Maximal abelian Borel stable subspaces and the CDSW conjecture for

70 Pages·2012·0.71 MB·English

by  Paolo Papi

Maximal abelian Borel stable subspaces and the
CDSW conjecture for symmetric spaces
Paolo Papi
SapienzaUniversit`adiRoma
joint work with Paola Cellini, Pierluigi M¨oseneder and Marco
Pasquali
PaoloPapi BorelstablesubspacesandtheCDSWconjecture
Special cases:
adjoint case: g = k⊕k, k simple, σ flip.
graded case: g simple.
Problem
Fix a Borel subalgebra b ⊂ g . Find the maximal dimension of a
0 0
b -stable abelian subalgebra of g .
0 1
It turned out that it was more natural to try solving the following
more general problem.
Problem’
Find the poset structure of certain special posets I of b -stable
α,µ 0
abelian subalgebra of g .
1
The main problem
g finite dimensional semisimple complex Lie algebra
σ indecomposable involution of g
g = g ⊕g eigenspace decomposition
0 1
PaoloPapi BorelstablesubspacesandtheCDSWconjecture
Problem
Fix a Borel subalgebra b ⊂ g . Find the maximal dimension of a
0 0
b -stable abelian subalgebra of g .
0 1
It turned out that it was more natural to try solving the following
more general problem.
Problem’
Find the poset structure of certain special posets I of b -stable
α,µ 0
abelian subalgebra of g .
1
The main problem
g finite dimensional semisimple complex Lie algebra
σ indecomposable involution of g
g = g ⊕g eigenspace decomposition
0 1
Special cases:
adjoint case: g = k⊕k, k simple, σ flip.
graded case: g simple.
PaoloPapi BorelstablesubspacesandtheCDSWconjecture
The main problem
g finite dimensional semisimple complex Lie algebra
σ indecomposable involution of g
g = g ⊕g eigenspace decomposition
0 1
Special cases:
adjoint case: g = k⊕k, k simple, σ flip.
graded case: g simple.
Problem
Fix a Borel subalgebra b ⊂ g . Find the maximal dimension of a
0 0
b -stable abelian subalgebra of g .
0 1
It turned out that it was more natural to try solving the following
more general problem.
Problem’
Find the poset structure of certain special posets I of b -stable
α,µ 0
abelian subalgebra of g .
1
PaoloPapi BorelstablesubspacesandtheCDSWconjecture
The CDSW conjecture
Let g be simple. Consider the g−map
c : g → (cid:86)2g, c(x) = (cid:80)dimg[x,x ]∧xi, where {x }, {xi} are a pair
i=1 i i
(cid:86)
of dual bases w.r.t. the Killing form of g. Set R = (g⊕g). Using
c, can define three g-maps with target R,
2 2
(cid:94) (cid:94)
c : g → g⊗1, c : g → 1⊗ g, c : g → g⊗g.
1 2 3
Define
B = R/(cid:104)Im(c ),Im(c )(cid:105), A = R/(cid:104)Im(c ),Im(c ).Im(c )(cid:105).
1 2 1 2 3
Cachazo-Douglas-Seiberg-Witten conjecture
If S is the image in A of (cid:80) x ⊗xi, then Ag is a truncated
i i
polynomial algebra of dimension h∨, h∨ being the dual Coxeter
number of g.
PaoloPapi BorelstablesubspacesandtheCDSWconjecture
The CDSW conjecture for symmetric spaces
Kumar has version of the conjecture for infinitesimal symmetric
spaces g = g ⊕g . Just mimic the above construction using the
0 1
g −map c˜ : g → (cid:86)2g , c˜(x) = (cid:80)dimg[x,x ]∧xi, where
0 0 1 i=1 i
{x }, {xi} are a pair of dual bases of g .Correspondingly, set
i 1
A˜ = (cid:86)(g ⊕g )/(cid:104)Im(c˜ ),Im(c˜ ).Im(c˜ )(cid:105) and S˜ = (cid:80)dimg1x ⊗xi
1 1 1 2 3 i=1 i
Proposition
If g is irreducible as a g -module, then A˜g is a truncated
1 0
polynomial algebra generated by S˜.
PaoloPapi BorelstablesubspacesandtheCDSWconjecture
The above statement turns out to be true
in the adjoint case, where it reduces to the CDSW conjecture;
in the graded case with g ,g simple, where it reduces to a
0 1
conjecture of Kumar;
for any g,σ with g irreducible such that dimg ≤ 16
1 1
(MAGMA computations, done with the help of John Cannon).
Speculation
Recall the special posets I which appeared (undefined!) a few
α,µ
slides ago. We prove that they have minimum.
Speculation
dimA˜g = nilpotency class of S˜ = min(dimminI ).
α,µ
α,µ
PaoloPapi BorelstablesubspacesandtheCDSWconjecture
Speculation
Recall the special posets I which appeared (undefined!) a few
α,µ
slides ago. We prove that they have minimum.
Speculation
dimA˜g = nilpotency class of S˜ = min(dimminI ).
α,µ
α,µ
The above statement turns out to be true
in the adjoint case, where it reduces to the CDSW conjecture;
in the graded case with g ,g simple, where it reduces to a
0 1
conjecture of Kumar;
for any g,σ with g irreducible such that dimg ≤ 16
1 1
(MAGMA computations, done with the help of John Cannon).
PaoloPapi BorelstablesubspacesandtheCDSWconjecture
Malcev 1945: Maximal dimension of commutative subalgebras
of any simple Lie algebra (case by case calculations).
Kostant 1965: Link between commutative subalgebras and
eigenvalues of a Casimir operator.
Peterson 1999: The abelian ideals of a Borel subalgebra of g
are 2rank(g).
Panyushev 2003: Natural bijection between maximal abelian
ideals and long simple roots.
Suter 2004: conceptual approach to Malcev’s results.
Problem’: a short historical perspective
Schur 1905: There exist at most (cid:98)N2(cid:99)+1 linearly
4
independent commuting matrices in gl(N).
PaoloPapi BorelstablesubspacesandtheCDSWconjecture
Kostant 1965: Link between commutative subalgebras and
eigenvalues of a Casimir operator.
Peterson 1999: The abelian ideals of a Borel subalgebra of g
are 2rank(g).
Panyushev 2003: Natural bijection between maximal abelian
ideals and long simple roots.
Suter 2004: conceptual approach to Malcev’s results.
Problem’: a short historical perspective
Schur 1905: There exist at most (cid:98)N2(cid:99)+1 linearly
4
independent commuting matrices in gl(N).
Malcev 1945: Maximal dimension of commutative subalgebras
of any simple Lie algebra (case by case calculations).
PaoloPapi BorelstablesubspacesandtheCDSWconjecture