) − 1;
3) if A is a unitary algebra, then
expL (A) = expp (A) = exp(A) − 1.
– p. 1
G.-Zaicev (2007). Let A be a finitely generated
PI-algebra.
Then
1) expL (A) and expp (A) exist and are integers;
2) expL (A) = expp (A) = exp(A) or exp(A) − 1;
3) if A is a unitary algebra, then
expL (A) = expp (A) = exp(A) − 1.
G= the infinite dimensional Grassmann algebra over F .
– p. 1
G.-Zaicev (2007). Let A be a finitely generated
PI-algebra.
Then
1) expL (A) and expp (A) exist and are integers;
2) expL (A) = expp (A) = exp(A) or exp(A) − 1;
3) if A is a unitary algebra, then
expL (A) = expp (A) = exp(A) − 1.
G= the infinite dimensional Grassmann algebra over F .

0 if n odd,
cn (G) = 2n−1 , cpn (G) =
, cLn (G) = 0, n ≥ 3.
1 if n even.
– p. 1
If A = G ⊕ B ⊕ N where B is a finite dimensional
algebra with Jacobson radical of codimension 1 and N is a
nilpotent algebra, then, for n large enough cpn (A) = cpn (G).
Hence expp (A) does not exist.
Remark.
– p. 2
If A = G ⊕ B ⊕ N where B is a finite dimensional
algebra with Jacobson radical of codimension 1 and N is a
nilpotent algebra, then, for n large enough cpn (A) = cpn (G).
Hence expp (A) does not exist.
Remark.
G.-Zaicev (2007). Let A be a PI-algebra with
exp(A) = d ≥ 1. Then expp (A) exists, is an integer and
expp (A) = d or d − 1, unless exp(A) = 2 and A is
PI-equivalent to an algebra of the type G ⊕ B ⊕ N where B
is a finite dimensional algebra whose Jacobson radical is
of codimension 1 and N is a nilpotent algebra.
– p. 2
Example

A = M1,1 (G) = 
G0 G1
G1 G0

.
– p. 2
Example

A = M1,1 (G) = 
G0 G1
G1 G0

.
It can be shown that asymptotically cn (A) ≤ n3 . Hence
expL (A) = 1. Also exp(A) = 4 and expp (A) = 3.
– p. 2
Max T-id
Notation. If Q = Id(A) is a T-ideal, exp(Q) = exp(A).
– p. 2
Max T-id
Notation. If Q = Id(A) is a T-ideal, exp(Q) = exp(A).
Problem. Try to classify T-ideals according to their
exponent.
– p. 2
Max T-id
Notation. If Q = Id(A) is a T-ideal, exp(Q) = exp(A).
Problem. Try to classify T-ideals according to their
exponent.
A T-ideal P is verbally prime if f g ∈ P with
f (x1 , . . . , xn ), g(xn+1 , . . . , xm ) ∈ F hXi, implies that f ∈ P or
g ∈ P.
Definition.
– p. 2
Max T-id
Notation. If Q = Id(A) is a T-ideal, exp(Q) = exp(A).
Problem. Try to classify T-ideals according to their
exponent.
A T-ideal P is verbally prime if f g ∈ P with
f (x1 , . . . , xn ), g(xn+1 , . . . , xm ) ∈ F hXi, implies that f ∈ P or
g ∈ P.
Definition.
Classification of verbally prime T-ideals (Kemer):
the only verbally prime T-ideals are
F hXi, 0, Id(Mk (F )), Id(Mk,l (G)), Id(Mk (G)),
k ≥ l ≥ 1.
– p. 2
where G = G0 ⊕ G1 ,
G0 = spanF {ei1 · · · ei2t | 1 ≤ i1 < · · · < i2t },
G1 = spanF {ei1 · · · ei2t+1 | 1 ≤ i1 < · · · < i2t+1 }
and
k
Mk,l (G) =
k
l


l
G0 G1
G1 G0

 .
– p. 2
where G = G0 ⊕ G1 ,
G0 = spanF {ei1 · · · ei2t | 1 ≤ i1 < · · · < i2t },
G1 = spanF {ei1 · · · ei2t+1 | 1 ≤ i1 < · · · < i2t+1 }
and
k
Mk,l (G) =
k
l


l
G0 G1
G1 G0

 .
Kemer. If Q is a T-ideal, there exist verbally prime T-ideals
P1 , . . . , Pt such that
P1 · · · Pt ⊆ Q ⊆ P1 ∩ · · · ∩ Pt .
– p. 2
A T-ideal Q is maximal of exponent d ≥ 2 if
exp(Q) = d and exp(S) < d for every T-ideal S ! Q.
Definition.
– p. 2
A T-ideal Q is maximal of exponent d ≥ 2 if
exp(Q) = d and exp(S) < d for every T-ideal S ! Q.
Definition.
- Every proper verbally prime T-ideal is maximal
– p. 2
A T-ideal Q is maximal of exponent d ≥ 2 if
exp(Q) = d and exp(S) < d for every T-ideal S ! Q.
Definition.
- Every proper verbally prime T-ideal is maximal
- Id(G) and Id(U T2 ) are the only T-ideals maximal of
exponent 2
– p. 2
A T-ideal Q is maximal of exponent d ≥ 2 if
exp(Q) = d and exp(S) < d for every T-ideal S ! Q.
Definition.
- Every proper verbally prime T-ideal is maximal
- Id(G) and Id(U T2 ) are the only T-ideals maximal of
exponent 2

- Id(
G
G
0 G0

),

Id(
G0 G
0
G

),
Id(U T3 )
are the only T-ideals maximal of exponent 3.
– p. 2
Let A be an F -algebras. Then exp(A) > 2 if and
only if Id(A) ⊆ Id(B), where B is one of the following
algebras:




G G
G G
 , U T3 , M2 (F ), M1,1 (G).

,  0
0 G0
0 G
Theorem.
– p. 2
Let A be an F -algebras. Then exp(A) > 2 if and
only if Id(A) ⊆ Id(B), where B is one of the following
algebras:




G G
G G
 , U T3 , M2 (F ), M1,1 (G).

,  0
0 G0
0 G
Theorem.
exp(A) = 2 if and only if Id(A) 6⊆ Id(B), where
B is any of the above algebras, and either Id(A) ⊆ Id(G)
or Id(A) ⊆ Id(U T2 ).
Corollary.
– p. 2
G.-Zaicev (2003). A T-ideal Q is maximal of exponent d if
and only if Q = P1 · · · Pt , for some verbally prime T-ideals
P1 , . . . Pt .
– p. 2
G.-Zaicev (2003). A T-ideal Q is maximal of exponent d if
and only if Q = P1 · · · Pt , for some verbally prime T-ideals
P1 , . . . Pt .
Moreover Q = Id(A) where


A1
∗




A2


A=
,
.


..


0
At
Id(A1 ) = P1 , . . . , Id(At ) = Pt , and
exp(Q) = d = exp(P1 ) + · · · + exp(Pt ).
– p. 2
Special case
Let A be a finitely generated algebra. Then Id(A) is
maximal of exponent d if and only if Id(A) = Id(B) where




B=


Mk1 (F )
0
∗



Mk2 (F )

,
..

.

Mkt (F )
and exp(A) = exp(B) = d = k12 + · · · + kt2 .
– p. 2
Nonassociative algebras and exponential growth
– p. 2
F {X}= the free non-associative algebra on a countable set X.
– p. 2
F {X}= the free non-associative algebra on a countable set X.
A= a non-necessarily associative algebra over F .
– p. 2
F {X}= the free non-associative algebra on a countable set X.
A= a non-necessarily associative algebra over F .
Id(A) = {f ∈ F {X} | f ≡ 0 in A} = the T-ideal of F {X} of polynomial
identities of A.
– p. 2
F {X}= the free non-associative algebra on a countable set X.
A= a non-necessarily associative algebra over F .
Id(A) = {f ∈ F {X} | f ≡ 0 in A} = the T-ideal of F {X} of polynomial
identities of A.
For every n ≥ 1, let Vn = the subspace of F {X} of all multilinear
polynomials in the variables x1 , . . . , xn .
– p. 2
F {X}= the free non-associative algebra on a countable set X.
A= a non-necessarily associative algebra over F .
Id(A) = {f ∈ F {X} | f ≡ 0 in A} = the T-ideal of F {X} of polynomial
identities of A.
For every n ≥ 1, let Vn = the subspace of F {X} of all multilinear
polynomials in the variables x1 , . . . , xn .
Definition.
cn (A) = dimF
Vn
Pn ∩Id(A)
is the n-th codimension of A.
– p. 2
F {X}= the free non-associative algebra on a countable set X.
A= a non-necessarily associative algebra over F .
Id(A) = {f ∈ F {X} | f ≡ 0 in A} = the T-ideal of F {X} of polynomial
identities of A.
For every n ≥ 1, let Vn = the subspace of F {X} of all multilinear
polynomials in the variables x1 , . . . , xn .
Definition.
cn (A) = dimF
Vn
Pn ∩Id(A)
is the n-th codimension of A.
EXAMPLES
- The number of distinct arrangements of parentheses on a monomial
1 2n−2
of length n is the Catalan number n n−1 . Hence
2n−2
cn (F {X}) = dimF Vn = n−1 (n − 1)!.
- If F hXi is the free associative algebra, then cn (F hXi) = n!
- For LhXi = the free Lie algebra, we have cn (LhXi) = (n − 1)!.
– p. 2
In general for nonassociative algebras cn (A), n = 1, 2, . . .
has overexponential growth.
Remark.
– p. 3
In general for nonassociative algebras cn (A), n = 1, 2, . . .
has overexponential growth.
Remark.
Petrogradsky (1997) constructed a scale of overexponential functions
behaving like the codimension sequences of suitable Lie algebras.
– p. 3
In general for nonassociative algebras cn (A), n = 1, 2, . . .
has overexponential growth.
Remark.
Petrogradsky (1997) constructed a scale of overexponential functions
behaving like the codimension sequences of suitable Lie algebras.
Bahturin-Drensky (2002). if dim A = d < ∞, then cn (A) ≤ dn .
– p. 3
In general for nonassociative algebras cn (A), n = 1, 2, . . .
has overexponential growth.
Remark.
Petrogradsky (1997) constructed a scale of overexponential functions
behaving like the codimension sequences of suitable Lie algebras.
Bahturin-Drensky (2002). if dim A = d < ∞, then cn (A) ≤ dn .
OPEN PROBLEM. In case the sequence of codimensions is
p
exponentially bounded, does lim n cn (A) exist?
n→∞
– p. 3
In general for nonassociative algebras cn (A), n = 1, 2, . . .
has overexponential growth.
Remark.
Petrogradsky (1997) constructed a scale of overexponential functions
behaving like the codimension sequences of suitable Lie algebras.
Bahturin-Drensky (2002). if dim A = d < ∞, then cn (A) ≤ dn .
OPEN PROBLEM. In case the sequence of codimensions is
p
exponentially bounded, does lim n cn (A) exist?
n→∞
p
G-Zaicev (1999). For any associative PI-algebra A, lim n cn (A)
n→∞
exists and is a nonnegative integer.
– p. 3
In general for nonassociative algebras cn (A), n = 1, 2, . . .
has overexponential growth.
Remark.
Petrogradsky (1997) constructed a scale of overexponential functions
behaving like the codimension sequences of suitable Lie algebras.
Bahturin-Drensky (2002). if dim A = d < ∞, then cn (A) ≤ dn .
OPEN PROBLEM. In case the sequence of codimensions is
p
exponentially bounded, does lim n cn (A) exist?
n→∞
p
G-Zaicev (1999). For any associative PI-algebra A, lim n cn (A)
n→∞
exists and is a nonnegative integer.
Zaicev (2001). The same conclusion holds for any finite dimensional
Lie algebra.
– p. 3
In general for nonassociative algebras cn (A), n = 1, 2, . . .
has overexponential growth.
Remark.
Petrogradsky (1997) constructed a scale of overexponential functions
behaving like the codimension sequences of suitable Lie algebras.
Bahturin-Drensky (2002). if dim A = d < ∞, then cn (A) ≤ dn .
OPEN PROBLEM. In case the sequence of codimensions is
p
exponentially bounded, does lim n cn (A) exist?
n→∞
p
G-Zaicev (1999). For any associative PI-algebra A, lim n cn (A)
n→∞
exists and is a nonnegative integer.
Zaicev (2001). The same conclusion holds for any finite dimensional
Lie algebra.
G.-Mishchenko-Zaicev (2007). For any real number α > 1, we
construct an algebra Aα whose sequence of codimensions grows
p
exponentially and lim n cn (Aα ) = α.
n→∞
– p. 3
Let w = w1 w2 . . . be an infinite (associative) word in the alphabet
{0, 1}. Fix an integer m ≥ 2 and let Km,w = {ki }i≥1 be the sequence
defined by

 m,
if wi = 0
ki =
 m + 1, if wi = 1
– p. 3
Let w = w1 w2 . . . be an infinite (associative) word in the alphabet
{0, 1}. Fix an integer m ≥ 2 and let Km,w = {ki }i≥1 be the sequence
defined by

 m,
if wi = 0
ki =
 m + 1, if wi = 1
Definition.
Let A(m, w) be the algebra over F with basis
{a, b} ∪ Z1 ∪ Z2 ∪ . . .
(i)
where Zi = {zj | 1 ≤ j ≤ ki },
i = 1, 2, . . . .
– p. 3
Let w = w1 w2 . . . be an infinite (associative) word in the alphabet
{0, 1}. Fix an integer m ≥ 2 and let Km,w = {ki }i≥1 be the sequence
defined by

 m,
if wi = 0
ki =
 m + 1, if wi = 1
Definition.
Let A(m, w) be the algebra over F with basis
{a, b} ∪ Z1 ∪ Z2 ∪ . . .
(i)
where Zi = {zj | 1 ≤ j ≤ ki },
The multiplication table is
(i)
(i)
i = 1, 2, . . . .
(i)
(i)
(i)
(i)
z2 a = z3 , . . . , zki −1 a = zki , zki a = z1 ,
(i)
(i+1)
z1 b = z2
,
i = 1, 2 . . . ,
i = 1, 2, . . .
and all the remaining products are zero.
– p. 3
(i)
Remark. The only non-zero products are of the type zj f (a, b) for
some left-normed monomial f (a, b).
– p. 3
(i)
Remark. The only non-zero products are of the type zj f (a, b) for
some left-normed monomial f (a, b).
We further specialize the algebra A(m, w) by choosing the word w in a
suitable way.
– p. 3
(i)
Remark. The only non-zero products are of the type zj f (a, b) for
some left-normed monomial f (a, b).
We further specialize the algebra A(m, w) by choosing the word w in a
suitable way.
Given an infinite word w in a finite alphabet, the complexity
Compw of w is the function Compw : N → N, where Compw (n) is the
number of distinct subwords of w of length n.
Definition.
– p. 3
(i)
Remark. The only non-zero products are of the type zj f (a, b) for
some left-normed monomial f (a, b).
We further specialize the algebra A(m, w) by choosing the word w in a
suitable way.
Given an infinite word w in a finite alphabet, the complexity
Compw of w is the function Compw : N → N, where Compw (n) is the
number of distinct subwords of w of length n.
Definition.
An infinite word w = w1 w2 · · · in the alphabet {0, 1} is
periodic with period T if wi = wi+T for i = 1, 2, . . . .
Definition.
– p. 3
(i)
Remark. The only non-zero products are of the type zj f (a, b) for
some left-normed monomial f (a, b).
We further specialize the algebra A(m, w) by choosing the word w in a
suitable way.
Given an infinite word w in a finite alphabet, the complexity
Compw of w is the function Compw : N → N, where Compw (n) is the
number of distinct subwords of w of length n.
Definition.
An infinite word w = w1 w2 · · · in the alphabet {0, 1} is
periodic with period T if wi = wi+T for i = 1, 2, . . . .
Definition.
If w is a periodic word of period T , Compw (n) ≤ T . Moreover
for any aperiodic word, Compw (n) ≥ n + 1.
Remark.
– p. 3
(i)
Remark. The only non-zero products are of the type zj f (a, b) for
some left-normed monomial f (a, b).
We further specialize the algebra A(m, w) by choosing the word w in a
suitable way.
Given an infinite word w in a finite alphabet, the complexity
Compw of w is the function Compw : N → N, where Compw (n) is the
number of distinct subwords of w of length n.
Definition.
An infinite word w = w1 w2 · · · in the alphabet {0, 1} is
periodic with period T if wi = wi+T for i = 1, 2, . . . .
Definition.
If w is a periodic word of period T , Compw (n) ≤ T . Moreover
for any aperiodic word, Compw (n) ≥ n + 1.
Remark.
An infinite word w is a Sturmian word if Compw (n) = n + 1
for all n ≥ 1.
Definition.
– p. 3
For a finite word x, the height h(x) of x is the number of
letters 1 appearing in x.
Definition. If |x| denotes the length of the word x, the slope of x is
defined as π(x) = h(x)
|x| .
Definition.
– p. 3
For a finite word x, the height h(x) of x is the number of
letters 1 appearing in x.
Definition. If |x| denotes the length of the word x, the slope of x is
defined as π(x) = h(x)
|x| .
Definition.
Extension to infinite words: let w be an infinite word and let w(1, n)
denote its prefix subword of length n. If the limit
h(w(1, n))
n→∞
n
π(w) = lim
exists then π(w) is called the slope of w.
– p. 3
For a finite word x, the height h(x) of x is the number of
letters 1 appearing in x.
Definition. If |x| denotes the length of the word x, the slope of x is
defined as π(x) = h(x)
|x| .
Definition.
Extension to infinite words: let w be an infinite word and let w(1, n)
denote its prefix subword of length n. If the limit
h(w(1, n))
n→∞
n
π(w) = lim
exists then π(w) is called the slope of w.
For periodic words and Sturmian words the slope is well defined.
– p. 3
Theorem.
Let w be a Sturmian or periodic word. Then
1) the slope π(w) of w exists;
2) for any real number α ∈ (0, 1) there exists a word w with π(w) = α
and w is Sturmian or periodic according as α is irrational or
rational, respectively.
– p. 3
Theorem.
Let w be a Sturmian or periodic word. Then
1) the slope π(w) of w exists;
2) for any real number α ∈ (0, 1) there exists a word w with π(w) = α
and w is Sturmian or periodic according as α is irrational or
rational, respectively.
Let w be an infinite Sturmian or periodic word
with slope α, 0 < α < 1. If m ≥ 2 then for the algebra
A = A(m, w) the P I-exponent exists and exp(A) = Φ(β)
1
where β = m+α
and Φ : R → R is defined by
1
Φ(x) = xx (1−x)
1−x .
Theorem.
– p. 3
Remark.
the function Φ is continuous and Φ((0, 12 )) = (1, 2). Hence
– p. 3
Remark.
the function Φ is continuous and Φ((0, 12 )) = (1, 2). Hence
For any real number d, 1 < d < 2, there exists an algebra A
such that exp(A) = d.
Corollary.
– p. 3
Remark.
the function Φ is continuous and Φ((0, 12 )) = (1, 2). Hence
For any real number d, 1 < d < 2, there exists an algebra A
such that exp(A) = d.
Corollary.
The above theorem can be generalized to all real numbers > 1.
– p. 3
Remark.
the function Φ is continuous and Φ((0, 12 )) = (1, 2). Hence
For any real number d, 1 < d < 2, there exists an algebra A
such that exp(A) = d.
Corollary.
The above theorem can be generalized to all real numbers > 1.
For any real number t ≥ 1 there exists an algebra R such
that exp(R) = t.
Theorem.
– p. 3
Remark.
the function Φ is continuous and Φ((0, 12 )) = (1, 2). Hence
For any real number d, 1 < d < 2, there exists an algebra A
such that exp(A) = d.
Corollary.
The above theorem can be generalized to all real numbers > 1.
For any real number t ≥ 1 there exists an algebra R such
that exp(R) = t.
Theorem.
QUESTION: does the PI-exponent exist for any finite dimensional
algebra?
– p. 3
Constructing intermediate growth
An algebra A is of overpolynomial codimension growth if
for any constant k we have cn (A) > nk , for n large enough.
Definition.
– p. 3
Constructing intermediate growth
An algebra A is of overpolynomial codimension growth if
for any constant k we have cn (A) > nk , for n large enough.
Definition.
Let A be a finite dimensional algebra of overpolynomial
n
codimension growth and let dim A = d. Then cn (A) > n12 2 3d3 , for all n
large enough.
Theorem.
– p. 3
For a real number x, [x] denotes the integer part of x.
– p. 3
For a real number x, [x] denotes the integer part of x.
For any real number β with 0 < β < 1, let A = A(K) where
1
K = {ki }i≥1 is defined by the relation k1 + · · · + kt = [t β ], for all t ≥ 1.
Then the sequence of codimensions of A satisfies
Theorem.
lim logn logn cn (A) = β.
n→∞
nβ
Hence, cn (A) asymptotically equals n
.
– p. 3
For a real number x, [x] denotes the integer part of x.
For any real number β with 0 < β < 1, let A = A(K) where
1
K = {ki }i≥1 is defined by the relation k1 + · · · + kt = [t β ], for all t ≥ 1.
Then the sequence of codimensions of A satisfies
Theorem.
lim logn logn cn (A) = β.
n→∞
nβ
Hence, cn (A) asymptotically equals n
.
What about two dimensional algebras?
– p. 3
For a real number x, [x] denotes the integer part of x.
For any real number β with 0 < β < 1, let A = A(K) where
1
K = {ki }i≥1 is defined by the relation k1 + · · · + kt = [t β ], for all t ≥ 1.
Then the sequence of codimensions of A satisfies
Theorem.
lim logn logn cn (A) = β.
n→∞
nβ
Hence, cn (A) asymptotically equals n
.
What about two dimensional algebras?
Let A be a two dimensional algebra over a field of
characteristic zero. Then either cn (A) ≤ n + 1 or n12 2n ≤ cn (A) ≤ 2n .
Theorem.
– p. 3
Let α > 0 be a rational number and let Aα be the two dimensional
algebra with basis {a, b} and multiplication table
a2 = 0, ab = a, ba = 0, b2 = a + αb.
Theorem.
cn (Aα ) = n + 1 and Id(Aα ) 6= Id(Aβ ), if α 6= β.
– p. 3
Polynomial growth
– p. 3
Polynomial growth
Theorem.
For a variety V, the following conditions are equivalent:
1) cn (V) ≤ Cnt , for some constants C, t.
– p. 3
Polynomial growth
Theorem.
For a variety V, the following conditions are equivalent:
1) cn (V) ≤ Cnt , for some constants C, t.
2) exp(V) ≤ 1.
– p. 3
Polynomial growth
Theorem.
For a variety V, the following conditions are equivalent:
1) cn (V) ≤ Cnt , for some constants C, t.
2) exp(V) ≤ 1.
3) G, U T2 6∈ V.
– p. 3
Polynomial growth
Theorem.
For a variety V, the following conditions are equivalent:
1) cn (V) ≤ Cnt , for some constants C, t.
2) exp(V) ≤ 1.
3) G, U T2 6∈ V.
4) There exists a constant q such that
χn (V) =
X
mλ χλ
(5)
λ⊢n
|λ|−λ1 ≤q
for all n ≥ 1. When V = var(A) with A a finite dimensional algebra,
q is such that J(A)q = 0.
– p. 3
Polynomial growth
Theorem.
For a variety V, the following conditions are equivalent:
1) cn (V) ≤ Cnt , for some constants C, t.
2) exp(V) ≤ 1.
3) G, U T2 6∈ V.
4) There exists a constant q such that
χn (V) =
X
mλ χλ
(6)
λ⊢n
|λ|−λ1 ≤q
for all n ≥ 1. When V = var(A) with A a finite dimensional algebra,
q is such that J(A)q = 0.
5) There exists a constant k such that in (6)
X
mλ < k.
λ⊢n
– p. 3
Drensky-Regev (1996) If cn (A) is polynomially bounded,
then asymptotically
cn (A) = qnk + O(nk−1 ) ≈ qnk ,
n → ∞,
for some rational number q.
– p. 4
Drensky-Regev (1996) If cn (A) is polynomially bounded,
then asymptotically
cn (A) = qnk + O(nk−1 ) ≈ qnk ,
n → ∞,
for some rational number q.
If A is a unitary algebra and k > 1,
1
≤ q ≤
k!
k
X
(−1)j
j=2
j!
1
→ ,
e
k → ∞,
(8)
where e = 2, 71 . . ..
– p. 4
Drensky-Regev (1996) If cn (A) is polynomially bounded,
then asymptotically
cn (A) = qnk + O(nk−1 ) ≈ qnk ,
n → ∞,
for some rational number q.
If A is a unitary algebra and k > 1,
1
≤ q ≤
k!
k
X
(−1)j
j=2
j!
1
→ ,
e
k → ∞,
(9)
where e = 2, 71 . . ..
In the non-unitary case, for any 0 < q ∈ Q it is possible to construct an
algebra A, such that cn (A) ≈ qnk , for a suitable k.
– p. 4
G.- La Mattina - Petrogradsky (2007)
– p. 4
G.- La Mattina - Petrogradsky (2007)
Construction of an algebra of upper triangular matrices
Pk (−1)j
realizing the value q = j=2 j! .
– p. 4
G.- La Mattina - Petrogradsky (2007)
Construction of an algebra of upper triangular matrices
Pk (−1)j
realizing the value q = j=2 j! .
The above lower bound is reached only in case k is
even. For k odd the lower bound is given by k−1
.
k!
– p. 4
G.- La Mattina - Petrogradsky (2007)
Construction of an algebra of upper triangular matrices
Pk (−1)j
realizing the value q = j=2 j! .
The above lower bound is reached only in case k is
even. For k odd the lower bound is given by k−1
.
k!
Construction of algebras realizing such values.
– p. 4
G.- La Mattina - Petrogradsky (2007)
Construction of an algebra of upper triangular matrices
Pk (−1)j
realizing the value q = j=2 j! .
The above lower bound is reached only in case k is
even. For k odd the lower bound is given by k−1
.
k!
Construction of algebras realizing such values.
Let A be a unitary algebra. If the codimension
sequence cn (A), n = 0, 1, 2, . . ., is bounded by a
polynomial function, then cn (A) is a polynomial with
rational coefficients.
Remark.
– p. 4
Definition.
Let
Uk = {αE +
X
αij eij | α, αij ∈ F },
1≤i<j≤k
where E is the identity k × k matrix.
– p. 4
Definition.
Let
Uk = {αE +
X
αij eij | α, αij ∈ F },
1≤i<j≤k
where E is the identity k × k matrix.
Let A be a unitary algebra over an infinite field
F such that cn (A) ≈ qnk , n → ∞. Then Id(A) ⊇ Id(Uk+1 ).
Theorem.
– p. 4
Theorem.
Let F be an infinite field. Then
– p. 4
Theorem.
Let F be an infinite field. Then
1) A basis of the identities of Uk is given by all products of
commutators of total degree k
[x1 , . . . , xa1 ][xa1 +1 , . . . , xa2 ] · · · [xar−1 +1 , . . . , xar ]
(11)
with ar = k in case k is even, and by the polynomials in (10) plus
the polynomial of degree k + 1
[x1 , x2 ] · · · [xk , xk+1 ]
in case k is odd.
– p. 4
Theorem.
Let F be an infinite field. Then
1) A basis of the identities of Uk is given by all products of
commutators of total degree k
[x1 , . . . , xa1 ][xa1 +1 , . . . , xa2 ] · · · [xar−1 +1 , . . . , xar ]
(12)
with ar = k in case k is even, and by the polynomials in (10) plus
the polynomial of degree k + 1
[x1 , x2 ] · · · [xk , xk+1 ]
in case k is odd.
2)
cn (Uk ) =
k−1
X
j=0
where θi =
n!
θj ≈ θk−1 nk−1 ,
(n − j)!
(−1)j
j=0 j! ,
Pi
n → ∞,
for i ∈ N.
– p. 4
Let A be a unitary algebra over a field of
characteristic zero. If cn (A) = qnk + O(nk−1 ), for some odd
integer k > 1 and rational number q, then q ≥ k−1
.
k!
Theorem.
– p. 4
Let A be a unitary algebra over a field of
characteristic zero. If cn (A) = qnk + O(nk−1 ), for some odd
integer k > 1 and rational number q, then q ≥ k−1
.
k!
Theorem.
Definition.
Let k ≥ 3. For all l ∈ {1, 2, . . . , k − 1} let
Nk,l = spanF {E, J, J 2 ,
. . . , J k−l−1 ; e12 , e13 , . . . , e1,k−l ; eij | j − i ≥ k − l}.
Pk−1
where J = i=1 ei,i+1 ∈ Uk .
– p. 4
Let A be a unitary algebra over a field of
characteristic zero. If cn (A) = qnk + O(nk−1 ), for some odd
integer k > 1 and rational number q, then q ≥ k−1
.
k!
Theorem.
Definition.
Let k ≥ 3. For all l ∈ {1, 2, . . . , k − 1} let
Nk,l = spanF {E, J, J 2 ,
. . . , J k−l−1 ; e12 , e13 , . . . , e1,k−l ; eij | j − i ≥ k − l}.
Pk−1
where J = i=1 ei,i+1 ∈ Uk .
We have
Nk,1 = Nk,2 ⊂ · · · ⊂ Nk,k−1 = Uk .
Write Nk = Nk,1 .
– p. 4
Theorem.
Let 1 ≤ l ≤ k − 2, k > 4. If F is an infinite field
then
1) Nk,l and Uk generate different varieties.
2)
cn (Nk,l ) ≈
k−2
(k−1)!
nk−1 ,
n → ∞.
– p. 4
Theorem.
Let 1 ≤ l ≤ k − 2, k > 4. If F is an infinite field
then
1) Nk,l and Uk generate different varieties.
2)
cn (Nk,l ) ≈
Theorem.
k−2
(k−1)!
nk−1 ,
n → ∞.
Let k ≥ 3 and let F be an infinite field. Then
1) A basis of the identities of Nk is given by the
polynomials
[x1 , . . . , xk ], [x1 , x2 ][x3 , x4 ].
2)
cn (Nk ) = 1 +
Pk−1
j=2 (j − 1)
n
j
≈
k−2
(k−1)!
nk−1 ,
(14)
n → ∞.
– p. 4
Let G2k = h1, e1 , . . . , e2k | ei ej = −ej ei i be the Grassmann
algebra with unity on a 2k-dimensional vector space over a
field F of characteristic 6= 2.
– p. 4
Let G2k = h1, e1 , . . . , e2k | ei ej = −ej ei i be the Grassmann
algebra with unity on a 2k-dimensional vector space over a
field F of characteristic 6= 2.
Theorem.
Let F be an infinite field. Then
1) A basis of the identities of G2k is given by the polynomials
[x1 , x2 , x3 ], [x1 , x2 ] · · · [x2k+1 , x2k+2 ].
(16)
2)
cn (G2k ) =
k X
n
j=0
2j
1
n2k ,
≈
(2k)!
n → ∞.
– p. 4
Let G2k = h1, e1 , . . . , e2k | ei ej = −ej ei i be the Grassmann
algebra with unity on a 2k-dimensional vector space over a
field F of characteristic 6= 2.
Theorem.
Let F be an infinite field. Then
1) A basis of the identities of G2k is given by the polynomials
[x1 , x2 , x3 ], [x1 , x2 ] · · · [x2k+1 , x2k+2 ].
(17)
2)
cn (G2k ) =
k X
n
j=0
2j
1
n2k ,
≈
(2k)!
n → ∞.
Let A be a unitary algebra over a field F of
characteristic zero. If cn (A) ≤ an3 , for some a ≥ 1, then
either Id(A) = Id(F ) or Id(A) = Id(U3 ) or Id(A) = Id(U4 ).
Theorem.
– p. 4
G. - La Mattina (2005). Let A be an F -algebra. Then the
following conditions are equivalent:
1. cn (A) ≤ kn for all n ≥ 1, for some constant k;
2. A is PI-equivalent to either N or C ⊕ N or A2 ⊕ N or
A∗2 ⊕ N or A2 ⊕ A∗2 ⊕ N where N is a nilpotent algebra
and C is a commutative algebra.
– p. 4
G. - La Mattina (2005). Let A be an F -algebra. Then the
following conditions are equivalent:
1. cn (A) ≤ kn for all n ≥ 1, for some constant k;
2. A is PI-equivalent to either N or C ⊕ N or A2 ⊕ N or
A∗2 ⊕ N or A2 ⊕ A∗2 ⊕ N where N is a nilpotent algebra
and C is a commutative algebra.
Kemer (1978). For a PI-algebra A, cn (A) is polynomially
bounded if and only if G, U T2 6∈ var(A).
– p. 4
La Mattina (2007)
classified all subvarieties of var(G) and var(U T2 ).
– p. 4
La Mattina (2007)
classified all subvarieties of var(G) and var(U T2 ).
Definition.
For k ≥ 2 let
Ak = span{e11 , J, J 2 , . . . , J k−2 ; e12 , e13 , . . . , e1k }.
We denote by A∗k the algebra obtained by flipping Ak along
its secondary diagonal.
– p. 4
La Mattina (2007)
classified all subvarieties of var(G) and var(U T2 ).
Definition.
For k ≥ 2 let
Ak = span{e11 , J, J 2 , . . . , J k−2 ; e12 , e13 , . . . , e1k }.
We denote by A∗k the algebra obtained by flipping Ak along
its secondary diagonal.
Lemma.
If k ≥ 3, then
1) Id(Ak ) = h[x1 , x2 ][x3 , x4 ], [x1 , x2 ]x3 . . . xk+1 iT .
Pk−2 n
2) cn (Ak ) = l=0 l (n − l − 1) + 1 ≈ qnk−1 , where q ∈ Q.
Hence Id(A∗k ) = h[x1 , x2 ][x3 , x4 ], x3 . . . xk+1 [x1 , x2 ]iT and
Pk−2 n
∗
cn (Ak ) = l=0 l (n − l − 1) + 1.
– p. 4
A variety V is minimal of polynomial growth if
cn (V) ≈ qnk for some k ≥ 1, q > 0, and for any proper
subvariety U $ V we have that cn (U) ≈ q ′ nt with t < k.
Definition.
– p. 4
A variety V is minimal of polynomial growth if
cn (V) ≈ qnk for some k ≥ 1, q > 0, and for any proper
subvariety U $ V we have that cn (U) ≈ q ′ nt with t < k.
Definition.
Theorem.
The algebras
- Nk , k ≥ 3,
- G2k , k ≥ 1,
- Ak , k ≥ 2,
- A∗k , k ≥ 2
generate minimal variety.
– p. 4
Let A ∈ var(G). Then A is PI-equivalent to one
of the following algebras:
Theorem.
G, G2k ⊕ N, N, C ⊕ N,
where k ≥ 1, N is a nilpotent algebra and C is a
commutative algebra.
– p. 5
Let A ∈ var(G). Then A is PI-equivalent to one
of the following algebras:
Theorem.
G, G2k ⊕ N, N, C ⊕ N,
where k ≥ 1, N is a nilpotent algebra and C is a
commutative algebra.
If A ∈ var(U T2 ) then A is PI-equivalent to one of
the following algebras:
Theorem.
U T2 , N, Nt ⊕N,
Nt ⊕Ak ⊕N, Nt ⊕A∗r ⊕N, Nt ⊕Ak ⊕A∗r ⊕N,
where N is a nilpotent algebra and k, r ≥ 2, t ≥ 2.
– p. 5
A. Giambruno and M. Zaicev,
Polynomial Identities and Asymptotic Methods,
AMS, Mathematical Surveys and Monographs Vol. 122,
Providence, R.I., 2005.
– p. 5