Download Application of the graded Posner theorem

Application of the graded Posner theorem
Y. Karasik
1 Department of Mathematics
Technion - Israel Institute of Technology
Group graded algebras
Definition
Let G be a group and F be a field. An (associative) F-algebra A is
called G-graded if
M
A=
Ag ,
g ∈G
where Ag is an F-subspace of A and Ag Ah ⊆ Agh for every
g, h ∈ G.
Group graded algebras
Definition
Let G be a group and F be a field. An (associative) F-algebra A is
called G-graded if
M
A=
Ag ,
g ∈G
where Ag is an F-subspace of A and Ag Ah ⊆ Agh for every
g, h ∈ G.
A is G-simple if it does not have any non-trivial graded ideals.
Group graded algebras
Definition
Let G be a group and F be a field. An (associative) F-algebra A is
called G-graded if
M
A=
Ag ,
g ∈G
where Ag is an F-subspace of A and Ag Ah ⊆ Agh for every
g, h ∈ G.
A is G-simple if it does not have any non-trivial graded ideals.
Example (Fine grading)
A = Fα H =
L
g ∈H
F · ug , where H is a subgroup of G.
More examples
Example
Take G = Z/2Z = {0, 1} and consider

a
A0 =  c
0
b
d
0


0
0

0 ; A1 =  0
e
w
0
0
z
 

x
0


In short, 0
y
0
1
0
0
1

1
1 
0
More examples
Example
Take G = Z/2Z = {0, 1} and consider

a
A0 =  c
0
b
d
0


0
0

0 ; A1 =  0
e
w
0
0
z
 

x
0


In short, 0
y
0
1
0
0
1

1
1 
0
Definition
→
Let G be a group and −
g = (g1 , ..., gn ) ∈ G ×n . The G-graded F algebra
→
A = M−
(
F
)
is
the
algebra
Mn (F) graded by:
g
More examples
Example
Take G = Z/2Z = {0, 1} and consider

a
A0 =  c
0
b
d
0


0
0

0 ; A1 =  0
e
w
0
0
z
 

x
0


In short, 0
y
0
1
0
0
1

1
1 
0
Definition
→
Let G be a group and −
g = (g1 , ..., gn ) ∈ G ×n . The G-graded F algebra
→
A = M−
(
F
)
is
the
algebra
Mn (F) graded by:
g
g1−1 g1 = e
 g −1 g
 2 1



..


.

gn−1 g1
g1−1 g2
−1
g2 g2 = e
g1−1 g3
g2−1 g3
−1
g3 g3 = e
..
.
gn−1 g2
···
···
···
g1−1 gn
g2−1 gb
..
..
.
gn−1 gn = e
.








Posner’s Theorem for G-graded algebras
Theorem
Suppose A is a G-graded F-algebra such that:
Posner’s Theorem for G-graded algebras
Theorem
Suppose A is a G-graded F-algebra such that:
1
K := Z (A)e is a field.
Posner’s Theorem for G-graded algebras
Theorem
Suppose A is a G-graded F-algebra such that:
1
K := Z (A)e is a field.
2
A satisfies an ordinary PI.
Posner’s Theorem for G-graded algebras
Theorem
Suppose A is a G-graded F-algebra such that:
1
K := Z (A)e is a field.
2
A satisfies an ordinary PI.
3
A is G-semiprime (this holds if G is finite and J (A) = 0).
Posner’s Theorem for G-graded algebras
Theorem
Suppose A is a G-graded F-algebra such that:
1
K := Z (A)e is a field.
2
A satisfies an ordinary PI.
3
A is G-semiprime (this holds if G is finite and J (A) = 0).
Then, A is a G-simple finite dimensional K-algebra.
Posner’s Theorem for G-graded algebras
Theorem
Suppose A is a G-graded F-algebra such that:
1
K := Z (A)e is a field.
2
A satisfies an ordinary PI.
3
A is G-semiprime (this holds if G is finite and J (A) = 0).
Then, A is a G-simple finite dimensional K-algebra.
This holds for F of characteristic zero and G a residually finite
group.
Posner’s Theorem for G-graded algebras
Theorem
Suppose A is a G-graded F-algebra such that:
1
K := Z (A)e is a field.
2
A satisfies an ordinary PI.
3
A is G-semiprime (this holds if G is finite and J (A) = 0).
Then, A is a G-simple finite dimensional K-algebra.
This holds for F of characteristic zero and G a residually finite
group.
If one replaces “G-semiprime” by “G-semisimple”, the
theorem is true for every G.
Posner’s Theorem for G-graded algebras
Theorem
Suppose A is a G-graded F-algebra such that:
1
K := Z (A)e is a field.
2
A satisfies an ordinary PI.
3
A is G-semiprime (this holds if G is finite and J (A) = 0).
Then, A is a G-simple finite dimensional K-algebra.
This holds for F of characteristic zero and G a residually finite
group.
If one replaces “G-semiprime” by “G-semisimple”, the
theorem is true for every G.
The main ingredient in the proof is the existence of a special
kind of central polynomials:
Strong central polynomials
Definition
A polynomial f (x1 , ..., xn ) ∈ F hX i is a strong central
polynomial of exponent d 2 for the group G, If:
Strong central polynomials
Definition
A polynomial f (x1 , ..., xn ) ∈ F hX i is a strong central
polynomial of exponent d 2 for the group G, If:
1
f is central (non-identity) for every G-simple algebra A of
exponent d 2 .
Strong central polynomials
Definition
A polynomial f (x1 , ..., xn ) ∈ F hX i is a strong central
polynomial of exponent d 2 for the group G, If:
1
f is central (non-identity) for every G-simple algebra A of
exponent d 2 . Moreover, For every homogeneous elements
a1 , ..., an (i.e. they are inside ∪g ∈G Ag ),
f (a1 , ..., an ) 6= 0 ⇐⇒ f (a1 , ..., an )e 6= 0.
Strong central polynomials
Definition
A polynomial f (x1 , ..., xn ) ∈ F hX i is a strong central
polynomial of exponent d 2 for the group G, If:
1
f is central (non-identity) for every G-simple algebra A of
exponent d 2 . Moreover, For every homogeneous elements
a1 , ..., an (i.e. they are inside ∪g ∈G Ag ),
f (a1 , ..., an ) 6= 0 ⇐⇒ f (a1 , ..., an )e 6= 0.
2
f is an identity for every G-simple algebra of smaller exponent.
Strong central polynomials
Definition
A polynomial f (x1 , ..., xn ) ∈ F hX i is a strong central
polynomial of exponent d 2 for the group G, If:
1
f is central (non-identity) for every G-simple algebra A of
exponent d 2 . Moreover, For every homogeneous elements
a1 , ..., an (i.e. they are inside ∪g ∈G Ag ),
f (a1 , ..., an ) 6= 0 ⇐⇒ f (a1 , ..., an )e 6= 0.
2
f is an identity for every G-simple algebra of smaller exponent.
It turns out that when F is of characteristic zero, every
central polynomial for A is strong for every group G.
A theorem of Aljadeff and Haile
Theorem
Let G be a group and F an algebraically closed of 0 characteristic
and A, B two f.d. G-simple F-algebras.
A theorem of Aljadeff and Haile
Theorem
Let G be a group and F an algebraically closed of 0 characteristic
and A, B two f.d. G-simple F-algebras. Then, A is G-isomorphic
to B iff idG (A) = idG (B ).
A theorem of Aljadeff and Haile
Theorem
Let G be a group and F an algebraically closed of 0 characteristic
and A, B two f.d. G-simple F-algebras. Then, A is G-isomorphic
to B iff idG (A) = idG (B ).
Notice that the theorem is easy when G = {e }.
A theorem of Aljadeff and Haile
Theorem
Let G be a group and F an algebraically closed of 0 characteristic
and A, B two f.d. G-simple F-algebras. Then, A is G-isomorphic
to B iff idG (A) = idG (B ).
Notice that the theorem is easy when G = {e }. Indeed,
A = Mn (F), B = Mm (F) and n2 = exp(A) is determined by
id (A).
A theorem of Aljadeff and Haile
Theorem
Let G be a group and F an algebraically closed of 0 characteristic
and A, B two f.d. G-simple F-algebras. Then, A is G-isomorphic
to B iff idG (A) = idG (B ).
Notice that the theorem is easy when G = {e }. Indeed,
A = Mn (F), B = Mm (F) and n2 = exp(A) is determined by
id (A).
We now show how one can use Posner’s theorem in order to
get a quick proof of this theorem.
Generic algebras
Let A be a G-graded F-algebra. The (G-graded) relatively
free algebra of A is defined to be
UA := F hXG i /idG (A).
Generic algebras
Let A be a G-graded F-algebra. The (G-graded) relatively
free algebra of A is defined to be
UA := F hXG i /idG (A).
Every graded algebra B with the same graded identities as A,
is a homomorphic image of UA .
Generic algebras
Let A be a G-graded F-algebra. The (G-graded) relatively
free algebra of A is defined to be
UA := F hXG i /idG (A).
Every graded algebra B with the same graded identities as A,
is a homomorphic image of UA .
If A is also f.d., then it is possible to embed UA in a form of A:
Generic algebras
Let A be a G-graded F-algebra. The (G-graded) relatively
free algebra of A is defined to be
UA := F hXG i /idG (A).
Every graded algebra B with the same graded identities as A,
is a homomorphic image of UA .
If A is also f.d., then it is possible to embed UA in a form of A:
That is, there is a field LA for which UA is embedded inside
ALA := A ⊗F LA .
Proof of the theorem
We know that idG (A) = idG (B ), so
F hX i /idG (B ) ' F hX i /idG (B ).
Proof of the theorem
We know that idG (A) = idG (B ), so
F hX i /idG (B ) ' F hX i /idG (B ).
In other words,
/ U B
UA _
'
_
ALA
BLB
Proof of the theorem
We know that idG (A) = idG (B ), so
F hX i /idG (B ) ' F hX i /idG (B ).
In other words,
/ U B
UA _
'
_
ALA
BLB
We want to use Posner’s theorem, but the center is not be a field...
Proof of the theorem
We know that idG (A) = idG (B ), so
F hX i /idG (B ) ' F hX i /idG (B ).
In other words,
/ U B
UA _
'
_
ALA
BLB
We want to use Posner’s theorem, but the center is not be a field...
The evaluation of Z (UA )e to A, is inside Z (A)e = F · 1A . So, for
0 6= f ∈ Z (UA )e and 0 6= g ∈ UA , we have an evaluation
→
→
→
f (−
a ) · g (−
a ) = αg (−
a ) 6= 0, where 0 6= α ∈ F.
Proof of the theorem
We know that idG (A) = idG (B ), so
F hX i /idG (B ) ' F hX i /idG (B ).
In other words,
/ U B
UA _
'
_
ALA
BLB
We want to use Posner’s theorem, but the center is not be a field...
The evaluation of Z (UA )e to A, is inside Z (A)e = F · 1A . So, for
0 6= f ∈ Z (UA )e and 0 6= g ∈ UA , we have an evaluation
→
→
→
f (−
a ) · g (−
a ) = αg (−
a ) 6= 0, where 0 6= α ∈ F.
We may consider the algebras
SA := Z (UA )e−1 UA ' Z (UB )e−1 UB =: SB . Both are over K =the
quotient field of Z (UA )e = Z (UB )e and have the same G-graded
identities as A (and B).
Using Posner’s theorem
Now we can use Posner’s theorem:
Using Posner’s theorem
Now we can use Posner’s theorem:
We made Z (SA )e = K which is a field.
Using Posner’s theorem
Now we can use Posner’s theorem:
We made Z (SA )e = K which is a field.
Clearly A is PI (since f.d.), so also SA .
Using Posner’s theorem
Now we can use Posner’s theorem:
We made Z (SA )e = K which is a field.
Clearly A is PI (since f.d.), so also SA .
J (UA ) = 0, since an evaluation of f ∈ J (UA ) into A is inside
J (A) = 0. Hence, also J (SA ) = 0.
Using Posner’s theorem
Now we can use Posner’s theorem:
We made Z (SA )e = K which is a field.
Clearly A is PI (since f.d.), so also SA .
J (UA ) = 0, since an evaluation of f ∈ J (UA ) into A is inside
J (A) = 0. Hence, also J (SA ) = 0.
So, SA and SB are G-simple and f.d. over K.
Using Posner’s theorem
Now we can use Posner’s theorem:
We made Z (SA )e = K which is a field.
Clearly A is PI (since f.d.), so also SA .
J (UA ) = 0, since an evaluation of f ∈ J (UA ) into A is inside
J (A) = 0. Hence, also J (SA ) = 0.
So, SA and SB are G-simple and f.d. over K.
We have the picture:
/ U B
UA
'
_
_
_
/ S B
_
SA
ALA
'
BLB
Finishing the proof
What is left to show is that SA is a form of A. Since then it
follows that A and B are graded forms of each other.
Finishing the proof
What is left to show is that SA is a form of A. Since then it
follows that A and B are graded forms of each other.
For this it is enough to show that the natural map
SA ⊗K LA → SA · LA
is a graded isomorphism and
SA · LA = ALA .
Finishing the proof
What is left to show is that SA is a form of A. Since then it
follows that A and B are graded forms of each other.
For this it is enough to show that the natural map
SA ⊗K LA → SA · LA
is a graded isomorphism and
SA · LA = ALA .
The first part follows from the fact that SA ⊗K LA is
G-simple.
Finishing the proof
What is left to show is that SA is a form of A. Since then it
follows that A and B are graded forms of each other.
For this it is enough to show that the natural map
SA ⊗K LA → SA · LA
is a graded isomorphism and
SA · LA = ALA .
The first part follows from the fact that SA ⊗K LA is
G-simple.
The second part follows from
dimLA ALA = expG (ALA ) = expG (SA · LA ) = dimLA SA LA .
The End!
Thank you!