Introduction
Amitsur Conjecture
Pi Exponent and structure of Lie algebra with
an algebra acting
Geoffrey Janssens
January 2, 2016
Geoffrey Janssens
1 / 14
Introduction
Amitsur Conjecture
Outline
1
Introduction
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
2
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive
results
Geoffrey Janssens
2 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
Polynomial identities with coefficients from a field
Let F be a field of charF =0, F = F , A a f.d. associative algebra
and L a f.d. Lie algebra.
definition
A polynomial f = f (x1 , . . . , xn ) ∈ F hX i is a polynomial identity
(PI) of A iff f (a1 , . . . , an ) = 0 for all a1 , . . . , an ∈ A. Noted f ≡ 0.
If f ∈ L hX i then f is called a Lie identity.
Examples:
[x, y ] = xy − yx ≡ 0 is a PI of A iff A is commutative.
x m ≡ 0 is PI of A iff A is nil of bounded index.
x1 . . . . .xm ≡ 0 iff A is nilpotent.
All matrix algebras
Mn (F ) satisfy the ’standard identity’
P
sn2 +1 =
sgn(σ)xσ(1) . . . xσ(n2 +1) .
σ∈Sn2 +1
2
[[x, y ] , z] ≡ 0 for
M2 (F ) (due to Cayley-Hamilton)
Geoffrey Janssens
3 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
Polynomial identities with coefficients from a field
Let F be a field of charF =0, F = F , A a f.d. associative algebra
and L a f.d. Lie algebra.
definition
A polynomial f = f (x1 , . . . , xn ) ∈ F hX i is a polynomial identity
(PI) of A iff f (a1 , . . . , an ) = 0 for all a1 , . . . , an ∈ A. Noted f ≡ 0.
If f ∈ L hX i then f is called a Lie identity.
Examples:
[x, y ] = xy − yx ≡ 0 is a PI of A iff A is commutative.
x m ≡ 0 is PI of A iff A is nil of bounded index.
x1 . . . . .xm ≡ 0 iff A is nilpotent.
All matrix algebras
Mn (F ) satisfy the ’standard identity’
P
sn2 +1 =
sgn(σ)xσ(1) . . . xσ(n2 +1) .
σ∈Sn2 +1
2
[[x, y ] , z] ≡ 0 for
M2 (F ) (due to Cayley-Hamilton)
Geoffrey Janssens
3 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
Polynomial identities with coefficients from a field
Let F be a field of charF =0, F = F , A a f.d. associative algebra
and L a f.d. Lie algebra.
definition
A polynomial f = f (x1 , . . . , xn ) ∈ F hX i is a polynomial identity
(PI) of A iff f (a1 , . . . , an ) = 0 for all a1 , . . . , an ∈ A. Noted f ≡ 0.
If f ∈ L hX i then f is called a Lie identity.
Examples:
[x, y ] = xy − yx ≡ 0 is a PI of A iff A is commutative.
x m ≡ 0 is PI of A iff A is nil of bounded index.
x1 . . . . .xm ≡ 0 iff A is nilpotent.
All matrix algebras
Mn (F ) satisfy the ’standard identity’
P
sn2 +1 =
sgn(σ)xσ(1) . . . xσ(n2 +1) .
σ∈Sn2 +1
2
[[x, y ] , z] ≡ 0 for
M2 (F ) (due to Cayley-Hamilton)
Geoffrey Janssens
3 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
Polynomial identities with coefficients from a field
Let F be a field of charF =0, F = F , A a f.d. associative algebra
and L a f.d. Lie algebra.
definition
A polynomial f = f (x1 , . . . , xn ) ∈ F hX i is a polynomial identity
(PI) of A iff f (a1 , . . . , an ) = 0 for all a1 , . . . , an ∈ A. Noted f ≡ 0.
If f ∈ L hX i then f is called a Lie identity.
Examples:
[x, y ] = xy − yx ≡ 0 is a PI of A iff A is commutative.
x m ≡ 0 is PI of A iff A is nil of bounded index.
x1 . . . . .xm ≡ 0 iff A is nilpotent.
All matrix algebras
Mn (F ) satisfy the ’standard identity’
P
sn2 +1 =
sgn(σ)xσ(1) . . . xσ(n2 +1) .
σ∈Sn2 +1
2
[[x, y ] , z] ≡ 0 for
M2 (F ) (due to Cayley-Hamilton)
Geoffrey Janssens
3 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
Polynomial identities with coefficients from a field
Let F be a field of charF =0, F = F , A a f.d. associative algebra
and L a f.d. Lie algebra.
definition
A polynomial f = f (x1 , . . . , xn ) ∈ F hX i is a polynomial identity
(PI) of A iff f (a1 , . . . , an ) = 0 for all a1 , . . . , an ∈ A. Noted f ≡ 0.
If f ∈ L hX i then f is called a Lie identity.
Examples:
[x, y ] = xy − yx ≡ 0 is a PI of A iff A is commutative.
x m ≡ 0 is PI of A iff A is nil of bounded index.
x1 . . . . .xm ≡ 0 iff A is nilpotent.
All matrix algebras
Mn (F ) satisfy the ’standard identity’
P
sn2 +1 =
sgn(σ)xσ(1) . . . xσ(n2 +1) .
σ∈Sn2 +1
2
[[x, y ] , z] ≡ 0 for
M2 (F ) (due to Cayley-Hamilton)
Geoffrey Janssens
3 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
With a grading or action
Let H be a f.d. associative algebra with 1 acting on A.
Generalized action
H acts in a generalized way if for all h ∈ H, a, b ∈ A :
X
h · (a.b) =
(hi0 · a).(hi00 · b) + (hi000 · b).(hi0000 · a)
i
for some hi0 , hi00 , hi000 , hi0000 ∈ A.
Examples:
1−1
(i) T a semigroup, then T -gradings ←→ (FT )∗ -actions.
(ii) if A is a H-module algebra, i.e h · (a.b) = (h(1) · a)(h(2) · b),
for some Hopf algebra H.
(iii) G acts by automorphism and anti-automorphism at the
same time on A.
Geoffrey Janssens
4 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
With a grading or action
Let H be a f.d. associative algebra with 1 acting on A.
Generalized action
H acts in a generalized way if for all h ∈ H, a, b ∈ A :
X
h · (a.b) =
(hi0 · a).(hi00 · b) + (hi000 · b).(hi0000 · a)
i
for some hi0 , hi00 , hi000 , hi0000 ∈ A.
Examples:
1−1
(i) T a semigroup, then T -gradings ←→ (FT )∗ -actions.
(ii) if A is a H-module algebra, i.e h · (a.b) = (h(1) · a)(h(2) · b),
for some Hopf algebra H.
(iii) G acts by automorphism and anti-automorphism at the
same time on A.
Geoffrey Janssens
4 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
H-polynomials
Let (γβ )β∈Λ be a basis of H. Let F {X | H}
Let F {X | H} be the free (non)associative algebra over F with
γ
formal generators xi β , β ∈ Λ, i ∈ Z≥0 .
Example
(0)
(1) with
Let G = (Z2
, +), gl2 (F
2 (F )
) = gl2 (F ) ⊕ gl
F 0
0 F
(0)
(1)
gl2 (F ) =
and gl2 (F ) =
. Then
0 F
F 0
[x (0) , y (0) ] ∈ Id G (gl2 (F )).
Thus, these give information about the H-, graded-structure of
the algebra.
Geoffrey Janssens
5 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
H-polynomials
Let (γβ )β∈Λ be a basis of H. Let F {X | H}
Let F {X | H} be the free (non)associative algebra over F with
γ
formal generators xi β , β ∈ Λ, i ∈ Z≥0 .
Example
(0)
(1) with
Let G = (Z2
, +), gl2 (F
2 (F )
) = gl2 (F ) ⊕ gl
F 0
0 F
(0)
(1)
gl2 (F ) =
and gl2 (F ) =
. Then
0 F
F 0
[x (0) , y (0) ] ∈ Id G (gl2 (F )).
Thus, these give information about the H-, graded-structure of
the algebra.
Geoffrey Janssens
5 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
H-polynomials
Let (γβ )β∈Λ be a basis of H. Let F {X | H}
Let F {X | H} be the free (non)associative algebra over F with
γ
formal generators xi β , β ∈ Λ, i ∈ Z≥0 .
Example
(0)
(1) with
Let G = (Z2
, +), gl2 (F
2 (F )
) = gl2 (F ) ⊕ gl
F 0
0 F
(0)
(1)
gl2 (F ) =
and gl2 (F ) =
. Then
0 F
F 0
[x (0) , y (0) ] ∈ Id G (gl2 (F )).
Thus, these give information about the H-, graded-structure of
the algebra.
Geoffrey Janssens
5 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
PI’s follow from multilinear ones
definition
The space of multilinear polynomials:
h1
hn
PnH (F ) = spanF {xσ(1)
. . . xσ(n)
| hi ∈ H, σ ∈ Sn }.
Multilinear Lie polynomials:
h1
hn
VnH (F ) = spanf {[xσ(1)
, . . . , xσ(n)
] | hi ∈ H, σ ∈ Sn }
Denote, IdH (A) the set of all H-PI’s of A over F .
proposition
If char F = 0 then all H-PI’s, IdH (A), are consequences of
multilinear PI’s, i.e IdH (A) ∩ PnH (F ).
Geoffrey Janssens
6 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
PI’s follow from multilinear ones
definition
The space of multilinear polynomials:
h1
hn
PnH (F ) = spanF {xσ(1)
. . . xσ(n)
| hi ∈ H, σ ∈ Sn }.
Multilinear Lie polynomials:
h1
hn
VnH (F ) = spanf {[xσ(1)
, . . . , xσ(n)
] | hi ∈ H, σ ∈ Sn }
Denote, IdH (A) the set of all H-PI’s of A over F .
proposition
If char F = 0 then all H-PI’s, IdH (A), are consequences of
multilinear PI’s, i.e IdH (A) ∩ PnH (F ).
Geoffrey Janssens
6 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
Codimensions over a field
More interesting to look at is the F -module:
PnH (F )
PnH (F ) ∩ IdH (A)
.
definition
The non-negative integer cnH (A) = dimF
PnH (F )
PnH (F )∩IdH (A)
is called
the nth H-codimension of the algebra A.
Remark:
let H = {e}, then cn (A) < n! iff A is PI.
In this spirit, Regev: if A1 and A2 are PI, then A1 ⊗ A2 is PI.
If A is finite dimensional Lie or associative, then cnH (A) is
exponentially bounded.
Geoffrey Janssens
7 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
Codimensions over a field
More interesting to look at is the F -module:
PnH (F )
PnH (F ) ∩ IdH (A)
.
definition
The non-negative integer cnH (A) = dimF
PnH (F )
PnH (F )∩IdH (A)
is called
the nth H-codimension of the algebra A.
Remark:
let H = {e}, then cn (A) < n! iff A is PI.
In this spirit, Regev: if A1 and A2 are PI, then A1 ⊗ A2 is PI.
If A is finite dimensional Lie or associative, then cnH (A) is
exponentially bounded.
Geoffrey Janssens
7 / 14
Introduction
Amitsur Conjecture
Polynomial identities with coefficients from a field
H-polynomials
Codimensions
Codimensions over a field
More interesting to look at is the F -module:
PnH (F )
PnH (F ) ∩ IdH (A)
.
definition
The non-negative integer cnH (A) = dimF
PnH (F )
PnH (F )∩IdH (A)
is called
the nth H-codimension of the algebra A.
Remark:
let H = {e}, then cn (A) < n! iff A is PI.
In this spirit, Regev: if A1 and A2 are PI, then A1 ⊗ A2 is PI.
If A is finite dimensional Lie or associative, then cnH (A) is
exponentially bounded.
Geoffrey Janssens
7 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Conjectures
Amitsur conjecture
If A is a f.d. algebra withqa generalized action by H, then there
exists expH (A) := lim
n→∞
n
cnH (A, F ) ∈ Z+ .
Examples:
1
2
3
A is nilpotent iff exp(A) = 0.
p
lim n cn (Mn (F )) = n2 .
n→∞
p
lim n cn (sln (F )) = n2 − 1.
n→∞
4
exp(A) = 1 iff UT2 , G ∈
/ var(A)
Geoffrey Janssens
8 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Conjectures
Amitsur conjecture
If A is a f.d. algebra withqa generalized action by H, then there
exists expH (A) := lim
n→∞
n
cnH (A, F ) ∈ Z+ .
Examples:
1
2
3
A is nilpotent iff exp(A) = 0.
p
lim n cn (Mn (F )) = n2 .
n→∞
p
lim n cn (sln (F )) = n2 − 1.
n→∞
4
exp(A) = 1 iff UT2 , G ∈
/ var(A)
Geoffrey Janssens
8 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Conjectures
Amitsur conjecture
If A is a f.d. algebra withqa generalized action by H, then there
exists expH (A) := lim
n→∞
n
cnH (A, F ) ∈ Z+ .
Examples:
1
2
3
A is nilpotent iff exp(A) = 0.
p
lim n cn (Mn (F )) = n2 .
n→∞
p
lim n cn (sln (F )) = n2 − 1.
n→∞
4
exp(A) = 1 iff UT2 , G ∈
/ var(A)
Geoffrey Janssens
8 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Conjectures
Amitsur conjecture
If A is a f.d. algebra withqa generalized action by H, then there
exists expH (A) := lim
n→∞
n
cnH (A, F ) ∈ Z+ .
Examples:
1
2
3
A is nilpotent iff exp(A) = 0.
p
lim n cn (Mn (F )) = n2 .
n→∞
p
lim n cn (sln (F )) = n2 − 1.
n→∞
4
exp(A) = 1 iff UT2 , G ∈
/ var(A)
Geoffrey Janssens
8 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Positive results
S.A. Amitsur’s conjecture was proved
in 1999 by Giambruno and Zaicev for codimensions of
associative algebras; in 2002 for f.d. Lie algebras (Zaicev)
in 2010-2011 for G-graded associative algebras, G finite
group, by Aljadeff, Giambruno and La Mattina;
in 2012-2013 for actions of f.d semisimple Hopf algebras
and arbitrary group-gradings on f.d associative and Lie
algebras by Gordienko
Geoffrey Janssens
9 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Positive results
S.A. Amitsur’s conjecture was proved
in 1999 by Giambruno and Zaicev for codimensions of
associative algebras; in 2002 for f.d. Lie algebras (Zaicev)
in 2010-2011 for G-graded associative algebras, G finite
group, by Aljadeff, Giambruno and La Mattina;
in 2012-2013 for actions of f.d semisimple Hopf algebras
and arbitrary group-gradings on f.d associative and Lie
algebras by Gordienko
Geoffrey Janssens
9 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Positive results
S.A. Amitsur’s conjecture was proved
in 1999 by Giambruno and Zaicev for codimensions of
associative algebras; in 2002 for f.d. Lie algebras (Zaicev)
in 2010-2011 for G-graded associative algebras, G finite
group, by Aljadeff, Giambruno and La Mattina;
in 2012-2013 for actions of f.d semisimple Hopf algebras
and arbitrary group-gradings on f.d associative and Lie
algebras by Gordienko
Geoffrey Janssens
9 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Positive results
S.A. Amitsur’s conjecture was proved
in 1999 by Giambruno and Zaicev for codimensions of
associative algebras; in 2002 for f.d. Lie algebras (Zaicev)
in 2010-2011 for G-graded associative algebras, G finite
group, by Aljadeff, Giambruno and La Mattina;
in 2012-2013 for actions of f.d semisimple Hopf algebras
and arbitrary group-gradings on f.d associative and Lie
algebras by Gordienko
Geoffrey Janssens
9 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Information behind Exponent
Let A = B1 ⊕ . . . ⊕ Bt ⊕ J(A) Weddeburn-Malcev decomposition
For associative algebras:
exp(A) = max{dim(Bi1 ⊕ . . . ⊕ Bir ) | Bi1 J . . . JBir 6= 0}
For expH (A) one needs a H-version of W-M decomposition (i.e
Bi are H-simple semisimple subalgebras).
Geoffrey Janssens
10 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Information behind Exponent
Let A = B1 ⊕ . . . ⊕ Bt ⊕ J(A) Weddeburn-Malcev decomposition
For associative algebras:
exp(A) = max{dim(Bi1 ⊕ . . . ⊕ Bir ) | Bi1 J . . . JBir 6= 0}
For expH (A) one needs a H-version of W-M decomposition (i.e
Bi are H-simple semisimple subalgebras).
Geoffrey Janssens
10 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Information behind Exponent
Let L = B ⊕ R the Levi decomposition (sum of H-submod.) and
I1 , . . . , Ir , J1 , . . . , Jr H-invariant ideals of L s.t. Jk ⊆ Ik and
(i): Ik /Jk irreducible (H, L)-module
(ii): for all H-inv. B-submodules Tk s.t. Ik = Jk ⊕ Tk , there exists
qi ≥ 0 s.t.
[[T1 , L, . . . , L], . . . , [Tr , L, . . . , L]] 6= 0
| {z }
| {z }
q1
qr
For Lie algebras:
expH (L) = max{dim
L
Ann(I1 /J1 )∩···∩Ann(Ir /Jr )
}
Remark: if N = R then an ’associative like formula’ holds.
Geoffrey Janssens
11 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Structural theorems hidden behind
If L is a H-module for a Hopf algebra H. Then needed structural
results:
(i) Invariant N, nilpotent radical, and R, solvable radical
(ii) L = B ⊕ R with B invariant (Levi decomp.)
(iii) A1 = A˜1 ⊕ J(A1 ) with A1 ⊆ EndF (W ) H-mod. subalg. and
W ⊆ L H-submod. (Weddeburn-Malcev decomp.)
(iv) L is completely reducible (H, L0 )-module with L0 ⊆ gl(L)
H-inv. subalg. s.t. L is completely reducible L0 -module
(Weyl th.)
Remark: examples without (ii) but integer Exponent exist.
Geoffrey Janssens
12 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Structural theorems hidden behind
If L is a H-module for a Hopf algebra H. Then needed structural
results:
(i) Invariant N, nilpotent radical, and R, solvable radical
(ii) L = B ⊕ R with B invariant (Levi decomp.)
(iii) A1 = A˜1 ⊕ J(A1 ) with A1 ⊆ EndF (W ) H-mod. subalg. and
W ⊆ L H-submod. (Weddeburn-Malcev decomp.)
(iv) L is completely reducible (H, L0 )-module with L0 ⊆ gl(L)
H-inv. subalg. s.t. L is completely reducible L0 -module
(Weyl th.)
Remark: examples without (ii) but integer Exponent exist.
Geoffrey Janssens
12 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Structural theorems hidden behind
If L is a H-module for a Hopf algebra H. Then needed structural
results:
(i) Invariant N, nilpotent radical, and R, solvable radical
(ii) L = B ⊕ R with B invariant (Levi decomp.)
(iii) A1 = A˜1 ⊕ J(A1 ) with A1 ⊆ EndF (W ) H-mod. subalg. and
W ⊆ L H-submod. (Weddeburn-Malcev decomp.)
(iv) L is completely reducible (H, L0 )-module with L0 ⊆ gl(L)
H-inv. subalg. s.t. L is completely reducible L0 -module
(Weyl th.)
Remark: examples without (ii) but integer Exponent exist.
Geoffrey Janssens
12 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Cases where it go wrong
0 1
0 0
Let t = e12 =
, v = e21 =
and
0 0
1 0
1 0
u = e11 − e22 =
.
0 −1
Then [u, v ] = −2v , [u, t] = 2t, [v , t] = −u and
I = spanF {u, v , t} ∼
= sl2 (F ).
Let L = I ⊕ hu, v iL with (Z2 , .)-grading:
L0 = (sl2 (F ), 0) and L1 = {(a, a) | a ∈ hu, v i(−) }.
Theorem (J.)
Let L be the Lie algebra
as before. Then
q
√
n
expZ2 (L) = lim
cnZ2 (L) = 2 + 2 2.
n→∞
Geoffrey Janssens
13 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
Cases where it go wrong
0 1
0 0
Let t = e12 =
, v = e21 =
and
0 0
1 0
1 0
u = e11 − e22 =
.
0 −1
Then [u, v ] = −2v , [u, t] = 2t, [v , t] = −u and
I = spanF {u, v , t} ∼
= sl2 (F ).
Let L = I ⊕ hu, v iL with (Z2 , .)-grading:
L0 = (sl2 (F ), 0) and L1 = {(a, a) | a ∈ hu, v i(−) }.
Theorem (J.)
Let L be the Lie algebra
as before. Then
q
√
n
expZ2 (L) = lim
cnZ2 (L) = 2 + 2 2.
n→∞
Geoffrey Janssens
13 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
positive results
Let L be f.d. Lie algebra with generalized action by H f.d.
assoc. with 1
Proposition (J.)
Let L = L1 ⊕ . . . ⊕ Lm a sum of H-simple semisimple Lie
algebras Li . Then expH (L) = max{expH (Li ) | 1 ≤ i ≤ m} and
expH (Li ) = dim Li .
Remark: actually more generally for ’H-nice’ Lie alg. Li the
answer to following questions are true:
q
(Q. 1): Suppose lim sup n cnH (L) ≤ d(L) ∈ Z. Do there exists a
constant K ∈ Z s.t χn (L) ⊆ H(d(L), K )?
(Q. 2): Suppose L = L1 ⊕ . . . ⊕ Lm with expH (Li ) ∈ Z. Is
expH (L) = max{expH (Li ) | 1 ≤ i ≤ m}?
Geoffrey Janssens
14 / 14
Introduction
Amitsur Conjecture
Overview
Structure theory behind, non-integer Exp and positive results
positive results
Let L be f.d. Lie algebra with generalized action by H f.d.
assoc. with 1
Proposition (J.)
Let L = L1 ⊕ . . . ⊕ Lm a sum of H-simple semisimple Lie
algebras Li . Then expH (L) = max{expH (Li ) | 1 ≤ i ≤ m} and
expH (Li ) = dim Li .
Remark: actually more generally for ’H-nice’ Lie alg. Li the
answer to following questions are true:
q
(Q. 1): Suppose lim sup n cnH (L) ≤ d(L) ∈ Z. Do there exists a
constant K ∈ Z s.t χn (L) ⊆ H(d(L), K )?
(Q. 2): Suppose L = L1 ⊕ . . . ⊕ Lm with expH (Li ) ∈ Z. Is
expH (L) = max{expH (Li ) | 1 ≤ i ≤ m}?
Geoffrey Janssens
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