The Extended Affine Lie Algebra Associated with
a Connected Non-negative Unit Form
Gustavo Jasso Ahuja
Universidad Nacional
Autónoma de México
ICRA XIV
August 12, 2010
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
1 / 17
Connected positive definite unit
forms (corank 0).
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
2 / 17
Connected positive definite unit
forms (corank 0).
Gustavo Jasso Ahuja (UNAM)
↔
Simply-laced simple Lie algebras of finite type.
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
2 / 17
Connected positive definite unit
forms (corank 0).
↔
Simply-laced simple Lie algebras of finite type.
Connected non-negative unit
forms of corank 1.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
2 / 17
Connected positive definite unit
forms (corank 0).
↔
Simply-laced simple Lie algebras of finite type.
Connected non-negative unit
forms of corank 1.
↔
Simply-laced Kac-Moody algebras of affine type.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
2 / 17
Connected positive definite unit
forms (corank 0).
↔
Simply-laced simple Lie algebras of finite type.
Connected non-negative unit
forms of corank 1.
↔
Simply-laced Kac-Moody algebras of affine type.
Connected non-negative unit
forms of corank > 2.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
2 / 17
Connected positive definite unit
forms (corank 0).
↔
Simply-laced simple Lie algebras of finite type.
Connected non-negative unit
forms of corank 1.
↔
Simply-laced Kac-Moody algebras of affine type.
Connected non-negative unit
forms of corank > 2.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
?
ICRA XIV. August 12, 2010
2 / 17
Zn
Let q :
→ Z be a connected non-negative
unit form unit form.
Connected positive definite unit
forms (corank
0).
Connected nonnegative
unit
forms of corank
1.
Connected nonnegative
unit
forms of corank
> 2.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
↔
↔
Simply-laced
simple Lie algebras of finite
type.
Simply-laced
Kac-Moody algebras of affine
type.
?
ICRA XIV. August 12, 2010
3 / 17
Zn
Let q :
→ Z be a connected non-negative
unit form unit form.
We associate with q a matrix C given by
Cij = q(ci + cj ) − q(ci ) − q(cj ).
Connected positive definite unit
forms (corank
0).
Connected nonnegative
unit
forms of corank
1.
Connected nonnegative
unit
forms of corank
> 2.
↔
↔
Simply-laced
simple Lie algebras of finite
type.
Simply-laced
Kac-Moody algebras of affine
type.
?
Where {c1 , . . . , cn } denotes the standard
basis of Zn .
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
3 / 17
Zn
Let q :
→ Z be a connected non-negative
unit form unit form.
We associate with q a matrix C given by
Cij = q(ci + cj ) − q(ci ) − q(cj ).
Connected positive definite unit
forms (corank
0).
Connected nonnegative
unit
forms of corank
1.
Connected nonnegative
unit
forms of corank
> 2.
↔
↔
Simply-laced
simple Lie algebras of finite
type.
Simply-laced
Kac-Moody algebras of affine
type.
?
Where {c1 , . . . , cn } denotes the standard
basis of Zn . Also, let
R0 = q −1 (0)
Gustavo Jasso Ahuja (UNAM)
R× = q −1 (1).
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
3 / 17
Zn
Let q :
→ Z be a connected non-negative
unit form unit form.
We associate with q a matrix C given by
Cij = q(ci + cj ) − q(ci ) − q(cj ).
Connected positive definite unit
forms (corank
0).
Connected nonnegative
unit
forms of corank
1.
Connected nonnegative
unit
forms of corank
> 2.
↔
↔
Simply-laced
simple Lie algebras of finite
type.
Simply-laced
Kac-Moody algebras of affine
type.
?
Where {c1 , . . . , cn } denotes the standard
basis of Zn . Also, let
R0 = q −1 (0)
R× = q −1 (1).
We call R = R0 ∪ R× the root system of q.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
3 / 17
Construction [Barot, Kussin,
Lenzing]
Let q : Zn → Z be a connected non-negative
unit form unit form.
Let F L be the free Lie algebra with 3n
generators
e−i , hi , ei
i ∈ {1, . . . , n}
We associate with q a matrix C given by
Cij = q(ci + cj ) − q(ci ) − q(cj ).
Where {c1 , . . . , cn } denotes the standard basis
of Zn . Also, let
R
0
=q
−1
(0)
R
×
=q
−1
(1).
We call R = R0 ∪ R× the root system of q.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
4 / 17
Construction [Barot, Kussin,
Lenzing]
Let q : Zn → Z be a connected non-negative
unit form unit form.
Let F L be the free Lie algebra with 3n
generators
e−i , hi , ei
i ∈ {1, . . . , n}
We associate with q a matrix C given by
Cij = q(ci + cj ) − q(ci ) − q(cj ).
Where {c1 , . . . , cn } denotes the standard basis
of Zn . Also, let
homogeneous of degrees
−ci , 0, ci
Gustavo Jasso Ahuja (UNAM)
R
i ∈ {1, . . . , n}
0
=q
−1
(0)
R
×
=q
−1
(1).
We call R = R0 ∪ R× the root system of q.
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
4 / 17
The Algebra G̃(q)
Construction [Barot,
Let corank q ∈ Z> 0 and let G(q) be the
quotient of F L by the ideal generated by the
following generalized Serre relations:
Kussin, Lenzing]
Let F L be the free Lie algebra with 3n generators
i ∈ {1, . . . , n}
e−i , hi , ei
homogeneous of degrees
−ci , 0, ci
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
i ∈ {1, . . . , n}
ICRA XIV. August 12, 2010
5 / 17
The Algebra G̃(q)
Construction [Barot,
Let corank q ∈ Z> 0 and let G(q) be the
quotient of F L by the ideal generated by the
following generalized Serre relations:
For all i, j ∈ {1, . . . , n} and ε, δ = ±1 let
Kussin, Lenzing]
Let F L be the free Lie algebra with 3n generators
homogeneous of degrees
−ci , 0, ci
(S1)
i ∈ {1, . . . , n}
e−i , hi , ei
i ∈ {1, . . . , n}
[hi , hj ] =0.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
5 / 17
The Algebra G̃(q)
Construction [Barot,
Let corank q ∈ Z> 0 and let G(q) be the
quotient of F L by the ideal generated by the
following generalized Serre relations:
For all i, j ∈ {1, . . . , n} and ε, δ = ±1 let
Kussin, Lenzing]
Let F L be the free Lie algebra with 3n generators
homogeneous of degrees
−ci , 0, ci
(S1)
[hi , hj ] =0.
(S2)
[hi , eεj ] =εCij eεj .
Gustavo Jasso Ahuja (UNAM)
i ∈ {1, . . . , n}
e−i , hi , ei
The EALA Associated with a Unit Form
i ∈ {1, . . . , n}
ICRA XIV. August 12, 2010
5 / 17
The Algebra G̃(q)
Construction [Barot,
Let corank q ∈ Z> 0 and let G(q) be the
quotient of F L by the ideal generated by the
following generalized Serre relations:
For all i, j ∈ {1, . . . , n} and ε, δ = ±1 let
Kussin, Lenzing]
Let F L be the free Lie algebra with 3n generators
homogeneous of degrees
−ci , 0, ci
(S1)
[hi , hj ] =0.
(S2)
[hi , eεj ] =εCij eεj .
(S3)
i ∈ {1, . . . , n}
e−i , hi , ei
i ∈ {1, . . . , n}
[eεi , e−εi ] =εhi .
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
5 / 17
The Algebra G̃(q)
Construction [Barot,
Let corank q ∈ Z> 0 and let G(q) be the
quotient of F L by the ideal generated by the
following generalized Serre relations:
For all i, j ∈ {1, . . . , n} and ε, δ = ±1 let
Kussin, Lenzing]
Let F L be the free Lie algebra with 3n generators
homogeneous of degrees
−ci , 0, ci
(S1)
[hi , hj ] =0.
(S2)
[hi , eεj ] =εCij eεj .
(S3)
(S∞)
i ∈ {1, . . . , n}
e−i , hi , ei
i ∈ {1, . . . , n}
[eεi , e−εi ] =εhi .
[eε1 i1 , . . . , eεt it ] =0,
P
whenever q( tk=1 εk ck ) > 1, for εk = ±1
and ik ∈ {1, . . . , n}.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
5 / 17
The Algebra G̃(q)
The Algebra G̃(q)
Let corank q ∈ Z> 0 and let G(q) be the
quotient of F L by the ideal generated by the
following generalized Serre relations:
Remark
Every monomial [eε1 i1 , . . . , eεt it ] ∈ G(q) has
a well defined degree,
For all i, j ∈ {1, . . . , n} and ε, δ = ±1 let
(S1)
(S2)
(S3)
(S∞)
[hi , hj ] =0.
[hi , eεj ] =εCij eεj .
[eεi , e−εi ] =εhi .
[eε1 i1 , . . . , eεt it ] =0,
P
whenever q( tk=1 εk ck ) > 1, for εk = ±1
and ik ∈ {1, . . . , n}.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
6 / 17
The Algebra G̃(q)
The Algebra G̃(q)
Let corank q ∈ Z> 0 and let G(q) be the
quotient of F L by the ideal generated by the
following generalized Serre relations:
Remark
Every monomial [eε1 i1 , . . . , eεt it ] ∈ G(q) has
a well defined degree, namely
For all i, j ∈ {1, . . . , n} and ε, δ = ±1 let
(S1)
(S2)
α=
t
X
(S3)
εk ck .
(S∞)
[hi , hj ] =0.
[hi , eεj ] =εCij eεj .
[eεi , e−εi ] =εhi .
[eε1 i1 , . . . , eεt it ] =0,
k=1
P
whenever q( tk=1 εk ck ) > 1, for εk = ±1
and ik ∈ {1, . . . , n}.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
6 / 17
The Algebra G̃(q)
The Algebra G̃(q)
Let corank q ∈ Z> 0 and let G(q) be the
quotient of F L by the ideal generated by the
following generalized Serre relations:
Remark
Every monomial [eε1 i1 , . . . , eεt it ] ∈ G(q) has
a well defined degree, namely
For all i, j ∈ {1, . . . , n} and ε, δ = ±1 let
(S1)
(S2)
α=
t
X
(S3)
εk ck .
(S∞)
[hi , hj ] =0.
[hi , eεj ] =εCij eεj .
[eεi , e−εi ] =εhi .
[eε1 i1 , . . . , eεt it ] =0,
k=1
P
whenever q( tk=1 εk ck ) > 1, for εk = ±1
and ik ∈ {1, . . . , n}.
Definition
G̃(q) := G(q) ⊕ (rad q)∗ .
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
6 / 17
Properties of G̃(q)
The Algebra G̃(q)
(EA2) It has a finite dimensional abelian
subalgebra H which equals it’s own
centralizer in G̃(q) and such that ad h is
diagonalizable for all h ∈ H.
Remark
Every monomial [eε1 i1 , . . . , eεt it ] ∈ G(q) has
a well defined degree, namely
α=
t
X
εk ck .
k=1
Definition
∗
G̃(q) := G(q) ⊕ (rad q) .
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
7 / 17
Properties of G̃(q)
The Algebra G̃(q)
(EA2) It has a finite dimensional abelian
subalgebra H which equals it’s own
centralizer in G̃(q) and such that ad h is
diagonalizable for all h ∈ H.
Remark
Every monomial [eε1 i1 , . . . , eεt it ] ∈ G(q) has
a well defined degree, namely
α=
t
X
εk ck .
k=1
(EA3) ad xα acts locally nilpotently for
α ∈ R× .
Definition
∗
G̃(q) := G(q) ⊕ (rad q) .
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
7 / 17
Properties of G̃(q)
The Algebra G̃(q)
(EA2) It has a finite dimensional abelian
subalgebra H which equals it’s own
centralizer in G̃(q) and such that ad h is
diagonalizable for all h ∈ H.
Remark
Every monomial [eε1 i1 , . . . , eεt it ] ∈ G(q) has
a well defined degree, namely
α=
t
X
εk ck .
k=1
(EA3) ad xα acts locally nilpotently for
α ∈ R× .
Definition
∗
G̃(q) := G(q) ⊕ (rad q) .
(EA4) R is discrete.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
7 / 17
Properties of G̃(q)
The Algebra G̃(q)
(EA2) It has a finite dimensional abelian
subalgebra H which equals it’s own
centralizer in G̃(q) and such that ad h is
diagonalizable for all h ∈ H.
Remark
Every monomial [eε1 i1 , . . . , eεt it ] ∈ G(q) has
a well defined degree, namely
α=
t
X
εk ck .
k=1
(EA3) ad xα acts locally nilpotently for
α ∈ R× .
Definition
∗
G̃(q) := G(q) ⊕ (rad q) .
(EA4) R is discrete.
(EA5) R is irreducible.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
7 / 17
Extended Affine Lie Algebras
Properties of G̃(q)
(EA2) It has a finite dimensional abelian
[Høegh-Krohn & Torresani. Allison, Azam,
Berman, Gao, Pianzola]
An extended affine Lie algebra (EALA) is a
complex Lie algebra which satisfies axioms
(EA2)-(EA5) together with the following
axiom:
Gustavo Jasso Ahuja (UNAM)
subalgebra H which equals it’s own centralizer in
G̃(q) and such that ad h is diagonalizable for all
h ∈ H.
(EA3) ad xα acts locally nilpotently for α ∈ R× .
(EA4) R is discrete.
(EA5) R is irreducible.
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
8 / 17
Extended Affine Lie Algebras
Properties of G̃(q)
(EA2) It has a finite dimensional abelian
[Høegh-Krohn & Torresani. Allison, Azam,
Berman, Gao, Pianzola]
An extended affine Lie algebra (EALA) is a
complex Lie algebra which satisfies axioms
(EA2)-(EA5) together with the following
axiom:
subalgebra H which equals it’s own centralizer in
G̃(q) and such that ad h is diagonalizable for all
h ∈ H.
(EA3) ad xα acts locally nilpotently for α ∈ R× .
(EA4) R is discrete.
(EA5) R is irreducible.
(EA1) The algebra has a non-degenerate
invariant symmetric bilinear form.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
8 / 17
Extended Affine Lie
Algebras
[Høegh-Krohn & Torresani. Allison, Azam,
Remark
If corank q > 2, then the algebra G̃(q) is not
an EALA.
Gustavo Jasso Ahuja (UNAM)
Berman, Gao, Pianzola]
An extended affine Lie algebra (EALA) is a
complex Lie algebra which satisfies axioms
(EA2)-(EA5) together with the following axiom:
(EA1) The algebra has a non-degenerate invariant
symmetric bilinear form.
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
9 / 17
Extended Affine Lie
Algebras
[Høegh-Krohn & Torresani. Allison, Azam,
Remark
If corank q > 2, then the algebra G̃(q) is not
an EALA.
One cannot define a non-degenerate
symmetric invariant bilinear form on G̃(q).
Gustavo Jasso Ahuja (UNAM)
Berman, Gao, Pianzola]
An extended affine Lie algebra (EALA) is a
complex Lie algebra which satisfies axioms
(EA2)-(EA5) together with the following axiom:
(EA1) The algebra has a non-degenerate invariant
symmetric bilinear form.
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
9 / 17
Remark
How do we fix it?
The algebra G̃(q) is an H ∗ -graded
H-module, hence it contains a unique
maximal ideal I with respect to
I ∩ H = {0}.
Gustavo Jasso Ahuja (UNAM)
If corank q > 2, then the algebra G̃(q) is not an
EALA.
One cannot define a non-degenerate symmetric
invariant bilinear form on G̃(q).
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
10 / 17
Remark
How do we fix it?
The algebra G̃(q) is an H ∗ -graded
H-module, hence it contains a unique
maximal ideal I with respect to
I ∩ H = {0}.
If corank q > 2, then the algebra G̃(q) is not an
EALA.
One cannot define a non-degenerate symmetric
invariant bilinear form on G̃(q).
The algebra E(q) := G̃(q)/I is an EALA.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
10 / 17
Main result
How do we fix ∗it?
Theorem
Zn
Let q :
→ Z be a connected non-negative
unit form with associated root system R.
Then the Lie algebra E(q) is a centerless
tame EALA with root system R.
Gustavo Jasso Ahuja (UNAM)
The algebra G̃(q) is an H -graded H-module,
hence it contains a unique maximal ideal I with
respect to I ∩ H = {0}.
The algebra E(q) := G̃(q)/I is an EALA.
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
11 / 17
Main result
How do we fix ∗it?
Theorem
Zn
Let q :
→ Z be a connected non-negative
unit form with associated root system R.
Then the Lie algebra E(q) is a centerless
tame EALA with root system R.
Furthermore, if q 0 is a connected
non-negative unit form which is equivalent
to q then E(q) and E(q 0 ) are isomorphic as
EALAs.
Gustavo Jasso Ahuja (UNAM)
The algebra G̃(q) is an H -graded H-module,
hence it contains a unique maximal ideal I with
respect to I ∩ H = {0}.
The algebra E(q) := G̃(q)/I is an EALA.
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
11 / 17
Main result
Theorem
n
Remark
In order to show that E(q) is an EALA it is
useful to introduce an alternative
construction of this algebra.
Gustavo Jasso Ahuja (UNAM)
Let q : Z → Z be a connected non-negative unit
form with associated root system R. Then the Lie
algebra E(q) is a centerless tame EALA with root
system R. Furthermore, if q 0 is a connected
non-negative unit form which is equivalent to q
then E(q) and E(q 0 ) are isomorphic as EALAs.
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
12 / 17
The Algebra Ê(q)
Construct a Lie algebra Ê(q)
Remark
In order to show that E(q) is an EALA it is useful
to introduce an alternative construction of this
algebra.
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
13 / 17
The Algebra Ê(q)
such that there is a projection
Remark
Ê(q) o o
Gustavo Jasso Ahuja (UNAM)
p
G̃(q)
In order to show that E(q) is an EALA it is useful
to introduce an alternative construction of this
algebra.
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
13 / 17
The Algebra Ê(q)
which factors
Ê(q) eKo o
Remark
p
K
K
K
K
tt
ttt
t
t
ty tt
G̃(q)
In order to show that E(q) is an EALA it is useful
to introduce an alternative construction of this
algebra.
G̃(q)/ ker p
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
13 / 17
The Algebra Ê(q)
exactly through E(q).
Remark
Ê(q) o o
cG
p
G
G
G
G
ww
ww
ww
w
{w
w
G̃(q)
In order to show that E(q) is an EALA it is useful
to introduce an alternative construction of this
algebra.
G̃(q)/I
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
13 / 17
The Algebra Ê(q)
It suffices to show that Ê(q) is an EALA.
p
Ê(q) o o
bE
E
E
E
G̃(q)
yy
yy
y
y
|y
y
Remark
In order to show that E(q) is an EALA it is useful
to introduce an alternative construction of this
algebra.
E(q)
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
13 / 17
The Algebra Ê(q)
The Algebra Ê(q)
It suffices to show that Ê(q) is an EALA.
For α ∈ R we let
Ê(q)α =





Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
Ê(q)
oo
bE
p
G̃(q)
E
E
E
yy
yy
y
y
y| y
E(q)
ICRA XIV. August 12, 2010
14 / 17
The Algebra Ê(q)
The Algebra Ê(q)
It suffices to show that Ê(q) is an EALA.
For α ∈ R we let
Ê(q)α =


Ceα
if α ∈ R× ,


Gustavo Jasso Ahuja (UNAM)
Ê(q)
oo
bE
p
G̃(q)
E
E
E
yy
yy
y
y
y| y
E(q)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
14 / 17
The Algebra Ê(q)
The Algebra Ê(q)
It suffices to show that Ê(q) is an EALA.
For α ∈ R we let


Ceα
Ê(q)α = Cn / rad q


Gustavo Jasso Ahuja (UNAM)
if α ∈ R× ,
if α ∈ R0 \ {0},
Ê(q)
oo
bE
p
G̃(q)
E
E
E
yy
yy
y
y
y| y
E(q)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
14 / 17
The Algebra Ê(q)
The Algebra Ê(q)
It suffices to show that Ê(q) is an EALA.
For α ∈ R we let


Ceα
Ê(q)α = Cn / rad q

 n
C ⊕ (rad q)∗
Gustavo Jasso Ahuja (UNAM)
if α ∈ R× ,
if α ∈ R0 \ {0},
if α = 0.
The EALA Associated with a Unit Form
Ê(q)
oo
bE
p
G̃(q)
E
E
E
yy
yy
y
y
y| y
E(q)
ICRA XIV. August 12, 2010
14 / 17
The Algebra Ê(q)
The Algebra Ê(q)
It suffices to show that Ê(q) is an EALA.
For α ∈ R we let


Ceα
Ê(q)α = Cn / rad q

 n
C ⊕ (rad q)∗
An let Ê(q) =
space.
L
if α ∈ R× ,
if α ∈ R0 \ {0},
if α = 0.
α∈R Ê(q)α
Gustavo Jasso Ahuja (UNAM)
Ê(q)
oo
bE
p
G̃(q)
E
E
E
yy
yy
y
y
y| y
E(q)
as a vector
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
14 / 17
The Algebra Ê(q)
The Algebra Ê(q)
Let α, β ∈ R× , σ, τ ∈ R0 , v, w ∈ Cn :
For α ∈ R we let
Ê(q)α =


Ceα
Cn / rad q

Cn ⊕ (rad q)∗
An let Ê(q) =
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
L
α∈R
if α ∈ R× ,
if α ∈ R0 \ {0},
if α = 0.
Ê(q)α as a vector space.
ICRA XIV. August 12, 2010
15 / 17
The Algebra Ê(q)
The Algebra Ê(q)
Let α, β ∈ R× , σ, τ ∈ R0 , v, w ∈ Cn :
(B1)
[πσ (v), πτ (w)] = q(v, w)πσ+τ (σ).
For α ∈ R we let
Ê(q)α =


Ceα
Cn / rad q

Cn ⊕ (rad q)∗
An let Ê(q) =
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
L
α∈R
if α ∈ R× ,
if α ∈ R0 \ {0},
if α = 0.
Ê(q)α as a vector space.
ICRA XIV. August 12, 2010
15 / 17
The Algebra Ê(q)
The Algebra Ê(q)
Let α, β ∈ R× , σ, τ ∈ R0 , v, w ∈ Cn :
(B1)
[πσ (v), πτ (w)] = q(v, w)πσ+τ (σ).
(B2)
[πσ (v), eβ ] = q(v, β)eβ+σ .
For α ∈ R we let
Ê(q)α =


Ceα
Cn / rad q

Cn ⊕ (rad q)∗
An let Ê(q) =
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
L
α∈R
if α ∈ R× ,
if α ∈ R0 \ {0},
if α = 0.
Ê(q)α as a vector space.
ICRA XIV. August 12, 2010
15 / 17
The Algebra Ê(q)
The Algebra Ê(q)
Let α, β ∈ R× , σ, τ ∈ R0 , v, w ∈ Cn :
(B1)
[πσ (v), πτ (w)] = q(v, w)πσ+τ (σ).
(B2)
[πσ (v), eβ ] = q(v, β)eβ+σ .
For α ∈ R we let
Ê(q)α =


Ceα
Cn / rad q

Cn ⊕ (rad q)∗
An let Ê(q) =
L
α∈R
if α ∈ R× ,
if α ∈ R0 \ {0},
if α = 0.
Ê(q)α as a vector space.
(B3)
[eα , eβ ] =


(α, β)eα+β
if α + β ∈ R× ,


Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
15 / 17
The Algebra Ê(q)
The Algebra Ê(q)
Let α, β ∈ R× , σ, τ ∈ R0 , v, w ∈ Cn :
(B1)
[πσ (v), πτ (w)] = q(v, w)πσ+τ (σ).
(B2)
[πσ (v), eβ ] = q(v, β)eβ+σ .
For α ∈ R we let
Ê(q)α =


Ceα
Cn / rad q

Cn ⊕ (rad q)∗
An let Ê(q) =
L
α∈R
if α ∈ R× ,
if α ∈ R0 \ {0},
if α = 0.
Ê(q)α as a vector space.
(B3)


(α, β)eα+β
[eα , eβ ] = (α, β)πα+β (α)


Gustavo Jasso Ahuja (UNAM)
if α + β ∈ R× ,
if α + β ∈ R0 ,
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
15 / 17
The Algebra Ê(q)
The Algebra Ê(q)
Let α, β ∈ R× , σ, τ ∈ R0 , v, w ∈ Cn :
(B1)
[πσ (v), πτ (w)] = q(v, w)πσ+τ (σ).
(B2)
[πσ (v), eβ ] = q(v, β)eβ+σ .
For α ∈ R we let
Ê(q)α =


Ceα
Cn / rad q

Cn ⊕ (rad q)∗
An let Ê(q) =
L
α∈R
if α ∈ R× ,
if α ∈ R0 \ {0},
if α = 0.
Ê(q)α as a vector space.
(B3)


(α, β)eα+β
[eα , eβ ] = (α, β)πα+β (α)


0
Gustavo Jasso Ahuja (UNAM)
if α + β ∈ R× ,
if α + β ∈ R0 ,
otherwise.
The EALA Associated with a Unit Form
ICRA XIV. August 12, 2010
15 / 17
The Algebra Ê(q)
The Algebra Ê(q)
Let β ∈ R× , τ ∈ R0 , w ∈ Cn and
ξ, ζ ∈ (rad q)∗ :
(B4)
Let α, β ∈ R× , σ, τ ∈ R0 , v, w ∈ Cn :
(B1)
[πσ (v), πτ (w)] = q(v, w)πσ+τ (σ).
(B2)
[πσ (v), eβ ] = q(v, β)eβ+σ .
[ξ, eβ ] = −[eβ , ξ] = ξρ(β)eβ .
(B3)


(α, β)eα+β
[eα , eβ ] = (α, β)πα+β (α)

0
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
if α + β ∈ R× ,
if α + β ∈ R0 ,
otherwise.
ICRA XIV. August 12, 2010
16 / 17
The Algebra Ê(q)
The Algebra Ê(q)
Let β ∈ R× , τ ∈ R0 , w ∈ Cn and
ξ, ζ ∈ (rad q)∗ :
(B4)
(B5)
Let α, β ∈ R× , σ, τ ∈ R0 , v, w ∈ Cn :
(B1)
[πσ (v), πτ (w)] = q(v, w)πσ+τ (σ).
(B2)
[πσ (v), eβ ] = q(v, β)eβ+σ .
[ξ, eβ ] = −[eβ , ξ] = ξρ(β)eβ .
[ξ, πτ (w)] = ξρ(β)πτ (w).
Gustavo Jasso Ahuja (UNAM)
(B3)


(α, β)eα+β
[eα , eβ ] = (α, β)πα+β (α)

0
The EALA Associated with a Unit Form
if α + β ∈ R× ,
if α + β ∈ R0 ,
otherwise.
ICRA XIV. August 12, 2010
16 / 17
The Algebra Ê(q)
The Algebra Ê(q)
Let β ∈ R× , τ ∈ R0 , w ∈ Cn and
ξ, ζ ∈ (rad q)∗ :
(B4)
(B5)
(B6)
Let α, β ∈ R× , σ, τ ∈ R0 , v, w ∈ Cn :
(B1)
[πσ (v), πτ (w)] = q(v, w)πσ+τ (σ).
(B2)
[πσ (v), eβ ] = q(v, β)eβ+σ .
[ξ, eβ ] = −[eβ , ξ] = ξρ(β)eβ .
[ξ, πτ (w)] = ξρ(β)πτ (w).
[ξ, ζ] = 0.
Gustavo Jasso Ahuja (UNAM)
(B3)


(α, β)eα+β
[eα , eβ ] = (α, β)πα+β (α)

0
The EALA Associated with a Unit Form
if α + β ∈ R× ,
if α + β ∈ R0 ,
otherwise.
ICRA XIV. August 12, 2010
16 / 17
The Algebra Ê(q)
Thanks for your attention!
Let β ∈ R× , τ ∈ R0 , w ∈ Cn and
ξ, ζ ∈ (rad q)∗ :
(B4)
(B5)
(B6)
Gustavo Jasso Ahuja (UNAM)
The EALA Associated with a Unit Form
[ξ, eβ ] = −[eβ , ξ] = ξρ(β)eβ .
[ξ, πτ (w)] = ξρ(β)πτ (w).
[ξ, ζ] = 0.
ICRA XIV. August 12, 2010
17 / 17