Plan
Erice, June 24 – July 3, 2013
Some properties of the Iwasawa polynomials for irreducible symmetric spaces in supergravity
Abstract
• The Iwasawa decomposition for symmetric spaces
We analyse the polynomial part of the Iwasawa realization of symmetric spaces
B. L. Cerchiai (Università di Milano)
based on work with S. Cacciatori, S. Ferrara and A. Marrani, to appear soon
• Konstant’s analysis of the nilpotent subalgebra for the adjoint
representation of the split form and the principal su(2)P
appearing in supergravity. We first study the role of the principal SL(2)P
subgroup and how it determines the structure of the nilpotent subalgebras.
This allows us to compute the maximal degree of the polynomials for all the
• Generalization to all the faithful representations for all the
non compact real forms
faithful representations of the algebra. In particular, we find the degree of
the metric coefficients. Since the coset representative directly enters in the
• Tables with summary of the results
Lagrangian of the corresponding supergravity theory, this is important for the
investigation of its features, and should be helpful e.g. in approaching such
• Applications to supergravity theories
issues as its ultraviolet properties.
• Conclusions and outlook
Konstant’s analysis of the nilpotent subalgebra for the adjoint
representation of the split form and the principal su(2)P
The Iwasawa decomposition for symmetric spaces
[B. Kostant, Am. J. Math. 81, 973–1032 (1959)]
It can be applied only to a non compact group Gnc. Let’s start
by studying an irreducible representation of the algebra g.
1) Identification of the Lie algebra h corresponding to the symmetrically embedded maximal compact subgroup H:
g = h ⊕ p, with p the vector space orthogonal to h and
h =dim(H).
M[~
y, ~
x] = exp 
Nilpotent part of the coset representative M[~
x, ~
y ]:
N (~
x) = exp
a
x λαa
a=1
⇒ Terms of degree n:
1
Mn =
n!
X
d1 +...+dh =n, di ≥0
∞
X
1
=
n!
n=0
(x1 )d1
(xh )dh
···
d1 !
dh !
h
X
a
x λαa
a=1
permutations σ
Problem: Determine the largest n so that Mn 6= 0 and Mn+1 = 0.
Answer: Let α1, . . . , αl be the simple roots, such that all other
positive roots are linear combinations of these with non-negative
integer coefficients.
Let q = n1 +. . .+nl , with n1, . . . , nl the coefficients of the longest
root αL = n1α1 + . . . + nl αl . Notice that
An :
Bn :
Cn :
◦ − ◦ − ··· − ◦ − ◦
1
2
n
n−1
2
n
n−1
◦ − ◦ − ··· − ◦ ⇐ ◦
1
2
n
n−1
◦n
|
Dn : ◦ − ◦ − · · · − ◦ − ◦
1
E6 :
E7 :
E8 :
2
n−2 n−1
◦6
|
◦−◦−◦−◦−◦
1
2
3
4
2
3
4
5
n−1
!
2n
i
n−2
, 2n−1 , 2n−1
i=1
!
(27, 351, 2925, 351, 27, 78)
5
◦7
|
◦−◦−◦−◦−◦−◦
1
i=1
2n + 1
, 2n
i
i=1
n 2n
2n
−
i
i−2
i=1
◦ − ◦ − ··· − ◦ ⇒ ◦
1
n+1
i
n (133, 8645, 365750, 27664, 1539, 56, 912)
6
◦8
(3875, 6696000, 6899079264,
|
◦ − ◦ − ◦ − ◦ − ◦ − ◦ − ◦ 146325270, 2450240, 30380, 248, 147250)
1
2
3
4
5
6
7
F4 :
◦−◦⇒◦−◦
(52, 1274, 273, 26)
G2 :
◦V◦
(14, 7)
1
1
2
3
2
4
y ici exp 
h−k
X
a=1

Adj (G) =
xaλαa  .
l
X
(2jA + 1) =
A=1
l
X
A=1
SjA ,
j1 = 1 < .... < jl
with SjA the su(2)P -irrep. of spin s = jA.
Tables with summary of the results
The first table shows for the simple Lie algebras the corresponding Dynkin
diagram and the dimensions of the fundamental representations associated to
each of the nodes, by applying the Weyl character formula (see e.g. [Bourbaki,
”Groupes et algèbres de Lie”, Chap. VIII.13, Herman, Paris, 1975]).
In the second table the main ingredients necessary for the Iwasawa construction of the non-compact, irreducible, Riemannian, globally symmetric
spaces T = Gnc /H are listed (see also [Cacciatori, Dalla Piazza, Scotti,
arXiv:1207.1262 [math.GR]]), where
T is the coset of the non compact real form Gnc of the Lie group G with
respect to its maximal compact subgroup H in the classification from [Araki,
Journal of Mathematics, Osaka City University, 13 (1962) 1; Helgason, ”Differential Geometry, Lie Groups and Symmetric Spaces”, Academic Press, New
York, 1978)].
ΛG/H specifies the type of the root system,
(n1 , ..., nr ) are the coefficients of the longest root αL ,
m
~ λ, m
~ 2λ the multiplicities for the roots and the double roots.
For reducible representations: the degree is determined by the
maximal spin among all sub representations,
for semisimple groups: by the maximal degree among the simple
factors.
Then the nilpotency of the elements of N is 2q + 1, i.e. the
maximal degree is 2q.

digT = 2jgi T , where jgi T = ēi · C −1ε̄T .
Here C is the Cartan matrix of g.
q = jl = CG − 1
with CG the Coxeter number of G, and jl the maximal spin of the
sl(2)P -irreps. into which the adjoint irrep. Adj of G branches.
Fundamental Representations of the Exceptional Lie groups

Let g be a complex simple Lie algebra of rank l and gT be its
real form corresponding to a given symmetric space of type T
and rank l. Consider its Satake diagram in the classification by
[Araki, Journal of Mathematics, Osaka City University, 13 (1962) 1] and let
ε̄T be the column vector in Rl with entry 1 if the corresponding
simple root in the diagram is associated to a white dot, i.e. if
it belongs to the quotient, and zero otherwise (Satake vector ).
Let {ēi}li=1 the canonical basis of Rl . Then the nilpotent N (~
x)
in the i-th fundamental representation is a polynomial of degree
λσ1 · · · λσn
A=1
(2δA − 1) = dim (G)
Principal su(2)P : Unique su(2) embedded in g generally nonsymmetrically (exception: su(3)) and generally maximally (exception: e6) in such a way that δA = jA + 1, where:
Generalization to all the faithful representations for all the non
compact real forms
!n
X
i=1
with
l
X
Invariant symmetric primitive polynomials:
l polynomials of order δA (A = 1, . . . , l) ⇒
l primitive Racah-Casimir polynomials CδA of order δA and
l skewsymmetric invariant primitive tensors Ω(2δA−1) of order
2δA −1. ⇒ Homology behaves as a product of l spheres S (2δA−1).
A representative for the coset G/H can be written as:
r
X
1 + t2δA−1
It determines the topology of the compact form G.
⇓

l Y
A=1
with H the fiber,
A=Abelian subgroup generated by ap,
N =nilpotent subgroup generated by the eigenmatrices {λαi }
of the adjoint action of A on the system of positive roots.
3) Calculation of the corresponding system of positive roots {αi}
and of the corresponding eigenmatrices {λαi }, i = 1, . . . h − k,
with k the dimension of the normalizer of ap in h.
!
fG (t) =
g = HAN
2) Choice of a maximally non compact Cartan subalgebra a as a
pivot. It is generated by l commuting generators {c1, . . . , cl },
where l is the rank of the group. Out of these, at most r
can be chosen non compact, i.e. {c1, . . . , cr }, where r is the
rank of the coset G/H:
a = ah ⊕ ap, with ap = a ∩ p, ah ⊂ h, dim(ap) = r, dim(ah) =
s = l − r.
h
X
Poincaré polynomial of G:
4) Computation of a realization of a generic element g of the
group by using the above fibration to construct a well-defined
parametrization of the group:
The third table lists the rank of the group, the vector of the degrees of
the polynomials in the fundamental representations, and the degree of the
polynomial part in the biinvariant metric.
Non compact Lie groups with the list of data necessary for the analysis Summary of the analysis: fundamental degrees and degree of the metric
T
Gnc
H
ΛG/H
(n1 , ..., nr )
m
~ λ, m
~ 2λ
T
l
(d1 , . . . , dl )
AI
AII
AIIIa
AIIIb
AIV
BIa
BIb
BII
CI
CIIa
CIIb
DIa
DIb
DIc
DII
DIIIa
DIIIb
G
FI
FII
EI
EII
EIII
EIV
EV
EVI
EVII
EVIII
EIX
SL(n + 1, R)
SU ∗ (2k)
SU (p, q)
SU (p, p)
SU (1, n)
SO(n, n + 1)
SO(p, q)
SO(1, 2n)
Sp(2n, R)
U Sp(2p, 2q)
U Sp(2k, 2k)
SO(n, n)
SO(n − 1, n + 1)
SO(p, q)
SO(1, 2n − 1)
SO∗ (4k + 2)
SO∗ (4k)
G2(2)
F4(4)
F4(−20)
E6(6)
E6(2)
E6(−14)
E6(−26)
E7(7)
E7(−5)
E7(−25)
E8(8)
E8(−24)
SO(n + 1)
U Sp(2k)
S(U (p) × U (q))
S(U (p) × U (p))
S(U (1) × U (n))
SO(n) × SO(n + 1)
SO(p) × SO(q)
SO(2n)
U (n)
U Sp(2p) × U Sp(2q)
U Sp(2k) × U Sp(2k)
SO(n) × SO(n)
SO(n − 1) × SO(n + 1)
SO(p) × SO(q)
SO(2n − 1)
U (2k + 1)
U (2k)
SO(4)/Z2
U Sp(6) × U Sp(2)
SO(9)
U Sp(8)/Z2
(U Sp(2) × SU (6))/Z2
(U (1) × SO(10))/Z4
F4
SU (8)/Z2
(SU (2) × SO(12))/Z2
(U (1) × E6 )/Z3
Ss (16)
(SU (2) × E7 )/Z2
An (n ≥ 1)
Ak−1 (k > 1)
Bp (1 < p < q)
Cp (p > 1)
A1
Bn (n ≥ 2)
Bp (1 < p < n)
A1
Cn (n ≥ 3)
Bp (1 ≤ p ≤ (n − 1)/2)
Ck
Dn (n > 3)
Bn−1 (n > 2)
Bp (1 < p < n − 1)
A1
Bk (k ≥ 2)
Ck (k ≥ 2)
G2
F4
A1
E6
F4
B2
A2
E7
F4
C3
E8
F4
(1,1,. . . ,1)
(1,1,. . . ,1)
(2,2,. . . ,2)
(2, 2, . . . , 2, 1)
(2)
(1,2,. . . ,2)
(1,2,. . . ,2)
(1)
(2,2,. . . ,2,1)
(2,2,. . . ,2)
(2, 2, . . . , 2, 1)
(1,2,. . . ,2,1,1)
(1,2,. . . ,2)
(1,2,. . . ,2)
(1)
(2,2,. . . ,2)
(2,2,. . . ,2,1)
(3,2)
(2,3,4,2)
(2)
(1,2,2,3,2,1)
(2,3,4,2)
(2,2)
(1,1)
(2,2,3,4,3,2,1)
(2,3,4,2)
(2,2,1)
(2,3,4,6,5,4,3,2)
(2,3,4,2)
(1), (0)
(4), (0)
2(1, q − p), (0, 1)
(1,2), (0,0)
(2n-2), (1)
(1,1), (0,0)
(1,2(n-p)+1), (0,0)
(2n-1), (0)
(1,1), (0,0)
(4,4n-8p), (0,3)
(3,4), (0,0)
(1), (0)
(1,2), (0,0)
(1,2(n-p)), (0,0)
(2n-2), (0)
(4,4), (0,1)
(1,4), (0,0)
(1,1), (0,0)
(1,1), (0,0)
(8), (7)
(1),(0)
(1,2), (0,0)
(6,8), (0,1)
(8), (0)
(1),(0)
(1,4), (0,0)
(1,8), (0,0)
(1), (0)
(1,8), (0,0)
AI
n
({i(n + 1 − i)}ni=1 )
AII
AIIIa
AIIIb
AIV
BIa
BIb
BII
CI
CIIa
CIIb
DIa
DIb
2k − 1
p + q − 1 = 2n − 1
2p-1
n
n
n = (p + q − 1)/2
n
n
p + q = n = 2k
2k
n
n
DIc
DII
n = (p + q)/2
n
DIIIa
2k + 1
DIIIb
G
FI
FII
EI
EII
EIII
EIV
EV
EVI
EVII
EVIII
EIX
2k
2
4
4
6
6
6
6
7
7
7
8
8
dg
({2(k − i)(i − 1) + k − 1, 2i(k −
({i(2p + 1 −
i)}p−1
i=1 , {p(p
+ 1)}qi=p , {(2n −
({i(2p − i)}2p−1
i=1 )
i)}k−1
i=1 , k
− 1)
i)(i + p − q +
2(n − 1)
1)}2n−1
i=q+1 )
2k − 4
4p − 2
(2,. . . ,2)
({i(2n + 1 − i)}n−1
i=1 , n(n + 1)/2)
p
({i(2p + 1 − i)}i=1 , {p(p + 1)}ni=p+1 )
(2, . . . , 2, 1)
({i(2n − i)}ni=1 )
({2(2i − 1)(p + 1) − 2i2 , 2i(2p − i + 1)}pi=1 , {2p(p + 1)}ni=p+1 )
({2(2i − 1)(2k − 1) − 2i2 + 2i − 1, 2i(2k − i)}ki=1 )
({i(2n − 1 − i)}n−2
i=1 , n(n − 1)/2, n(n − 1)/2)
({i(2n − 1 − i)}n−2
i=1 , n(n − 1)/2, n(n − 1)/2)
4p − 4
2
4n − 4
4p − 4
0
4n − 4
4p − 2
4k − 4
4n − 8
4n − 8
2
2
({2ki − 2b 2i c2 }2k−1
i=1 , k + k − 1, k + k − 1)
4k − 2
2
n
({2pi − i2 + i}p−1
i=1 , {p + p}i=p )
(2,. . . ,2,1,1)
2
2
({2ki − 2b 2i c2 − i}2k−2
i=1 , k − 1, k )
(10, 6)
(22, 42, 30, 16)
(4, 8, 6, 4)
(16, 30, 42, 30, 16, 22)
(16, 30, 42, 30, 16, 22)
(6, 10, 14, 10, 6, 8)
(4, 6, 8, 6, 4, 4)
(34, 66, 96, 75, 52, 27, 49)
(22, 42, 60, 46, 32, 16, 30)
(10, 18, 26, 21, 16, 9, 13)
(92, 182, 270, 220, 168, 114, 58, 136)
(32, 62, 92, 76, 60, 42, 22, 46)
4p − 4
0
4k − 4
8
20
2
20
20
6
2
32
20
8
56
20
Conclusions and outlook
Applications to supergravity
From the results in tables 1–3:
1) Simplest case: 4-dim. N = 4 pure supergravity and N = 2
supergravity minimally coupled to one Abelian vector multiplet
G = SL(2,R) in the representation 2
H
SO(2)
⇒ Result: d=1
G = SL(2,R) in the rep. 4 = weight 3λ
2) The t3 model: H
1
SO(2)
⇒ Result: d=3
This agrees with the findings of [Ceresole, Ferrara, Gnecchi, Marrani,
arXiv:1210.5983 [hep-th]].
It is an example of a representation which does not correspond
to a fundamental weight.
3) N = 2 Magic theories associated to the algebra L3 (AS , J3(B))
[Günaydin, Sierra, Townsend, Phys. Lett. 133B (1983) 72]
In this table Gnc describes the electric magnetic duality of the
corresponding theory, which is defined in D dimensions. Each
subsequent row can be obtained by dimensional reduction of the
preceding one, each column from the following one by truncation.
AS \ B
R
C
H
O
D
d
dg
CS
SL(3, R)
SL(3, C)
SU ∗ (6)
E6(−26)
5
4
2
HS
Sp(6, R)
SU (3, 3)
SO∗ (12)
E7(−25)
4
9
8
OS
F4(4)
E6(2)
E7(−5)
E8(−24)
3
22
20
Notice that the values of d, dg depend only on the space-time
dimension D. This is consistent with the Tits-Satake projection
[Fré, Sorin, Trigiante, arXiv:1107.5986 [hep-th]].
• From the point of view of mathematics, we have extended
the methods by Kostant [B. Kostant, Am. J. Math. 81, 973–1032
(1959)] for studying the maximal nilpotency of subalgebras in
the adjoint representation of the split form to all the faithful
representations of the algebra for all the non compact forms.
This technique is based on the embedding of the principal
sl(2)P , determining the dimensions of the Hopf spheres which
enter in the cohomology of the group. Therefore, such an
approach should lead to a deeper understanding of its cohomological structure.
• From the point of view of physics, it is well-known that irreducible symmetric spaces play a prominent role in supergravity theories (see e.g. [Ferrara, Marrani, arXiv:0808.3567 [hep-th]]).
Then the Iwasawa parametrization provides a way to choose
the fields so that they enter in a polynomial way in the Lagrangian. A knowledge of their structure could thus provide
useful insights on the quantum properties of the theory, including the ultraviolet behaviour.