- Email: [email protected]
From hydrogen atom to generalized Dynkin diagrams
16 April 1998
Physics Letters B 425 Ž1998. 135–144
From hydrogen atom to generalized Dynkin diagrams Claudia Daboul a , Jamil Daboul a
b,1
Mathematisches Seminar, UniÕersitat ¨ Hamburg, Bundesstr. 55, D-20146 Hamburg, Germany b Physics Department, Ben Gurion UniÕersity of the NegeÕ, 84105 Beer SheÕa, Israel Received 23 September 1997 Editor: R. Gatto
Abstract We identify the ‘‘dynamical algebras’’ H D of the D-dimensional hydrogen atom as positiÕe subalgebras of twisted and Ž2.q untwisted affine Kac-Moody algebras: For odd D G 5 we obtain H 2 lq1 , D lq1 . But for even D G 6, H 2 l is a parabolic subalgebra of BlŽ1.. H 4 is a parabolic subalgebra of C2Ž1., H 3 , D 2Ž2.q, AŽ1.q , while H 2 is isomorphic to the Borel 1 subalgebra of AŽ1. . Along the way we prove a theorem on the untwisting of positive subalgebras of twisted affine algebras, 1 and introduce generalized Dynkin diagrams which enable us to represent graphically automorphisms and parabolic subalgebras of finite and affine algebras. q 1998 Elsevier Science B.V.
1. Introduction The dynamical symmetry of hydrogen atom has been associated with the Lie algebras, soŽ4., soŽ3,1. and eŽ3., which correspond to negative, positive and zero energies E, respectively w1–3x. Three years ago w4x we noted that this dynamical algebra, which is generated by the angular momentum L and the Runge Lenz A vectors, is not a closed 6-dimensional algebra, but actually an infinite-dimensional algebra w5–8x, which we called the H-Algebra H . We then identified H as the positiÕe subalgebra of the twisted affine Kac-Moody $ Ž4.t , and associated the twisting automorphism t with parity. Later we discovered that Ža. the algebra algebra so $ Žsee Eq. Ž36. below., soŽ4.t is not genuinely twisted, since it is isomorphic to the untwisted affine algebra AŽ1. 1 Ž . and b there are other ‘physical’ systems which can be treated similarly, such as a charged particle moving in the combined field of electric and magnetic monopoles w9x. The affine Kac Moody algebras are encountered in physics usually in connection with infinite-dimensional systems, such as current algebras w8x. The H-algebra is especially interesting, since it provides one of the simplest manifestations of affine Kac Moody algebras in a finite-dimensional physical system. In this letter we shall explain the problematics of identifying the dynamical algebra H D of the D-dimensional hydrogen atom, which is defined in Ž1. below. Along the way, we shall also review and $ illustrate the relevant concepts, especially in Section 3. As expected, the algebras H D can be identified as soŽ D q 1.tq,
1
E-mail: [email protected].
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 0 7 8 - 1
C. Daboul, J. Daboulr Physics Letters B 425 (1998) 135–144
136
$
which are the positiÕe subalgebras of the twisted affine algebras soŽ D q 1.t . However, it turns out that these algebras are fundamentally different for even and odd D: For odd D s 2 l y 1, i.e. for Dl ' soŽ2 l ., the ‘‘parity’’ involution t Ž13. is an outer automorphism and therefore it leads to genuine twisting for D G 5 Ž l G 3., namely H 2 ly1 , D lŽ2.q. However, Bl ' soŽ2 l q 1. has no outer automorphisms!. This means Žsee Section 5 below. that Bˆlt is isomorphic to the untwisted BlŽ1., i.e. f Ž Bˆlt . s BlŽ1., where f is called ‘‘untwist isomorphism’’ w5,6x. But since Bˆltq is a subalgebra of Bˆlt , its image must also be a subalgebra of BlŽ1., which we shall identify as a parabolic subalgebra. As a byproduct, we prove in general that by untwisting positive subalgebras one always gets parabolic subalgebras Žsee theorem 3 below.. We also introduce generalized Dynkin diagrams, which provide a graphical representation of all the automorphisms of finite simple algebras and also representations of parabolic subalgebras of finite and affine algebras.
2. The D-dimensional hydrogen atom We define the Kepler problem in D dimensions by the Hamiltonian H :s
1 2m
p2 y
a ,
r
r :s < r < s Ž x 12 q PPP qx D2 .
where
1r2
,
Ž 1.
where r and p are the D-dimensional position and linear momentum vectors in the center of mass, m is the reduced mass and a :s e 2 is a coupling constant. Its bound-states energies are w10x: En s y
E0 n q Ž D y 3 . r2
2
,
where
E0 :s m a 2r2 " 2 ,
Ž 2.
E0rn2 where n is the principal quantum number. This formula reduces to the famous binding energies En s yE w x for D s 3 1,4 . As in the 3-dimensional case, the angular momentum operators L i j and the D-dimensional Runge-Lenz vector A are conserved, w H, L i j x s w H, A j x s 0, where L i j :s x i pj y x j pi A i :s
1 2
where
i , j s 1, . . . , D .
D
Ý Ž L i j pj q pj L i j . y m a js1
xi r
,
Ž 3. Ž 4.
These operators satisfy the following commutation relations: L i j , L k l s i" Ž d i k L jl y d jk L i l q d jl L i k y d i l L jk . ,
Ž 5.
L i j , A k s i" Ž d i k A j y d jk A i . ,
Ž 6.
ˆ ij , A i , A j s i"HL
where
for
i , j,k ,l s 1, . . . , D ,
Hˆ ' y2 mH .
Ž 7.
Because of the commutation relations Ž7., L i j and A i do not form a closed algebra, since Hˆ is an operator and not a constant. To overcome this difficulty, the Hamiltonian operator H is usually replaced by its energy eigenvalues, E s yh < E <, where h s 1,y 1,0 for bound states E - 0, for scattering states E ) 0 and for the threshold energy E s 0, respectively. With this trick the commutation relations Ž7. become A i , A j s h i" Ž 2 m < E < . L i j ,
where h s "1,0 ,
Ž 8.
and the resulting algebras are isomorphic to the three real algebras soŽ D q 1., soŽ D,1. and eŽ D . for h s 1,y 1,0.
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However, if we keep Hˆ as it is, an operator, we get an infinite-dimensional Lie algebra, which is generated ˆ as follows: Since w H, ˆ L i j x s w H, ˆ A j x s 0, we can generate the new operators HA ˆ iNis by L i j , A i , and Ž!. H, ˆ i j with A j , where HL ˆ i j appear on the r.h.s. of Ž7.. Similar commutations generate 1, . . . , D4 by commuting HL the following infinite dimensional Lie algebra: H D ' ² Hˆ n L i j , Hˆ nA i N n G 0; i , j s 1, . . . , D :R ,
for
DG2 .
Ž 9.
To identify H D we shall first review some basic definitions which are connected with twisted and untwisted affine Kac Moody algebras.
3. Mathematical background 3.1. Automorphisms and gradation Let g be a finite dimensional Lie algebra over the fields R or C , and s an automorphism of g . If s has a finite order m, then its eigenvalues have the form e mk :s expw ik 2prm x with k s 0,1, . . . , m y 1. The corresponding eigenspaces are gks :s x N x g g , s Ž x . s e mk x 4 ,
where
k s 0,1, . . . ,m y 1 .
Ž 10 .
Thus, s yields a Z rmZ -gradation of the algebra g : g s gs0 [ PPP [ gsmy1 ,
with
gsi , gsj ; gsiqj .
Ž 11 .
Since w gs0 , gjs x ; gsj , it follows that the elements of gjs provide a representation of the subalgebra g0s , which may be reducible. Hence, eÕery subspace gjs is a g0s -module, which we shall call eigenmodule. If g is semisimple, the following useful relation among these eigenmodules follows from Žw6x, Lemma 8.1.a.: gsmy i , gis ,
Ž 12 .
where gsi denotes the g0s -module contragredient to gsi . An example of an automorphism, which we shall need for identifying the H D algebras Ž9., is the ‘‘ parity inÕolution’’ Ž m s 2. w4x, which we define for the algebras soŽ D q 1., as follows:
t Ž L i j . s L i j , and t Ž L i , Dq1 . s yL i , Dq1 ,
i , j s 1, . . . , D .
Ž 13 .
The corresponding Z r2 Z -gradation is g ' so Ž D q 1 . s so Ž D . [ M ' g t0 [ g t1 ,
where
Mi ' L i , Dq1 .
Ž 14 .
Two automorphisms s and l of g are conjugate to each other, iff s s tlty1 , where t g Aut g . An automorphism s of g is called an inner automorphism of g , if it is a product of automorphisms of the form expŽad Ž x .., where x g g and ad x Ž y .:s w x, y x. Otherwise, s is called an outer automorphism. The inner automorphisms form a normal subgroup of Aut Ž g ., which is denoted by Int Ž g .. We define the remoteness r G 1 of a finite-order automorphism s to be the smallest integer such that s r is an inner automorphism. We shall use (m,r) automorphism to denote an automorphism of order m and remoteness r.
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3.2. Diagram automorphisms, marks, and Kac numbers If g is simple, the automorphism group Aut Ž g . can be described as follows: Let g be of rank l with Cartan matrix A ' Ž a i j .. The automorphism group Aut Ž A. of A is defined as the group of all permutations p g Sl such that ap Ž i.p Ž j. s a i j . This group is regarded as a subgroup of Aut Ž g . by requiring that p Ž Ea i . s Eap Ž i. and p Ž Eya i . s Eyap Ž i. for a standard set of generators of g corresponding to a basis a 1 , . . . , a l 4 of simple roots. Then it can be shown Žsee e.g. w13x, Ch.IX., that the automorphism group Aut Ž g . is the semidirect product of Aut Ž A. and Int Ž g .. The group Aut Ž A. can be visualized as the symmetry group of the Dynkin diagram of the algebra g . This is because any permutation which leaves the Cartan matrix invariant must also leave the Dynkin diagram invariant and vice versa. The Dynkin diagrams of A l , Dl , E6 have two-fold symmetry, whereas the Dynkin diagram of D4 has in addition a three-fold symmetry. For the aboÕe algebras, and only for these, one can define outer (2,2) and (3,3) automorphisms [6], respectiÕely, which are called diagram automorphisms m . For details see w6x Ž§7.9.. The following theorem gives a characterization of finite-order automorphisms up to conjugacy: Theorem 1: ŽProposition 8.1 in Kac w6x. Let g be a simple finite-dimensional Lie algebra, h its Cartan subalgebra and P s {a 1 , . . . , a N } its simple roots. Then eÕery automorphism s of g of order m is conjugate to an automorphism of the form 2p i s s m exp ad H s , with H s g h 0m s h l g 0m , Ž 15 . m where m is a diagram automorphism of g , such that ² a k ,H s : g Z, k s 1, . . . , N. Since Aut Ž g . is the semidirect product of Aut Ž A. and Int Ž g ., which implies in particular that Aut Ž A. l Int Ž g . s id4 , the automorphism s in Ž15. is inner iff m s id. Since m and exp 2mp i ad H s commute, the remoteness of s equals the order of m. It also follows that rank Ž g s0 . s rank Ž g . if s is an inner automorphism. Since the fixed-point algebra of a nontrivial diagram automorphism always has a lower rank than the original algebra, this yields the following useful criterion: rank Ž g s0 . s rank Ž g . ,
iff s is an inner automorphism . Ž 16 . For example, by applying this criterion to Ž14. we can immediately conclude that the parity involution t of soŽ D q 1. is an inner automorphism iff D is even. Two more interesting examples of gradations are given in appendix A; They are gradations of G 2 obtained by inner automorphisms. One of them yields an suŽ3. octet plus two triplet eigenmodules. For r s 1 we define u 0 to be the highest root of g . For r s 2,3 we define yu 0 as the lowest weight of the g0m-module g1m. u 0 can be expanded in terms of the basis of g0m as follows l
u0 s
Ý ak a k ,
where
ak G 0 .
Ž 17 .
ks1
The coefficients a k , called marks w7x, can be found as labels on the affine Dynkin diagrams in Tables Aff 1 and Aff 2 of w6x. They are needed for the following theorem, which is essentially Theorem 8.6. of Kac w6x: Theorem 2: Let s be an (m,r) automorphism of g ' X N , and let g 0m be the corresponding fixed point algebra of the diagram automorphism m of g , and l s rank (g 0m ). Then, it is always possible to find a basis a k of the root system of g 0m and nonnegatiÕe relatiÕely prime integers sk G 0, for k s 0,1, . . . , l, such that l
msr
Ý a k sk ,
where
a0 ' 1 ,
and
Ž 18 .
ks0
s Ž Eyu 0 . s e ms 0 Ey u 0 ,
s Ž Ea k . s e ms k Ea k , k s 1,2, . . . , l .
Ž 19 .
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The (l q 1)-tupel (s0 ,s1 , . . . , sl ) is unique up to permutation of its entries by automorphisms of the associated affine Dynkin diagram . On the other hand, giÕen any (l q 1)-tupel (s0 ,s1 , . . . , sl ) such that (18) holds, then relations (19) define an m-th order automorphism of g . We use s Ž s . s Ž s0 , s1 , . . . , sl . to denote the Ž l q 1.-tupel which is assigned to an automorphism s according to Ž19.. We refer to the sk as Kac numbers. We define a0 ' 1 ŽKac w6x makes an exception, and defines a0 s 2 for case 5 , cf.Section 8.3., so that all diagram automorphisms m will have the Kac numbers s Ž m . s Ž1,0, . . . , 0. for all r. It follows from Ž19. that m Ž Eyu 0 . s e r Ey u 0 , so that Ey u 0 g g 1m for r s 2,3. 3.3. Untwisted and twisted affine Kac-Moody algebras Since H D correspond to positiÕe subalgebras of affine Kac-Moody algebras, the central term K w5–8x will not appear. Hence, to simplify matters, we shall concentrate in this letter on loop algebras only. Let g be a finite-dimensional Lie algebra. The corresponding loop algebra is defined as the set of Laurent polynomials with coefficients in g : gˆ' L Ž g . s g w t ,ty1 x ' C w t ,ty1 x m g ' [j g Z t j m g .
Ž 20 .
For given g and s the subalgebra of L Ž g . , defined by gˆs ' L Ž g , s ,m . :s [j g Z t j m g sj ,
where
j ' j Ž mod m . ,
Ž 21 .
is called twisted loop algebra. By restricting the generation index j in Ž21. to nonnegative integers, we obtain an infinite-dimensional subalgebra, which we call the Žstandard. positiÕe subalgebra of L Ž g , s ,m. : q
gˆs q' L Ž g , s ,m . s [j G 0 t j m g sj
Ž 22 .
For example, the positive twisted subalgebra for the parity map t Ž13. is q
L Ž so Ž D q 1 . ,t ,2 . s [n G 0 Ž t 2 n m so Ž D . [ t 2 nq1 m M . .
Ž 23 .
3.4. Untwisting Every twisted algebra L Ž X N , s ,m. of an Ž m,r . automorphism s can be mapped isomorphically onto the corresponding standard affine algebra X NŽ r . :s L Ž X N , m ,r .. It is easy to check that the following map, called the ‘untwist map’ f w5,6x, is an isomorphism:
f Ž t n mqk m Ea . s t Ž n mqky a Ž Hs .. r r m m Ea g L Ž X N , m ,r . , fŽt
n mqk
mH . st
Ž n mqk . r r m
m H g L Ž X N , m ,r . ,
Ea g gsk ,
Hgh .
Ž 24 . Ž 25 .
3.5. Parabolic subalgebras of finite and affine algebras Definition: Let g be a finite, affine Lie algebra or eÕen a general Kac-Moody algebra. A subalgebra p of g will be called a parabolic subalgebra , if it contains a Borel subalgebra b :s t (h [n q ) for some t g Aut (g ), where n q is the sum of the root spaces of positiÕe roots. This generalizes the definition in the finite-dimensional case w12,14x. More formally, p s b [ ny0 ' Ž h [ n q . [ ny0 ' Ž h [ [a g D qg a . [ [a g D q0 g y a , s g 0 [ hX 0 [ Ž [a g D q _ D q0 g a . ,
where
q Dq 0 9D
Ž 26 .
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Fig. 1. Root and generalized Dynkin diagrams of B2 and its parabolic subalgebras.
where g 0 in Ž26. is the subalgebra, which is generated by Ea N a g D0 4 , h 0 ; h is the Cartan subalgebra of g 0 and hX 0 is a vector space complement of h 0 in h : g 0 :s h 0 [ nq0 [ ny0s h 0 [ [a g D 0 g a ,
where
h 0 :s ² w Ea , Ey a x N a g D0 :C .
Ž 27 .
Let I be an indexing set for the simple roots of g . The subalgebra g 0 in Ž27., or equivalently the set T Ž p . s k g I N a k g Dq 0 4 ; I,
Ž 28 .
is uniquely determined by p and determines p up to conjugation. We shall refer to the set T Ž p . as the type of p. A parabolic subalgebra is called a standard parabolic subalgebra if it contains the standard Borel subalgebra b s h [ n q of g . We denote the standard parabolic subalgebra of type T of g by pŽ g , T .. We shall also use pj Ž g . Žsee Fig. 1 and Eqs. Ž35. and Ž36.. to denote the special parabolic algebras, whose type T Ž p . is equal to I _ j4 ŽLater on, we shall be able to define these pj Ž g . more graphically, as the parabolic subalgebras whose generalized Dynkin diagrams haÕe just one box at the j-th node .. Root diagrams of Lie algebras of rank 2 provide an excellent tool for illustrating the above definitions: Consider the algebra B2 ' soŽ5., which has two simple roots, a long root a 1 and a short root a 2 , which we choose as in Fig. 1. The smallest parabolic subalgebra of any algebra g is its Borel subalgebra b s h [ n q. Here, n q is spanned by the four raising operators Ea , where a g Dq Ž B2 ., as illustrated in Fig. 1Žb.. The root diagrams of two nontrivial parabolic subalgebras of B2 , p 2 :s b [ g y a 1 and p 1:s b [ g y a 2 are indicated in Fig. 1Žc. and 1Žd., respectively. We now apply theorem 2 to prove the following result: Theorem 3: Let s be an (m,r) automorphism of g ' X N . Then the positiÕe subalgebra gˆs q' L Ž g , s ,m.q of the ’twisted’ affine algebra gˆs is a parabolic subalgebra p of X NŽ r . ' L Ž X N , m ,r .. More specifically, if f is the standard untwist map, and s Ž s . s Ž s0 , s1 , . . . , s l . are the corresponding Kac numbers of s , then f maps gˆs q onto the standard parabolic subalgebra of gˆm of type T s N Ž s Ž s .. s i g 0, . . . , l 4 N si s 04 : $
f Ž X N s q . s p Ž X NŽ r . , N Ž s Ž s . . . s : p Ž X NŽ r . ; s Ž s . . .
Ž 29 .
Proof: Ža. According to theorem 2, we may assume that a k Ž H s . s sk . Noting that the basis raising operators Ea k g gss k l g 0m ,k s 1, . . . , l , and Ey u 0 g gss 0 l g 1m , we get for n G 0 Ž!.:
f Ž t n mqs k m Ea k . s t n r m Ea k ,
for
a k g P Ž g 0m . ,
f Ž t n mqs 0 m Eyu 0 . s t n rq1 m Ey u 0 , f Ž t n m m Eya k . s t n r m Ey a k ,
for
f Ž t n mqmys k m Eya k . s t Ž nq1. r m Ey a k ,
Ž a. Ž b.
sk s 0 , for
Ž c. sk / 0 ,
Ž d.
Ž 30 .
where we used Eya k g gs0 for sk s 0 in Ž30a., but Ey a k g gsmys k for sk G 1 in Ž30b.. The generators involving Eu 0 can be treated similarly. Hence, f Ž gˆs q . is a parabolic subalgebra of gˆm, since it contains as images all the
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simple raising generators of gˆm, namely t 1 m Eyu 0 j Ea k ' t 0 m Ea k N a k g P Ž g m0 .4 . Since n G 0, we see from Ž30d. that 1 m Eya k g u f Ž g s q . if sk / 0. In contrast, Eq. Ž30.c tells us that all lowering operators with sk s 0 are included in f Ž gˆs q . . I
4. Generalized Dynkin diagrams Ž1. We define the generalized Dynkin diagram of a parabolic subalgebra p of a finite or affine algebra g by replacing the circle at the j-th node of the Dynkin diagram of g by a square Žbox. iff k g u T Ž p .. In particular, we use the special notation pj (g ) for the standard parabolic subalgebra of g whose Dynkin diagram has only one box at the j-th node. These are all the maximal standard parabolic subalgebras. For example, Dynkin diagrams of the parabolic subalgebras of B2 are given in Fig. 1, while those of the affine D4Ž3. and G Ž1. 2 are given in Ž31. and ŽA.1.. Ž2. Let g be a finite Lie algebra of type X N . With eÕery automorphism s of g of remoteness r and Kac numbers s(s ) we associate a ‘filled Dynkin diagram’, which is the Dynkin diagram X NŽ r . of the corresponding twisted affine algebra, in which the circle at the j-th node is replaced by a square containing the number s j iff s j / 0. Such a filled diagram enables us: Ža. To calculate the order m of s by using Eq. Ž18.. Žb. To read off immediately g s0 , the fixed-point algebra of s : It is the direct sum of a semisimple Lie algebra, whose diagram is defined by the remaining circles, plus a center, whose dimension is the number of squares minus 1. This result can be deduced from w6x ŽProp. 8.6 Žb... Žc. Once we determine orderŽ s . and g s0 , we can often easily guess the whole gradation, from the knowledge of the dimensions of possible modules, as illustrated in the examples below. Žd. To describe the parabolic subalgebras L Ž g , s ,m.q of L Ž g , s ,m. ( L Ž g .m , in accordance with theorem 3. 4.1. Applications and examples In physics one is often interested in knowing the possible subalgebras of a given algebra w2x. The above diagrams allow us to find all those subalgebras that can be obtained from a given finite algebra g , as fixed point subalgebras g s0 of some finite order automorphism s . Unfortunately, not all subalgebras can be found in this way: For example, G 2 is a subalgebra of B3 w12x, which cannot be obtained in this way. We now illustrate how fixed point subalgebras can be found: In Ž31. we give four filled Dynkin diagrams of D4Ž3., which describe 4 different automorphisms of remoteness 3 of D4 ' oŽ8,C ..
Ž31.
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From the above diagrams we can deduce the following information: Ži. We can calculate the order m of each automorphism, by using Eq. Ž18., where we wrote the marks Ž a 0 ,a1 ,a 2 . s Ž1,2,1. of D4Ž3. w6x above the Dynkin diagram Ža.: We get order Ž m . s order Ž s 2 . s r s 3 ,
order Ž s 1 . s 2 r s 6 ,
order Ž s 3 . s Ž 1 q 2 q 1 . r s 12 .
Ž 32 .
Žii. Each of the first 3 filled Dynkin diagrams has two circles. Hence, in each case the corresponding automorphism yields a fixed-point subalgebra g s0 s Ž D4 . 0s of rank 2. In particular, by removing the middle box in Žb. we end up with two disconnected circles, which constitute the Dynkin diagram of D 2 , A1 [ A1. Hence, Ž D4 . 0s 1 s D 2 . Žiii. Since dim D4 s 28, we conclude that the eigenmodules g 1 and g 2 of G 2 and of suŽ3. in Ža. and Žc. must have dimensions 7 and 10, respectively. To determine the eigenmodules of D 2 s suŽ2. [ suŽ2., we note that the irreducible representations of D 2 must be tensor products of the familiar suŽ2. multiplets, namely w j1 , j2 x:s < j1 ,m1 : m < j2 ,m 2 :4 , with dimensions Ž2 j1 q 1.Ž2 j2 q 1.. Hence, g 1 ' 4 ' w 12 , 12 x, g 2 ' 3 ' w0,1x, g 3 ' 8 ' w 12 , 32 x . But one might ask, why do you identify g 1 with 4and not with 3? The answer is that w g 1 , g 1 x ; g 2 and two multiplets of j s 1r2 can couple to triplets j s 1 or singlets j s 0, but not the other way around. Živ. The Kac numbers inside the boxes determine the type of the corresponding parabolic subalgebras of the twisted affine algebra D4Ž3.. Žv. Finally, in Žd. we have a filled Dynkin diagram , which yields a Borel subalgebra. Such diagrams can have only boxes. Since each box must be filled at least with 1, which is the smallest Kac number for a box, we conclude that m s 12 is the smallest order that an automorphism must have in order to yield a Borel subalgebra of D4Ž3.. Similar conclusions can be drawn for any other algebra g .
5. Identification of the H D algebras One can easily check that the following map c of the H D algebras Ž9.
c Hˆ n L i j s t 2 n m L i j , and
ž
/
c Ž Hˆ nA i . s t 2 nq1 m L i , Dq1 ,
nG0 ,
Ž 33 .
onto the positive subalgebras L Ž soŽ D q 1.,t ,2.q, defined in Ž23., is an isomorphism. The parity map t is identical to the diagram automorphism m , when applied to the algebras D l s soŽ2 l . for l G 3. Therefore, for odd dimensions D s 2 l y 1 the H D correspond to genuinely twisted algebras: q
q
H 2 ly1 , L Ž so Ž 2 l . ,t ,2 . ' L Ž D l , m ,2 . ' D lŽ2.q .
Ž 34 .
In contrast, the parity map t is an inner automorphism when applied to Bl s soŽ2 l q 1.. The corresponding Kac numbers are s Žt . s Ž s0 , s1 , PPP , sl . s Ž0,0, PPP ,0,1.. Hence, for even dimensions, D s 2 l, the H D algebras are isomorphic to parabolic subalgebras of the untwisted BlŽ1. :
f Ž c Ž H 2 l . . s f Ž L Ž so Ž 2 l q 1 . ,t ,2 .
q
. s p Ž BlŽ1. ; Ž 0, PPP ,0,1 . . ' pl Ž BlŽ1. . .
Ž 35 .
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The identification of the algebras H D can be summarized by the following generalized Dynkin diagrams:
Ž36. It is amazing that the simple model of the D-dimensional hydrogen atom requires such a sophisticated mathematical formalism to describe its symmetry properly. By identifying the H-algebras we obtained as byproducts: Ža. simple and intuitive manifestations of many aspects of affine Kac Moody algebras, Žb. theorem 3 and Žc. generalized Dynkin diagrams .
Acknowledgements It is a pleasure to thank Professor Peter Slodowy for helpful comments and suggestions.
Appendix A. Gradations and automorphisms of G 2 An example of physical interest is provided by the following gradations of the exceptional Lie algebra G 2 w2,12x, which has dimension 14:
ŽA.1. The gradation ŽA.1a. splits G 2 into an suŽ3. subalgebra Žoctet. plus two triplets, which transform like quarks and antiquarks w11x. This gradations can be seen clearly in Fig. 2, where we choose the long and short simple roots a 1 and a 2 of G 2 w2,12x, such that Ea 0 s Ey u 0 coincides with the raising isospin operator Iq of suŽ3.,
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Fig. 2. The root diagram of G 2 , illustrating the two gradations of ŽA.1.: The simple roots of G 2 and of its two fixed-point subalgebras, suŽ3. and D 2 , are given by Ž a 1 , a 2 ., Ž a 0 , a 1 . and Ž a 0 , a 2 ., respectively. The Cartan subalgebra and the long roots of G 2 constitute an suŽ3. subalgebra, whose roots are labeled by the elements of meson octet, while the six short roots of G 2 are labeled by the usual triplets of light quarks and antiquarks. The gradation ŽA.1b. yields another subalgebra of G 2 , namely D 2 s A1[A1 l Žpq,p 0 ,py .[Ž s,h 8 , s ..
where u 0 s 2 a 1 q 3 a 2 is the highest weight of G 2 . Hence, the familiar I3 and the hypercharge Y of the suŽ3. subalgebra w11x can be expressed in terms of the generators of G 2 , as follows: I3 s
1
w Iq , Iy x s
1
Ey u 0 , Eu 0 ,
and
Ys
1
Ha 2 s
1
E ,E . Ž A.2 . 2 2 3 3 a 2 ya 2 The gradation ŽA.1b. yields D 2 s suŽ2. [ suŽ2. l Žpq,p 0 ,py . [ Ž s,h 8 , s . as a subalgebra g 0l of G 2 plus an 8-dimensional module g 1l s 8 s w1r2,3r2x of D 2 . The two <1r2," 1r2: m <3r2, M : Ž‘Y-spin’ quartets. are labeled by Ž Kq,u,d, K 0 . and Ž K 0 ,d,u, Ky . . Note that in both quartets the charge Q of the labeling ‘particles’ increases by D Q s 1r3, as we move vertically upwards, as expected from the general relation Q s I3 q Yr2 w11x. The two automorphisms s and l in ŽA.1. can be represented by filled Dynkin diagrams of the untwisted affine algebra G Ž1. 2 . Hence, they are both inner automorphisms of order m s 3 and m s 2, respectively. They can be defined explicitly, as follows:
s ' exp w i2p ad Y x , so that
s Ž Ea 1 . s Ea 1 , and
s Ž Ea 2 . s e i4p r3 Ea 2 ,
Ž A.3 .
l ' exp w i2p ad I3 x , so that
l Ž Ea 2 . s Ea 2 , and
l Ž Ea 1 . s yEa 1 .
Ž A.4 .
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x
L.I. Schiff, Quantum Mechanics, 3rd ed., McGraw-Hill, 1968. B.G. Wybourne, Classical Groups for Physicists, Wiley, 1974. A. Sudbery, Quantum Mechanics and the Particles of Nature, Cambridge University Press, 1986. J. Daboul, P. Slodowy, C. Daboul, Phys. Lett. B 317 Ž1993. 321; C. Daboul, J. Daboul, P. Slodowy, in: A. Arima, T. Eguchi, N. Nakanishi ŽEds.., Group Theoretical Methods in Physics, World Scientific, Singapore, 1995, hep-thr9408080. P. Goddard, D. Olive, Inter. J. Mod. Phys. A 1 Ž1986. 303. V.G. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge Univ. Press, Cambridge, 1990. R.V. Moody, A. Pianzola, Lie Algebras With Triangular Decompositions, Wiley, New York, 1995. J. Mickelsson, Current Algebras and Groups, Plenum, New York, 1989. G.W. Gibbons, P.J. Ruback, Phys. Lett. B 188 Ž1987. 226. M.M. Nieto, Am. J. Phys. 47 Ž1979. 1067. W.E. Burcham, M. Jobes, Nuclear and Particle Physics, Longman, Singapore, 1995. J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. N. Jacobson, Lie Algebras, Wiley Interscience, New York-London, 1962. N. Bourbaki, Elements de Mathematique: Groupes et al.gebres de Lie, Hermann, Paris, 1975, Chapters 7, 8.
Physics Letters B 425 Ž1998. 135–144
From hydrogen atom to generalized Dynkin diagrams Claudia Daboul a , Jamil Daboul a
b,1
Mathematisches Seminar, UniÕersitat ¨ Hamburg, Bundesstr. 55, D-20146 Hamburg, Germany b Physics Department, Ben Gurion UniÕersity of the NegeÕ, 84105 Beer SheÕa, Israel Received 23 September 1997 Editor: R. Gatto
Abstract We identify the ‘‘dynamical algebras’’ H D of the D-dimensional hydrogen atom as positiÕe subalgebras of twisted and Ž2.q untwisted affine Kac-Moody algebras: For odd D G 5 we obtain H 2 lq1 , D lq1 . But for even D G 6, H 2 l is a parabolic subalgebra of BlŽ1.. H 4 is a parabolic subalgebra of C2Ž1., H 3 , D 2Ž2.q, AŽ1.q , while H 2 is isomorphic to the Borel 1 subalgebra of AŽ1. . Along the way we prove a theorem on the untwisting of positive subalgebras of twisted affine algebras, 1 and introduce generalized Dynkin diagrams which enable us to represent graphically automorphisms and parabolic subalgebras of finite and affine algebras. q 1998 Elsevier Science B.V.
1. Introduction The dynamical symmetry of hydrogen atom has been associated with the Lie algebras, soŽ4., soŽ3,1. and eŽ3., which correspond to negative, positive and zero energies E, respectively w1–3x. Three years ago w4x we noted that this dynamical algebra, which is generated by the angular momentum L and the Runge Lenz A vectors, is not a closed 6-dimensional algebra, but actually an infinite-dimensional algebra w5–8x, which we called the H-Algebra H . We then identified H as the positiÕe subalgebra of the twisted affine Kac-Moody $ Ž4.t , and associated the twisting automorphism t with parity. Later we discovered that Ža. the algebra algebra so $ Žsee Eq. Ž36. below., soŽ4.t is not genuinely twisted, since it is isomorphic to the untwisted affine algebra AŽ1. 1 Ž . and b there are other ‘physical’ systems which can be treated similarly, such as a charged particle moving in the combined field of electric and magnetic monopoles w9x. The affine Kac Moody algebras are encountered in physics usually in connection with infinite-dimensional systems, such as current algebras w8x. The H-algebra is especially interesting, since it provides one of the simplest manifestations of affine Kac Moody algebras in a finite-dimensional physical system. In this letter we shall explain the problematics of identifying the dynamical algebra H D of the D-dimensional hydrogen atom, which is defined in Ž1. below. Along the way, we shall also review and $ illustrate the relevant concepts, especially in Section 3. As expected, the algebras H D can be identified as soŽ D q 1.tq,
1
E-mail: [email protected].
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 0 7 8 - 1
C. Daboul, J. Daboulr Physics Letters B 425 (1998) 135–144
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$
which are the positiÕe subalgebras of the twisted affine algebras soŽ D q 1.t . However, it turns out that these algebras are fundamentally different for even and odd D: For odd D s 2 l y 1, i.e. for Dl ' soŽ2 l ., the ‘‘parity’’ involution t Ž13. is an outer automorphism and therefore it leads to genuine twisting for D G 5 Ž l G 3., namely H 2 ly1 , D lŽ2.q. However, Bl ' soŽ2 l q 1. has no outer automorphisms!. This means Žsee Section 5 below. that Bˆlt is isomorphic to the untwisted BlŽ1., i.e. f Ž Bˆlt . s BlŽ1., where f is called ‘‘untwist isomorphism’’ w5,6x. But since Bˆltq is a subalgebra of Bˆlt , its image must also be a subalgebra of BlŽ1., which we shall identify as a parabolic subalgebra. As a byproduct, we prove in general that by untwisting positive subalgebras one always gets parabolic subalgebras Žsee theorem 3 below.. We also introduce generalized Dynkin diagrams, which provide a graphical representation of all the automorphisms of finite simple algebras and also representations of parabolic subalgebras of finite and affine algebras.
2. The D-dimensional hydrogen atom We define the Kepler problem in D dimensions by the Hamiltonian H :s
1 2m
p2 y
a ,
r
r :s < r < s Ž x 12 q PPP qx D2 .
where
1r2
,
Ž 1.
where r and p are the D-dimensional position and linear momentum vectors in the center of mass, m is the reduced mass and a :s e 2 is a coupling constant. Its bound-states energies are w10x: En s y
E0 n q Ž D y 3 . r2
2
,
where
E0 :s m a 2r2 " 2 ,
Ž 2.
E0rn2 where n is the principal quantum number. This formula reduces to the famous binding energies En s yE w x for D s 3 1,4 . As in the 3-dimensional case, the angular momentum operators L i j and the D-dimensional Runge-Lenz vector A are conserved, w H, L i j x s w H, A j x s 0, where L i j :s x i pj y x j pi A i :s
1 2
where
i , j s 1, . . . , D .
D
Ý Ž L i j pj q pj L i j . y m a js1
xi r
,
Ž 3. Ž 4.
These operators satisfy the following commutation relations: L i j , L k l s i" Ž d i k L jl y d jk L i l q d jl L i k y d i l L jk . ,
Ž 5.
L i j , A k s i" Ž d i k A j y d jk A i . ,
Ž 6.
ˆ ij , A i , A j s i"HL
where
for
i , j,k ,l s 1, . . . , D ,
Hˆ ' y2 mH .
Ž 7.
Because of the commutation relations Ž7., L i j and A i do not form a closed algebra, since Hˆ is an operator and not a constant. To overcome this difficulty, the Hamiltonian operator H is usually replaced by its energy eigenvalues, E s yh < E <, where h s 1,y 1,0 for bound states E - 0, for scattering states E ) 0 and for the threshold energy E s 0, respectively. With this trick the commutation relations Ž7. become A i , A j s h i" Ž 2 m < E < . L i j ,
where h s "1,0 ,
Ž 8.
and the resulting algebras are isomorphic to the three real algebras soŽ D q 1., soŽ D,1. and eŽ D . for h s 1,y 1,0.
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However, if we keep Hˆ as it is, an operator, we get an infinite-dimensional Lie algebra, which is generated ˆ as follows: Since w H, ˆ L i j x s w H, ˆ A j x s 0, we can generate the new operators HA ˆ iNis by L i j , A i , and Ž!. H, ˆ i j with A j , where HL ˆ i j appear on the r.h.s. of Ž7.. Similar commutations generate 1, . . . , D4 by commuting HL the following infinite dimensional Lie algebra: H D ' ² Hˆ n L i j , Hˆ nA i N n G 0; i , j s 1, . . . , D :R ,
for
DG2 .
Ž 9.
To identify H D we shall first review some basic definitions which are connected with twisted and untwisted affine Kac Moody algebras.
3. Mathematical background 3.1. Automorphisms and gradation Let g be a finite dimensional Lie algebra over the fields R or C , and s an automorphism of g . If s has a finite order m, then its eigenvalues have the form e mk :s expw ik 2prm x with k s 0,1, . . . , m y 1. The corresponding eigenspaces are gks :s x N x g g , s Ž x . s e mk x 4 ,
where
k s 0,1, . . . ,m y 1 .
Ž 10 .
Thus, s yields a Z rmZ -gradation of the algebra g : g s gs0 [ PPP [ gsmy1 ,
with
gsi , gsj ; gsiqj .
Ž 11 .
Since w gs0 , gjs x ; gsj , it follows that the elements of gjs provide a representation of the subalgebra g0s , which may be reducible. Hence, eÕery subspace gjs is a g0s -module, which we shall call eigenmodule. If g is semisimple, the following useful relation among these eigenmodules follows from Žw6x, Lemma 8.1.a.: gsmy i , gis ,
Ž 12 .
where gsi denotes the g0s -module contragredient to gsi . An example of an automorphism, which we shall need for identifying the H D algebras Ž9., is the ‘‘ parity inÕolution’’ Ž m s 2. w4x, which we define for the algebras soŽ D q 1., as follows:
t Ž L i j . s L i j , and t Ž L i , Dq1 . s yL i , Dq1 ,
i , j s 1, . . . , D .
Ž 13 .
The corresponding Z r2 Z -gradation is g ' so Ž D q 1 . s so Ž D . [ M ' g t0 [ g t1 ,
where
Mi ' L i , Dq1 .
Ž 14 .
Two automorphisms s and l of g are conjugate to each other, iff s s tlty1 , where t g Aut g . An automorphism s of g is called an inner automorphism of g , if it is a product of automorphisms of the form expŽad Ž x .., where x g g and ad x Ž y .:s w x, y x. Otherwise, s is called an outer automorphism. The inner automorphisms form a normal subgroup of Aut Ž g ., which is denoted by Int Ž g .. We define the remoteness r G 1 of a finite-order automorphism s to be the smallest integer such that s r is an inner automorphism. We shall use (m,r) automorphism to denote an automorphism of order m and remoteness r.
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3.2. Diagram automorphisms, marks, and Kac numbers If g is simple, the automorphism group Aut Ž g . can be described as follows: Let g be of rank l with Cartan matrix A ' Ž a i j .. The automorphism group Aut Ž A. of A is defined as the group of all permutations p g Sl such that ap Ž i.p Ž j. s a i j . This group is regarded as a subgroup of Aut Ž g . by requiring that p Ž Ea i . s Eap Ž i. and p Ž Eya i . s Eyap Ž i. for a standard set of generators of g corresponding to a basis a 1 , . . . , a l 4 of simple roots. Then it can be shown Žsee e.g. w13x, Ch.IX., that the automorphism group Aut Ž g . is the semidirect product of Aut Ž A. and Int Ž g .. The group Aut Ž A. can be visualized as the symmetry group of the Dynkin diagram of the algebra g . This is because any permutation which leaves the Cartan matrix invariant must also leave the Dynkin diagram invariant and vice versa. The Dynkin diagrams of A l , Dl , E6 have two-fold symmetry, whereas the Dynkin diagram of D4 has in addition a three-fold symmetry. For the aboÕe algebras, and only for these, one can define outer (2,2) and (3,3) automorphisms [6], respectiÕely, which are called diagram automorphisms m . For details see w6x Ž§7.9.. The following theorem gives a characterization of finite-order automorphisms up to conjugacy: Theorem 1: ŽProposition 8.1 in Kac w6x. Let g be a simple finite-dimensional Lie algebra, h its Cartan subalgebra and P s {a 1 , . . . , a N } its simple roots. Then eÕery automorphism s of g of order m is conjugate to an automorphism of the form 2p i s s m exp ad H s , with H s g h 0m s h l g 0m , Ž 15 . m where m is a diagram automorphism of g , such that ² a k ,H s : g Z, k s 1, . . . , N. Since Aut Ž g . is the semidirect product of Aut Ž A. and Int Ž g ., which implies in particular that Aut Ž A. l Int Ž g . s id4 , the automorphism s in Ž15. is inner iff m s id. Since m and exp 2mp i ad H s commute, the remoteness of s equals the order of m. It also follows that rank Ž g s0 . s rank Ž g . if s is an inner automorphism. Since the fixed-point algebra of a nontrivial diagram automorphism always has a lower rank than the original algebra, this yields the following useful criterion: rank Ž g s0 . s rank Ž g . ,
iff s is an inner automorphism . Ž 16 . For example, by applying this criterion to Ž14. we can immediately conclude that the parity involution t of soŽ D q 1. is an inner automorphism iff D is even. Two more interesting examples of gradations are given in appendix A; They are gradations of G 2 obtained by inner automorphisms. One of them yields an suŽ3. octet plus two triplet eigenmodules. For r s 1 we define u 0 to be the highest root of g . For r s 2,3 we define yu 0 as the lowest weight of the g0m-module g1m. u 0 can be expanded in terms of the basis of g0m as follows l
u0 s
Ý ak a k ,
where
ak G 0 .
Ž 17 .
ks1
The coefficients a k , called marks w7x, can be found as labels on the affine Dynkin diagrams in Tables Aff 1 and Aff 2 of w6x. They are needed for the following theorem, which is essentially Theorem 8.6. of Kac w6x: Theorem 2: Let s be an (m,r) automorphism of g ' X N , and let g 0m be the corresponding fixed point algebra of the diagram automorphism m of g , and l s rank (g 0m ). Then, it is always possible to find a basis a k of the root system of g 0m and nonnegatiÕe relatiÕely prime integers sk G 0, for k s 0,1, . . . , l, such that l
msr
Ý a k sk ,
where
a0 ' 1 ,
and
Ž 18 .
ks0
s Ž Eyu 0 . s e ms 0 Ey u 0 ,
s Ž Ea k . s e ms k Ea k , k s 1,2, . . . , l .
Ž 19 .
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The (l q 1)-tupel (s0 ,s1 , . . . , sl ) is unique up to permutation of its entries by automorphisms of the associated affine Dynkin diagram . On the other hand, giÕen any (l q 1)-tupel (s0 ,s1 , . . . , sl ) such that (18) holds, then relations (19) define an m-th order automorphism of g . We use s Ž s . s Ž s0 , s1 , . . . , sl . to denote the Ž l q 1.-tupel which is assigned to an automorphism s according to Ž19.. We refer to the sk as Kac numbers. We define a0 ' 1 ŽKac w6x makes an exception, and defines a0 s 2 for case 5 , cf.Section 8.3., so that all diagram automorphisms m will have the Kac numbers s Ž m . s Ž1,0, . . . , 0. for all r. It follows from Ž19. that m Ž Eyu 0 . s e r Ey u 0 , so that Ey u 0 g g 1m for r s 2,3. 3.3. Untwisted and twisted affine Kac-Moody algebras Since H D correspond to positiÕe subalgebras of affine Kac-Moody algebras, the central term K w5–8x will not appear. Hence, to simplify matters, we shall concentrate in this letter on loop algebras only. Let g be a finite-dimensional Lie algebra. The corresponding loop algebra is defined as the set of Laurent polynomials with coefficients in g : gˆ' L Ž g . s g w t ,ty1 x ' C w t ,ty1 x m g ' [j g Z t j m g .
Ž 20 .
For given g and s the subalgebra of L Ž g . , defined by gˆs ' L Ž g , s ,m . :s [j g Z t j m g sj ,
where
j ' j Ž mod m . ,
Ž 21 .
is called twisted loop algebra. By restricting the generation index j in Ž21. to nonnegative integers, we obtain an infinite-dimensional subalgebra, which we call the Žstandard. positiÕe subalgebra of L Ž g , s ,m. : q
gˆs q' L Ž g , s ,m . s [j G 0 t j m g sj
Ž 22 .
For example, the positive twisted subalgebra for the parity map t Ž13. is q
L Ž so Ž D q 1 . ,t ,2 . s [n G 0 Ž t 2 n m so Ž D . [ t 2 nq1 m M . .
Ž 23 .
3.4. Untwisting Every twisted algebra L Ž X N , s ,m. of an Ž m,r . automorphism s can be mapped isomorphically onto the corresponding standard affine algebra X NŽ r . :s L Ž X N , m ,r .. It is easy to check that the following map, called the ‘untwist map’ f w5,6x, is an isomorphism:
f Ž t n mqk m Ea . s t Ž n mqky a Ž Hs .. r r m m Ea g L Ž X N , m ,r . , fŽt
n mqk
mH . st
Ž n mqk . r r m
m H g L Ž X N , m ,r . ,
Ea g gsk ,
Hgh .
Ž 24 . Ž 25 .
3.5. Parabolic subalgebras of finite and affine algebras Definition: Let g be a finite, affine Lie algebra or eÕen a general Kac-Moody algebra. A subalgebra p of g will be called a parabolic subalgebra , if it contains a Borel subalgebra b :s t (h [n q ) for some t g Aut (g ), where n q is the sum of the root spaces of positiÕe roots. This generalizes the definition in the finite-dimensional case w12,14x. More formally, p s b [ ny0 ' Ž h [ n q . [ ny0 ' Ž h [ [a g D qg a . [ [a g D q0 g y a , s g 0 [ hX 0 [ Ž [a g D q _ D q0 g a . ,
where
q Dq 0 9D
Ž 26 .
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Fig. 1. Root and generalized Dynkin diagrams of B2 and its parabolic subalgebras.
where g 0 in Ž26. is the subalgebra, which is generated by Ea N a g D0 4 , h 0 ; h is the Cartan subalgebra of g 0 and hX 0 is a vector space complement of h 0 in h : g 0 :s h 0 [ nq0 [ ny0s h 0 [ [a g D 0 g a ,
where
h 0 :s ² w Ea , Ey a x N a g D0 :C .
Ž 27 .
Let I be an indexing set for the simple roots of g . The subalgebra g 0 in Ž27., or equivalently the set T Ž p . s k g I N a k g Dq 0 4 ; I,
Ž 28 .
is uniquely determined by p and determines p up to conjugation. We shall refer to the set T Ž p . as the type of p. A parabolic subalgebra is called a standard parabolic subalgebra if it contains the standard Borel subalgebra b s h [ n q of g . We denote the standard parabolic subalgebra of type T of g by pŽ g , T .. We shall also use pj Ž g . Žsee Fig. 1 and Eqs. Ž35. and Ž36.. to denote the special parabolic algebras, whose type T Ž p . is equal to I _ j4 ŽLater on, we shall be able to define these pj Ž g . more graphically, as the parabolic subalgebras whose generalized Dynkin diagrams haÕe just one box at the j-th node .. Root diagrams of Lie algebras of rank 2 provide an excellent tool for illustrating the above definitions: Consider the algebra B2 ' soŽ5., which has two simple roots, a long root a 1 and a short root a 2 , which we choose as in Fig. 1. The smallest parabolic subalgebra of any algebra g is its Borel subalgebra b s h [ n q. Here, n q is spanned by the four raising operators Ea , where a g Dq Ž B2 ., as illustrated in Fig. 1Žb.. The root diagrams of two nontrivial parabolic subalgebras of B2 , p 2 :s b [ g y a 1 and p 1:s b [ g y a 2 are indicated in Fig. 1Žc. and 1Žd., respectively. We now apply theorem 2 to prove the following result: Theorem 3: Let s be an (m,r) automorphism of g ' X N . Then the positiÕe subalgebra gˆs q' L Ž g , s ,m.q of the ’twisted’ affine algebra gˆs is a parabolic subalgebra p of X NŽ r . ' L Ž X N , m ,r .. More specifically, if f is the standard untwist map, and s Ž s . s Ž s0 , s1 , . . . , s l . are the corresponding Kac numbers of s , then f maps gˆs q onto the standard parabolic subalgebra of gˆm of type T s N Ž s Ž s .. s i g 0, . . . , l 4 N si s 04 : $
f Ž X N s q . s p Ž X NŽ r . , N Ž s Ž s . . . s : p Ž X NŽ r . ; s Ž s . . .
Ž 29 .
Proof: Ža. According to theorem 2, we may assume that a k Ž H s . s sk . Noting that the basis raising operators Ea k g gss k l g 0m ,k s 1, . . . , l , and Ey u 0 g gss 0 l g 1m , we get for n G 0 Ž!.:
f Ž t n mqs k m Ea k . s t n r m Ea k ,
for
a k g P Ž g 0m . ,
f Ž t n mqs 0 m Eyu 0 . s t n rq1 m Ey u 0 , f Ž t n m m Eya k . s t n r m Ey a k ,
for
f Ž t n mqmys k m Eya k . s t Ž nq1. r m Ey a k ,
Ž a. Ž b.
sk s 0 , for
Ž c. sk / 0 ,
Ž d.
Ž 30 .
where we used Eya k g gs0 for sk s 0 in Ž30a., but Ey a k g gsmys k for sk G 1 in Ž30b.. The generators involving Eu 0 can be treated similarly. Hence, f Ž gˆs q . is a parabolic subalgebra of gˆm, since it contains as images all the
C. Daboul, J. Daboulr Physics Letters B 425 (1998) 135–144
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simple raising generators of gˆm, namely t 1 m Eyu 0 j Ea k ' t 0 m Ea k N a k g P Ž g m0 .4 . Since n G 0, we see from Ž30d. that 1 m Eya k g u f Ž g s q . if sk / 0. In contrast, Eq. Ž30.c tells us that all lowering operators with sk s 0 are included in f Ž gˆs q . . I
4. Generalized Dynkin diagrams Ž1. We define the generalized Dynkin diagram of a parabolic subalgebra p of a finite or affine algebra g by replacing the circle at the j-th node of the Dynkin diagram of g by a square Žbox. iff k g u T Ž p .. In particular, we use the special notation pj (g ) for the standard parabolic subalgebra of g whose Dynkin diagram has only one box at the j-th node. These are all the maximal standard parabolic subalgebras. For example, Dynkin diagrams of the parabolic subalgebras of B2 are given in Fig. 1, while those of the affine D4Ž3. and G Ž1. 2 are given in Ž31. and ŽA.1.. Ž2. Let g be a finite Lie algebra of type X N . With eÕery automorphism s of g of remoteness r and Kac numbers s(s ) we associate a ‘filled Dynkin diagram’, which is the Dynkin diagram X NŽ r . of the corresponding twisted affine algebra, in which the circle at the j-th node is replaced by a square containing the number s j iff s j / 0. Such a filled diagram enables us: Ža. To calculate the order m of s by using Eq. Ž18.. Žb. To read off immediately g s0 , the fixed-point algebra of s : It is the direct sum of a semisimple Lie algebra, whose diagram is defined by the remaining circles, plus a center, whose dimension is the number of squares minus 1. This result can be deduced from w6x ŽProp. 8.6 Žb... Žc. Once we determine orderŽ s . and g s0 , we can often easily guess the whole gradation, from the knowledge of the dimensions of possible modules, as illustrated in the examples below. Žd. To describe the parabolic subalgebras L Ž g , s ,m.q of L Ž g , s ,m. ( L Ž g .m , in accordance with theorem 3. 4.1. Applications and examples In physics one is often interested in knowing the possible subalgebras of a given algebra w2x. The above diagrams allow us to find all those subalgebras that can be obtained from a given finite algebra g , as fixed point subalgebras g s0 of some finite order automorphism s . Unfortunately, not all subalgebras can be found in this way: For example, G 2 is a subalgebra of B3 w12x, which cannot be obtained in this way. We now illustrate how fixed point subalgebras can be found: In Ž31. we give four filled Dynkin diagrams of D4Ž3., which describe 4 different automorphisms of remoteness 3 of D4 ' oŽ8,C ..
Ž31.
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From the above diagrams we can deduce the following information: Ži. We can calculate the order m of each automorphism, by using Eq. Ž18., where we wrote the marks Ž a 0 ,a1 ,a 2 . s Ž1,2,1. of D4Ž3. w6x above the Dynkin diagram Ža.: We get order Ž m . s order Ž s 2 . s r s 3 ,
order Ž s 1 . s 2 r s 6 ,
order Ž s 3 . s Ž 1 q 2 q 1 . r s 12 .
Ž 32 .
Žii. Each of the first 3 filled Dynkin diagrams has two circles. Hence, in each case the corresponding automorphism yields a fixed-point subalgebra g s0 s Ž D4 . 0s of rank 2. In particular, by removing the middle box in Žb. we end up with two disconnected circles, which constitute the Dynkin diagram of D 2 , A1 [ A1. Hence, Ž D4 . 0s 1 s D 2 . Žiii. Since dim D4 s 28, we conclude that the eigenmodules g 1 and g 2 of G 2 and of suŽ3. in Ža. and Žc. must have dimensions 7 and 10, respectively. To determine the eigenmodules of D 2 s suŽ2. [ suŽ2., we note that the irreducible representations of D 2 must be tensor products of the familiar suŽ2. multiplets, namely w j1 , j2 x:s < j1 ,m1 : m < j2 ,m 2 :4 , with dimensions Ž2 j1 q 1.Ž2 j2 q 1.. Hence, g 1 ' 4 ' w 12 , 12 x, g 2 ' 3 ' w0,1x, g 3 ' 8 ' w 12 , 32 x . But one might ask, why do you identify g 1 with 4and not with 3? The answer is that w g 1 , g 1 x ; g 2 and two multiplets of j s 1r2 can couple to triplets j s 1 or singlets j s 0, but not the other way around. Živ. The Kac numbers inside the boxes determine the type of the corresponding parabolic subalgebras of the twisted affine algebra D4Ž3.. Žv. Finally, in Žd. we have a filled Dynkin diagram , which yields a Borel subalgebra. Such diagrams can have only boxes. Since each box must be filled at least with 1, which is the smallest Kac number for a box, we conclude that m s 12 is the smallest order that an automorphism must have in order to yield a Borel subalgebra of D4Ž3.. Similar conclusions can be drawn for any other algebra g .
5. Identification of the H D algebras One can easily check that the following map c of the H D algebras Ž9.
c Hˆ n L i j s t 2 n m L i j , and
ž
/
c Ž Hˆ nA i . s t 2 nq1 m L i , Dq1 ,
nG0 ,
Ž 33 .
onto the positive subalgebras L Ž soŽ D q 1.,t ,2.q, defined in Ž23., is an isomorphism. The parity map t is identical to the diagram automorphism m , when applied to the algebras D l s soŽ2 l . for l G 3. Therefore, for odd dimensions D s 2 l y 1 the H D correspond to genuinely twisted algebras: q
q
H 2 ly1 , L Ž so Ž 2 l . ,t ,2 . ' L Ž D l , m ,2 . ' D lŽ2.q .
Ž 34 .
In contrast, the parity map t is an inner automorphism when applied to Bl s soŽ2 l q 1.. The corresponding Kac numbers are s Žt . s Ž s0 , s1 , PPP , sl . s Ž0,0, PPP ,0,1.. Hence, for even dimensions, D s 2 l, the H D algebras are isomorphic to parabolic subalgebras of the untwisted BlŽ1. :
f Ž c Ž H 2 l . . s f Ž L Ž so Ž 2 l q 1 . ,t ,2 .
q
. s p Ž BlŽ1. ; Ž 0, PPP ,0,1 . . ' pl Ž BlŽ1. . .
Ž 35 .
C. Daboul, J. Daboulr Physics Letters B 425 (1998) 135–144
143
The identification of the algebras H D can be summarized by the following generalized Dynkin diagrams:
Ž36. It is amazing that the simple model of the D-dimensional hydrogen atom requires such a sophisticated mathematical formalism to describe its symmetry properly. By identifying the H-algebras we obtained as byproducts: Ža. simple and intuitive manifestations of many aspects of affine Kac Moody algebras, Žb. theorem 3 and Žc. generalized Dynkin diagrams .
Acknowledgements It is a pleasure to thank Professor Peter Slodowy for helpful comments and suggestions.
Appendix A. Gradations and automorphisms of G 2 An example of physical interest is provided by the following gradations of the exceptional Lie algebra G 2 w2,12x, which has dimension 14:
ŽA.1. The gradation ŽA.1a. splits G 2 into an suŽ3. subalgebra Žoctet. plus two triplets, which transform like quarks and antiquarks w11x. This gradations can be seen clearly in Fig. 2, where we choose the long and short simple roots a 1 and a 2 of G 2 w2,12x, such that Ea 0 s Ey u 0 coincides with the raising isospin operator Iq of suŽ3.,
C. Daboul, J. Daboulr Physics Letters B 425 (1998) 135–144
144
Fig. 2. The root diagram of G 2 , illustrating the two gradations of ŽA.1.: The simple roots of G 2 and of its two fixed-point subalgebras, suŽ3. and D 2 , are given by Ž a 1 , a 2 ., Ž a 0 , a 1 . and Ž a 0 , a 2 ., respectively. The Cartan subalgebra and the long roots of G 2 constitute an suŽ3. subalgebra, whose roots are labeled by the elements of meson octet, while the six short roots of G 2 are labeled by the usual triplets of light quarks and antiquarks. The gradation ŽA.1b. yields another subalgebra of G 2 , namely D 2 s A1[A1 l Žpq,p 0 ,py .[Ž s,h 8 , s ..
where u 0 s 2 a 1 q 3 a 2 is the highest weight of G 2 . Hence, the familiar I3 and the hypercharge Y of the suŽ3. subalgebra w11x can be expressed in terms of the generators of G 2 , as follows: I3 s
1
w Iq , Iy x s
1
Ey u 0 , Eu 0 ,
and
Ys
1
Ha 2 s
1
E ,E . Ž A.2 . 2 2 3 3 a 2 ya 2 The gradation ŽA.1b. yields D 2 s suŽ2. [ suŽ2. l Žpq,p 0 ,py . [ Ž s,h 8 , s . as a subalgebra g 0l of G 2 plus an 8-dimensional module g 1l s 8 s w1r2,3r2x of D 2 . The two <1r2," 1r2: m <3r2, M : Ž‘Y-spin’ quartets. are labeled by Ž Kq,u,d, K 0 . and Ž K 0 ,d,u, Ky . . Note that in both quartets the charge Q of the labeling ‘particles’ increases by D Q s 1r3, as we move vertically upwards, as expected from the general relation Q s I3 q Yr2 w11x. The two automorphisms s and l in ŽA.1. can be represented by filled Dynkin diagrams of the untwisted affine algebra G Ž1. 2 . Hence, they are both inner automorphisms of order m s 3 and m s 2, respectively. They can be defined explicitly, as follows:
s ' exp w i2p ad Y x , so that
s Ž Ea 1 . s Ea 1 , and
s Ž Ea 2 . s e i4p r3 Ea 2 ,
Ž A.3 .
l ' exp w i2p ad I3 x , so that
l Ž Ea 2 . s Ea 2 , and
l Ž Ea 1 . s yEa 1 .
Ž A.4 .
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x
L.I. Schiff, Quantum Mechanics, 3rd ed., McGraw-Hill, 1968. B.G. Wybourne, Classical Groups for Physicists, Wiley, 1974. A. Sudbery, Quantum Mechanics and the Particles of Nature, Cambridge University Press, 1986. J. Daboul, P. Slodowy, C. Daboul, Phys. Lett. B 317 Ž1993. 321; C. Daboul, J. Daboul, P. Slodowy, in: A. Arima, T. Eguchi, N. Nakanishi ŽEds.., Group Theoretical Methods in Physics, World Scientific, Singapore, 1995, hep-thr9408080. P. Goddard, D. Olive, Inter. J. Mod. Phys. A 1 Ž1986. 303. V.G. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge Univ. Press, Cambridge, 1990. R.V. Moody, A. Pianzola, Lie Algebras With Triangular Decompositions, Wiley, New York, 1995. J. Mickelsson, Current Algebras and Groups, Plenum, New York, 1989. G.W. Gibbons, P.J. Ruback, Phys. Lett. B 188 Ž1987. 226. M.M. Nieto, Am. J. Phys. 47 Ž1979. 1067. W.E. Burcham, M. Jobes, Nuclear and Particle Physics, Longman, Singapore, 1995. J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. N. Jacobson, Lie Algebras, Wiley Interscience, New York-London, 1962. N. Bourbaki, Elements de Mathematique: Groupes et al.gebres de Lie, Hermann, Paris, 1975, Chapters 7, 8.