Affine Kac-Moody symmetric spaces associated with untwisted Kac-Moody algebras Saudamini Nayak, S.S.Rout and K.C.Pati∗ Department of Mathematics, National Institute of Technology, Rourkela, Odisha-769 008 India.
arXiv:1312.3784v1 [math-ph] 13 Dec 2013
In this paper we have computed all the affine Kac-Moody symmetric spaces which are tame Fr´ echet manifolds starting from the Vogan diagrams related to the affine untwisted Kac-Moody (1) algebras. The detail computation of affine Kac-Moody symmetric spaces associated with A1 and (1) A2 are shown algebraically to corroborate our method. PACS numbers:
I.
INTRODUCTION
Finite dimensional symmetric spaces [20, 25] are rather well understood mathematical objects which have recently gained much importance in both mathematics and physics due to their intimate connections with random matrix theories, Reimannian geometries and their applications to many integrable systems, quantum transport phenomena (disordered system etc) [4, 5, 9, 22]. A compact irreducible symmetric space is either a compact simple Lie group G or a quotient G/K of a compact simple Lie group by the fixed point set of an involution ρ (or an open subgroup of it) and g = t ⊕ p is the decomposition of the Lie algebra g of group G into +1 and -1 eigenvalue spaces of ρ then K acts on g by adjoint representation leaving the decomposition invariant. The restriction of this action to p can be identified with the isotropy representation of G/K and we know that the isotropy representation of a symmetric space is polar. With the advent of Kac-Moody algebras [15, 16, 21] which can be considered as the generalization of finite dimensional Lie algebras, naturally a search of infinite dimensional version of symmetric spaces began, the closest generalization being the affine Kac-Moody symmetric spaces. An affine Kac-Moody symmetric space is by definition ˆ ˆ G ˆ ρˆ of G ˆ by the fixed point set of an involution of either an affine Kac-Moody group G(group type) or a quotient G/ ˆ into the ±1 eigen spaces of ρˆ. Then ˆ the second kind. In fact if gˆ = t + pˆ in the splitting of the Lie algebra of G ˆ K ˆ is the left invariant metric obtained from the restriction of the inner product of gˆ to pˆ. In finite the metric of G/ dimension, the isotropy representation is polar itself while in infinite dimension it leaves invariant a co-dimension-2 sub-manifold which can be identified with a (pre-)Hilbert space and the induced action on this space is a polar action by affine isometries. The study on affine Kac-Moody symmetric spaces began with C.L.Terng [26] who conjectured the existence of infinite dimensional symmetric spaces. Important progresses towards their constructions and geometries are achieved by B. Popescu [23], W. Freyn [10], Heintze [13, 14], et al. Now it has been shown that affine Kac-Moody symmetric spaces are tame Fr´ echet manifolds. In particular let G be the simply connected Lie group with Lie algebra g and denote σ be the automorphism of G corresponding to σ ∈ Aut(g). Then the loop group L(G, σ) = {g : R → G | g ∈ C ∞ , g(t + 2π) = σ(g(t)) ∀t}
(1)
ˆ with point wise multiplication is a Fr´ echet Lie group with Lie algebra L(g, σ). The affine Kac-Moody group L(G, σ) 2 1 1 ˆ will be a T = S × S bundle over L(G, σ). By construction L(G, σ) is a Fr´ echet group and it is a Lorentz manifold. ˆ ˆ ˆ It is also well known that L(G, σ)/L(G, σ)ρˆ for any involution ρˆ of L(G, σ) and L(G, σ)/L(G, σ)ρ for any involution ρˆ are tame Fr´ echet. The classification of affine Kac-Moody symmetric spaces is essentially equivalent to the classification of involutions of affine Kac-Moody algebras upto conjugation. This has been achieved by a long series of papers by various authors Batra [1], Levstein [19], Kobayashi [17], Rousseau and Messaoud [24]. There is a one-to-one correspondence between real forms of Kac-Moody algebras, involutions. The diagramatic representation of real forms are the Vogan diagrams (to each real form there is a unique Vogan diagram) which are Dynkin diagrams together with some extra pieces of informations. We have collected/construted the Vogan diagrams related with untwisted classical algebras (1) (1) (1) (1) An , Bn , Cn , Dn explicitly. From the Vogan diagrams we have obtained the fixed point set as well as the real forms of the algebras and then we have constructed affine Kac-Moody symmetric spaces. In chapter-2 we have given a brief introduction to affine untwisted Kac-Moody algebras with their involutions, real forms and Vogan diagrams. In chapter-3 we have explicitly calculated the affine Kac-Moody symmetric spaces with two elementary untwisted (1) (1) Kac-Moody algebras A1 and A2 relating them with their Vogan diagrams to corroborate our technique to construct
∗ Electronic
address:
[email protected]
2 affine Kac-Moody symmetric spaces. Towards the end of the chapter we have given exhaustive list of all the affine Kac-Moody symmetric spaces together with their real forms and fixed algebras. Chapter-4 contains appendix. II. A.
PRELIMINARIES
Kac-Moody Lie algebras
Let I = [1, n + 1], n ∈ N, be an interval in N. A matrix A = (aij )i,j∈I with integer coefficients is called a generalized Cartan matrix if it satisfies the following conditions: 1. aii = 2 for i = 1, · · · , n + 1. 2. aij ≤ 0 for i 6= j. 3. aij = 0 iff aji = 0. A realization of A is a triple (h, π, π ˇ ), where h is a finite-dimensional complex vector space and π = {αi }i∈I ⊂ h∗ and π ˇ = {αi }i∈I ⊂ h are indexed subsets in h∗ and h respectively, and they satisfy 1. both π, π ˇ are linearly independent. 2. < αi , αˇj >= aji , for i, j = 1, · · · , n. 3. rank (A) = 2n - dim h. For any n × n matrix A there exits a unique(upto isomorphism) realization. Given two matrices A and A′ and their realizations (h, π, π ˇ ) and (h′ , π ′ , π ˇ ′ ), we obtain a realization of the direct ′ ′ ′ sum of two matrices (h ⊕ h , π ⊗ {0} ∪ {0} ⊗ π , π ˇ ⊗ {0} ∪ {0} ⊗ π ˇ ) which is called direct sum of the realizations. A matrix A is called decomposable if after reordering of indices A decomposes into a non-trivial direct sum. Otherwise A is called indecomposable. Let A = (aij ) be a generalized Cartan matrix and let (h, π, π ˇ ) be a realization of A. Let g(A) be a complex Lie algebra with generators ei , fi for i = 1 · · · n and h and the following defining relations: [ei , fj ] [h, h′ ] [h, ei ] [h, fi ]
= = = =
δij αˇi 0 < αi , h > ei − < αi , h > fi
and the Serre relations (ad ei )1−aij ej = 0,
(ad fi )1−aij fj = 0; ∀i 6= j,
(2)
The Lie algebra g = g(A) is called a Kac-Moody algebra. The subalgebra h of g is called the Cartan subalgebra. The matrix A is the Cartan matrix of g which is of rank n. The elements ei , fi for i = 1 · · · n, are called Chevalley generators and they generates the subalgebra g ′ = [g, g] and g = g ′ + h. B.
Affine Kac-Moody Algebras
Consider the generalized Cartan matrix A. It is called a Cartan matrix of affine type if A1 0 . • A is an indecomposable matrix, i.e. after the indices are reordered A cannot be written in the form 0 A2 n+1 • There exits a vector (ai )n+1 i=1 , with ai all positive such that A(ai )i=1 = 0.
Then, the algebra g associated with A is called an affine Kac-Moody algebra. Affine Kac-Moody algebras are of two types untwisted and twisted. In this paper we have confined ourselves to untwisted case only.
3 C.
A realization of non-twisted affine Kac-Moody Lie algebra
−1 PLet L j = C[t, t ] be the algebra of Laurent polynomials in t. The residue of the Laurent polynomial P = c t (where all but finite number of cj are zero) is ResP = c−1 . j∈Z j Let ˚ g be a finite-dimensional simple Lie algebra over C of type Xn , then L(˚ g) = L ⊗ ˚ g is an infinite-dimensional Lie algebra with the bracket
[P ⊗ X, Q ⊗ Y ] = P Q ⊗ [x, y]
P, Q ∈ L; X, Y ∈ ˚ g.
(3)
Fix a non-degenerate, invariant, symmetric bilinear form (., .) in ˚ g and extend this form to an L valued form (., .)t on L(˚ g) by (P ⊗ x, Q ⊗ y)t = P Q(x, y)
P, Q ∈ L; x, y ∈ ˚ g.
(4)
P ∈ L; X ∈ ˚ g.
(5)
j
The derivation t (d/dt) of L extends to L(˚ g) by tj
dP d (P ⊗ X) = tj ⊗ X, dt dt
Therefore ψ(a, b) = Res( da g) defines a two-cocycle on L(˚ g). dt , b)t for a, b ∈ L(˚ ˜ Now we denote by L(˚ g ) the central extension of the Lie algebra L(˚ g ) associated to the cocycle ψ. Explicitly ˜ g) = L(˚ L(˚ g ) ⊕ Cc with the bracket [a + λc, b + µc] = [a, b] + ψ(a, b)c a, b ∈ L(˚ g); λ, µ ∈ C.
(6)
ˆ g ) is the Lie algebra which is obtained by adjoining to L(˚ ˜ g ) a derivation d which acts on L(˚ Finally, denote by L(˚ g) as d ˆ g ) = L(˚ g ) ⊕ Cc ⊕ Cd with the bracket defined by t dt and kills c. Explicitly we have L(˚ [tk ⊗ x + λc + µd, tj ⊗ y + λ1 c + µd] = tj+k ⊗ [x, y] + µjtj ⊗ y − µ1 ktk ⊗ x + kδj,−k (x, y)c,
(7)
ˆ g) is a non- twisted affine Kac-Moody Lie algebra associated to the where x, y ∈ ˚ g; λ, µ, λ1 , µ1 ∈ C; j, k ∈ Z. This L(˚ (1) affine matrix A of type Xn . D.
Automorphisms and Real forms of non-twisted affine Kac-Moody algebras
Define a group G acting on the algebra g through adjoint representation Ad : G → Aut(g). It is generated by the subgroup Uα for α ∈ ±π and Ad(Uα ) = exp(ad(gα )). A maximal adg -diagonalizable subalgebra of g is called a Cartan subalgebra. Every Cartan subalgebra of g is Ad(G)-conjugate to the standard Cartan subalgebra h. A Borel subalgebra of g is maximal L completely solvable L subalgebra. It is conjugated by Ad(G) to b+ and b− where b+ = h ⊕ α>0 gα and b− = h ⊕ α
α1
α2
Thus the corresponding symmetric spaces are (1)
(1)
SL1 (3, R)/SO1 (3),
(1)
SU (1) (3)/SO1 (3).
(32)
Case IV:
0 0 1 U (t) = 0 1 0 , 1 0 0
u = −1,
ξ = 0,
U (t)−1
0 0 −1 = 0 −1 0 −1 0 0
(33)
Under the automorphism (8) with (33) the matrix transforms as (n) (n) (n) (n) (n) (n) A11 A12 A13 −A33 −A32 −A31 (n) (n) (n) (n) (n) (n) A21 A22 A23 −→ −A23 −A22 −A21 . (n) (n) (n) (n) (n) (n) n∈Z n∈Z A31 A32 A33 −A13 −A12 −A11
P
Here K = (n) (n) A11 − A33 (n) (n) 1 A21 − A23 2 (n) (n) n∈Z A31 − A13 and P = (n) (n) A11 + A33 (n) (n) 1 A21 + A23 2 (n) (n) n∈Z A31 + A13
P
P
P
(34)
P
(n) (n) (n) (n) b(n) )− (A b(n) )− (A b(n) )− A12 − A32 A13 − A31 (A 11 12 13 b(n) (n) (n) b(n) )− (A21 )− 0 A23 − A21 = 12 0 (A 23 (n) (n) (n) (n) (n) (n) (n) b b b n∈Z A32 − A12 A33 − A11 (A31 )− (A32 )− (A33 )− (n) (n) (n) (n) b(n) )+ (A b(n) )+ (A b(n) )+ A12 + A32 A13 + A31 (A 11 12 13 b(n) (n) (n) (n) b(n) )+ b(n) )+ (A Thus K + iP is (A21 )+ (A A23 + A21 = 12 A22 23 22 (n) (n) (n) (n) b(n) )+ (A b(n) )+ (A b(n) )+ n∈Z (A A32 + A12 A11 + A33 31 32 33 b(n) )− + i(A b(n) )+ (A b(n) )− + i(A b(n) )+ (A b(n) )− + i(A b(n) )+ (A 11 11 12 12 13 13 1 b(n) b(n) )+ b(n) )+ b(n) )− + i(A b(n) )+ (A21 )− + i(A i(A (A 21 22 23 23 2 b(n) )− + i(A b(n) )+ (A b(n) )− + i(A b(n) )+ (A b(n) )− + i(A b(n) )+ n∈Z (A 31 31 32 32 33 33
P
P
(35)
(1)
such that trace of this matrix is zero. Hence the non-compact real form is K + iP ⊕ Ric ⊕ Rid ∈ sl−1 (3, R). The Vogan diagram is (1) Here K = so−1 (3). Thus the corresponding symmetric spaces are (1)
(1)
SL−1 (3, R)/SO−1 (3),
(1)
SU (1) (3)/SO−1 (3).
(36)
12 ❄ ⑧⑧ ❄❄α❄0 ⑧ α1 ⑧ ❄❄ ❄ ⑧⑧ ⑧ _ α2 >
Thus we have completed the explicit (algebraically) calculation of affine Kac-Moody symmetric spaces associated (1) (1) with A1 and A2 and relating them with the Vogan diagrams which classify their real forms also. The next tables contain the real forms, Vogan diagrams, fixed algebras associated with the automorphism, related compact and noncompact affine kac-Moody symmetric spaces. In order to have a clear idea about the computation of fixed algebras and affine Kac-Moody symmetric spaces we have provided an appendix which contains all the necessary ingredients to understand the method of our calculations.
(1)
TABLE I: Affine Kac-Moody symmetric spaces associated with A2n−1 Dynkin Diagram
Real Forms
su(1) (2n)
♦❖❖α❖0❖ ❖❖❖ ♦♦♦ ♦ ♦ ❖❖❖ ♦ ♦ ♦ ❖ α ♦ α ♦α♦1 α2 αn 2n−2 ❖ 2n−1
(1)
su−1 (p, q), p + q = 2n
(1)
su1 (p, q), p + q = 2n
(1)
sls (n, H)
(1)
sl−1 (2n, R), n≥3
(1)
sl1 (2n, R), n≥4 (1)
slrn (n, H)
(1)
slrs (n, H)
Vogan Diagram
Fixed Algebra
❖α0 ♦♦♦ ❖❖❖❖❖ ♦ ♦ ❖❖❖ ♦♦ ❖❖ α ♦♦♦ α ♦α♦1 α2 αn 2n−2 ❖ 2n−1 •❖α0 ♦♦♦ ❖❖❖❖❖ ♦ ♦ ❖❖❖ ♦♦ ❖❖ α ♦♦♦ α ♦α♦1 α2 αn 2n−2 ❖ 2n−1
Compact affine Kac- Non-compact Moody Symmetric Kac-Moody spaces metric spaces
su(1) (2n)
(1)
su(2n)
SU (1) (p+q) SU (2n)
SU−1 (p,q)
•❖α0 ♦♦♦ ❖❖❖❖❖ ♦ ♦ ♦ ❖❖❖ ❖❖❖ α2n−1 ♦♦♦ α2 α2n−2 α♦1♦ αn ♦ •
su(p) ⊕ su(q)
SU (1) (p+q) SU (p)⊕SU (q)
SU1 (p,q) SU (p)⊕SU (q)
♦❖❖α❖0❖ ❖❖❖ ♦♦♦ ♦ ♦ ❖❖❖ ♦ ♦ ♦ ❖ α ♦ α b ♦α♦1 c α2 αn ; 2n−2 ❖ 3 Simple roots for t0 :
24 compact simple roots and ep−1 + ep when p ≥ 2 no other when p = 1 • so∗ (1) (2n) Vogan diagram: Dn , trivial automorphism, two imaginary simple roots (e1 − e2 ), (en−1 − en ) painted t0 = su(n) Simple roots for t0 : compact simple roots only (1)
• so−1 (2, 2n − 2) Vogan diagram: Dn , trivial automorphism, two imaginary simple roots (e1 − e2 ), −(e1 − e2 ) painted t0 = su(2n − 2) Simple roots for t0 : compact simple roots only (1)
• soσv (1, 2n − 1) Vogan diagram: Dn , nont-rivial automorphism, no imaginary simple root painted t0 = so(2) (2n − 2) Simple roots for t0 : −e2 , en−1 and all (ei − ei+1 ) for 2 ≤ i ≤ (n − 2) (1)
• soσv (2p + 1, 2q + 1), p + q = n − 1 Vogan diagram: Dn , non-trivial automorphism, pth simple root painted ep − ep+1 , ( su(3) ⊕ so(2n − 5), if p = 2 t0 = (2) (1) so (2p) ⊕ so (2q + 1), if p ≥ 3 Simple roots for t0 : −e2 , en−1 , ep and ei − ei+1 for 2 ≤ i ≤ p − 1 and p + 1 ≤ i ≤ n − 2 (1)
• soγ (1, 2n − 1) Vogan diagram: Dn , non-trivial automorphism, no imaginary simple root painted t0 = so(1) (2n − 1) Simple roots for t0 : (e1 − e2 ), −(e1 + e2 ), en−1 and all (ei − ei+1 ) for 2 ≤ i ≤ (n − 2) (1)
• soγ (2p + 1, 2q + 1), p + q = n − 1 Vogan diagram: Dn , non-trivial automorphism, pth simple root painted ep − ep+1 , (2) if p = 1 so (2n), (1) t0 = sp (2) ⊕ so(2q + 1), if p = 2 so(1) (2p + 1) ⊕ so(2q + 1), if p ≥ 3 Simple roots for t0 : For p = 1 ep , −(e1 + e2 ), en−1 , all (ei − ei+1 ) for p + 1 ≤ i ≤ (n − 2) For p 6= 1 (e1 − e2 ), −(e1 + e2 ), en−1 , ep and all (ei − ei+1 ) for 2 ≤ i ≤ (p − 1) and p + 1 ≤ i ≤ (n − 2)
25 (1)
• soσs (1, 2n − 1), n even Vogan diagram: Dn , non-trivial automorphism, no imaginary simple root painted t0 = su(2) (n) Simple roots for t0 : 1 1 n n 2 (e1 − e2 + en−1 − en ), 2 (−e1 − e2 + en−1 + en ), (e 2 − e 2 +1 ) and 1 all 2 (ei − ei+1 + en−i − en+1−i ) for 2 ≤ i ≤ (n − 2) (1)
• soσs (1, 2n − 1), n even Vogan diagram: Dn , non-trivial automorphism, unique imaginary simple root (e n2 − e n2 +1 ) painted t0 = so(1) (n) Simple roots for t0 : 1 1 2 (e1 − e2 + en−1 − en ), 2 (−e1 − e2 + en−1 + en ), 1 n n n n 2 (e 2 −1 + e 2 − e 2 +1 − e 2 +2 ) and 1 all 2 (ei − ei+1 + en−i − en+1−i ) for 2 ≤ i ≤ (n − 2) (1)
• soσs (1, 2n − 1), nodd Vogan diagram: Dn , non-trivial automorphism, no imaginary simple root painted t0 = so(1) (n) Simple roots for t0 : 1 1 2 (e1 − e2 + en−1 − en ), 2 (−e1 − e2 + en−1 + en ), and 1 all 2 (ei − ei+1 + en−i − en+1−i ) for 2 ≤ i ≤ (n − 2) D.
Notations used
• gl(n, C), (gl(n, R)): {all n × n complex (real matrices)} • sl(n, C), (sl(n, R)): {all n × n complex (real matrices) of trace zero} (1)
• sl1 (n, R): sl(n, R) ⊗ C[t, t−1 ] ⊕ Ric ⊕ Rid with u = 1 (1)
• sl−1 (n, R): sl(n, R) ⊗ C[t, t−1 ] ⊕ Ric ⊕ Rid with u = −1 • so(n) : {X ∈ gl(n, R) | X + X ∗ = 0} (1)
• so1 (n): so(n) ⊗ C[t, t−1 ] ⊕ Ric ⊕ Rid with u = 1 (1)
• so−1 (n): so(n) ⊗ C[t, t−1 ] ⊕ Ric ⊕ Rid with u = −1 • sp(n) : {X ∈ gl(n, H) | X + X ∗ = 0} • su(n) : {X ∈ gl(n, C) | X + X ∗ = 0 and T rX = 0} • su(1) (n) : {X ∈ su(n) ⊗ C[t, t−1 ] ⊕ Ric ⊕ Rid | X + X ∗ = 0 and T rX = 0} • su(2) (n) :
1 X
X
(2)
(2)
tj ⊗ su(n)p ⊕ Ric ⊕ Rid where su(n)0
(2)
and su(n)1
are the eigenspaces corresponding
p=0j mod 2=p (2)
to eigenvalues 1 and eπi respectively. That is a ∈ su(n)p if a ∈ su(n) and Ψτ (a) = eπip whereΨτ is the is the outer automorphism of su(n). ( ) Z1 Z2 Z1 , Z3 skew Hermitian of order p and q • u(p, q) : | t respectively, Z2 is arbitrary Z 2 Z3
26
• su(p, q) :
(1)
(
Z1 Z2 t Z 2 Z3
• su1 (p, q) :
(
Z1 skew Hermitian of order p | Z3 skew Hermitian of order q T rZ1 + T rZ3 = 0, Z2 is arbitrary
X1 X2 t X 2 X3
)
Xi = Zi ⊗ C[t, t−1 ] ) X1 skew Hermitian of order p | X3 skew Hermitian of order q T rX1 + T rX3 = 0 X2 is arbitrary
Xi = Zi ⊗ C[t, t−1 ] ) X1 X2 X1 skew Hermitian of order p (1) | • su−1 (p, q) : t X3 skew Hermitian of order q −X 2 X3 T rX1 + T rX3 = 0, X2 is arbitrary ( ) Z1 Z2 Z1 , Z2 n × n complex matrix ∗ • su (2n) : | −Z 2 Z1 T rZ1 + T rZ1 = 0 (
• sp(n, C) :
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
(
Z1 Z2 Z 3 −Z1t
Z n × n complex matrix | i Z2 , Z2 are symmetric
)
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