q-Deformation of the Real Forms

LetG be a real noncompact semisimple Lie algebra,(be the Cartan involution inG, andG =K⊕Qbe theCartan decompositionofG, so that(X=X,X∈K,(X= –X,X∈ Q;K is the maximal compact subalgebra ofG. LetA0be the maximal subspace of Q, which is an abelian subalgebra ofG;r₀ =dimA0is thereal rank(orsplit rank) of G, 1≤r₀≤ ℓ= rankG.

LetB⁰R be the root system of the pair (G,A0), also called (A0 -)restricted root system:

B⁰_R ={+∈A₀^∗|+= 0,̸ G₊⁰= 0̸ }, (1.104) G₊⁰ ={X∈G|[Y,X] =+(Y)X,∀Y∈A0}. (1.105) The elements of B⁰R = B⁰⁺R ∪ B^0–R is called (A0 -)restricted roots; if + ∈ B⁰R, G₊⁰ is called (A0-)restricted root space, dim_RG₊⁰≥1. Now we can introduce the subalgebras corresponding to the positive (B⁰⁺_R ) and negative (B^0–_R ) restricted roots:

̃N0= ⊕

+∈B⁰⁺_R G₊⁰=Ñ ¹₀⊕ ̃N²₀, (1.106) N0= ⊕

+∈B^0–_R G₊⁰=N₀¹⊕N₀²=(Ñ 0, (1.107) whereÑ ¹₀,Ñ ²₀is the direct sum ofG₊⁰with dim_RG₊⁰ = 1, dim_RG₊⁰ > 1, respectively, and analogously forN₀^a=(Ñ ^a₀. Then we have the Bruhat decompositions which we shall use for ourq-deformations:

G =Ñ 0⊕A0⊕M0⊕N0=ѹ₀⊕ ̃N²₀⊕A0⊕M0⊕N₀¹⊕N₀², (1.108) whereM₀is the centralizer ofA₀inK; that is,M₀= {X ∈K|[X,Y] = 0,∀Y ∈A₀}. In generalM0is a compact reductive Lie algebra, and we shall writeM0=M₀^s⊕Z₀^m, whereM₀^s = [M0,M0] is the semisimple part ofM0, andZ₀^m is the centre ofM0. Note thatP̃⁰₀ ≡ ̃N0⊕A0⊕M0,P₀⁰ ≡ A0⊕M0⊕N0are subalgebras ofG, the so-called minimal parabolic subalgebras ofG. IdentifyingP̃⁰₀,P₀⁰ is the first step of our procedure.

Further, letH₀^mbe the Cartan subalgebra ofM0; that is,H₀^m=H₀^ms⊕Z₀^m, where H₀^msis the Cartan subalgebra ofM₀^s. ThenH0≡H₀^m⊕A0is a Cartan subalgebra of G, the most noncompact one; dim_RH0 = dim_RH0^ms+ dim_RZ0^m+r₀. We chooseH0to be also the Cartan subalgebra ofU_q(G). LetH^ℂbe the complexification ofH0(ℓ= rankG^ℂ= dim_CH^ℂ); then it is a Cartan subalgebra of the complexificationG^ℂofG.

The second step in our procedure is to choose consistently the basis of the rest ofG andG^ℂ, and thus ofU_q(G). For this we use the classification of the roots from Bwith respect toH0. The setB⁰_r ≡ {! ∈ B|!|_H0^m = 0}is called the set ofreal roots,

B⁰i ≡ {!∈B|!|_A0 = 0}- the set ofcompact roots,B⁰c ≡B\(B⁰r ∪B⁰i) - the set ofcomplex roots(cf. Bourbaki [109]). ThusB = B⁰_r ∪B⁰_i ∪B⁰_c. Further, let! ∈ B⁺; letL_!^cbe the complex linear span ofH_!,X_!,X_–!; and letL_! = L_!^c∩G. Then dim_RL_! = 3 if the

! ∈ B⁰_r ∪B⁰_i [10]. If! ∈ B⁰_r thenX_! ∈ P^ℂandL_!is noncompact. Since the Cartan subalgebra isH0, thenX_! ∈ K^ℂandL_!is compact if! ∈ B⁰_i. The algebrasL_!are given by:

L_!=r.l.s.{H_!,X_!,X_–!}, !∈B⁰⁺_r , (1.109a) L_!=r.l.s.{iH_!,X_!–X_–!,i(X_!+X_–!)}, !∈B⁰⁺_i , (1.109b) where r.l.s. stands for real linear span.

Note that there is a one-to-one correspondence between the real roots ! ∈ B⁰_r and the restricted roots+ ∈ B⁰_R with dim_RG₊⁰ = 1 and naturally this correspond-ence is realized by the restriction: + = !|_A0. Thus the elements in (8a) X_!^± for! ∈ B⁰r we take also as elements ofU_q(G). Thus, following (1.19),(1.23) these generators obey:

[X_!,X_–!] = [H_!]_q

!, [H_!,X_±!] =±!(H_!)X_±!, for !∈B⁰⁺_r , (1.110) and the Hopf algebra structure is given exactly as for!∈B(cf. (1.22) and the text after that).

Remark 1.3.Formulae (1.109a) and (1.110) determine completely aq-deformation of any maximally split real form (or normal real form), when all roots are real,M0 = 0, andH0=A0. In this case the Bruhat decomposition is just

G =Ñ 0⊕A0⊕N0, (1.111)

that is, this is the restriction to ℝof the standard triangular decomposition G^ℂ = G₊^ℂ⊕H^ℂ⊕G_–^ℂ, and henceU_q(G) is just the restriction ofU_q(G^ℂ) toℝwithq ∈ ℝ. Thus we also inherit the property that U_q(Ñ 0 ⊕A0), U_q(N0 ⊕A0) are Hopf sub-algebra of U_q(G), sinceU_q(G_±^ℂ ⊕H^ℂ) is Hopf subalgebra of U_q(G^ℂ). Note that 3 here is an antilinear involution and co-algebra homomorphism such that3(Y) = Y

∀Y ∈ U_q(G^ℂ). For the classical complex Lie algebras these forms are U_q(sl(n,ℝ)), U_q(so(n,n)),U_q(so(n+1,n)),U_q(sp(n,ℝ)), which are dual to the matrix quantum groups SL_q(n,ℝ),SO_q(n,n),SO_q(n,n+ 1),Sp_q(n,ℝ), introduced in [272] from another point of

view than ours. ◊

Further note that the set of the compact rootsB⁰_i may be identified with the root system ofM0^sℂ. Thus the elements in (1.109b) give the Hopf algebraU_q(M0^s) by the formulae:

1.5 q-Deformations of Noncompact Lie Algebras 23

[C⁺_!,C^–_!] = sinh(H̃_!h_!/2)

sin(h_!/2) , (1.112)

[H̃_!, C^±_!] =±C^∓_!, q_!=q^(!,!)/2=e^–ih!, C⁺_!= (i/√2)(X_!+X_–!), C^–_!= (1/√2)(X_!–X_–!)

̃H_!= –iH_!,

$(C^±_!) =C^±_!⊗e^H̃!h!/4+e^–H̃!h!/4⊗C_!^±, !∈B⁺_i ∩B_S.

SinceM0=M₀^s⊕Z₀^mis a compact reductive Lie algebra we have to choose how to do the deformation in such cases. Our choice is to preserve the reductive structure, that is, writing in more detailM0=⊕jM₀^sj⊕ ⊕kZ₀^mk, whereM₀^sjis simple andZ₀^mkis one-dimensional; then we shall have the Hopf algebraU_q(M0) =⊗jU_q(M₀^sj)⊗ ⊗kU_q(Z₀^mk), where we also have to specify that ifZ0^mkis spanned byK, thenU_q(Z0^mk) is spanned byK,q^±K/4.

Remark 1.4.Formulae (1.109b) and (1.112) (with h_! ∈ ℝ) determine completely a Drinfeld-Jimboq-deformation of any compact semisimple Lie algebra [251] (when all roots ofBare compact). Here one may take3as an antilinear involution and coalgebra homomorphism such that3(X_!^±) = –X_!^∓,∀!∈B,3(H) = –H,∀H∈H. Note that in this case theq-deformation inherited fromU_q(G^ℂ) is often used in the physics literature

without the basis change (1.112). ◊

Returning to the general situation, so far we have chosen consistently the generat-ors ofѹ₀⊕A₀⊕M₀⊕N₀¹ (cf. (1.106)) as linear combinations of the generators of H0⊕ ⊕_!∈B⁰

r∪B⁰_iG_!. Now it remains to choose consistently the generators ofÑ ²₀,N₀²as linear combinations of the generators of the rest ofG^ℂ, that is, of⊕_!∈B⁰⁺_c G_!,⊕_!∈B^0–_c G_!, respectively. If!∈B⁰_c,+=!|_A0, then dim_RG₊⁰> 1. LetB₊={!∈B|!|_A0 =+}. If!∈B⁰_c, then we haveX_!=Y_!+Z_!, whereY_!∈Q^ℂ,Z_!∈K^ℂ. Now we can see thatG₊⁰= r.l.s.

{ ̃X_! = Y_!+iZ_!,∀!∈ B₊}. The actual choice of basis inG₊⁰is a matter of convenience (cf. the examples below) and is related to the choice of3andq, and to the general property thatU_q(P̃⁰0),U_q(P0⁰) are Hopf subalgebras ofU_q(G).

1.5.2.1 q-Deformations with Other Cartan Subalgebras

For the purposes of q-deformations we need also to consider Cartan subalgebras H which are not conjugate to H0. Cartan subalgebras which represent different conjugacy classes may be chosen asH =H_k⊕A, whereH_kis compact,A is non-compact, dimA < dimA0ifH is nonconjugate toH0. The Cartan subalgebras with maximal dimension ofA are conjugate toH0; also those with minimal dimension of A are conjugate to each other.

All notions introduced until now are easily generalized forH =Hk⊕A noncon-jugate toH0. We note the differences, and notationwise we drop all 0 subscripts and

superscripts. One difference is that the algebraM is the centralizer ofA inG (mod A) and thus is in general a noncompact reductive Lie algebra which has the compact H_kas Cartan subalgebra (besides, in general, other noncompact Cartan subalgebras);

in particular, ifG has a compact Cartan subalgebra then for the choiceA = 0 one hasM =G. For the purposes of theq-deformation we shall use this compact Cartan subalgebra, that is, we setH^m=Hk. Further, the classification of the roots ofBwith respect toH goes as before. The difference is that if!∈BithenL_!may also be non-compact. Thus for!∈Bithe root!is calledsingular root,!∈Bs, ifL_!is noncompact, and!is called as before compact root,!∈ Bk, ifL_!is compact. ThusBi = Bs∪Bk. Formulae (1.109b) hold forB_k, while for!∈B_swe have:

L_!=r.l.s.{iH_!,i(X_!–X_–!),X_!+X_–!}, !∈B⁺_s, [S⁺_!,S^–_!] = sinh(H̃_!h_!/2)

sin(h_!/2) ,

[H̃_!,S^±_!] =∓S^∓_!, q_!=q^(!,!)/2=e^–ih!, (1.113) S⁺_!= (1/√2)(X_!+X_–!), S^–_!= (i/√2)(X_!–X_–!),

̃H_!= –iH_!,

$(S^±_!) =S^±_!⊗e^H̃!h!/4+e^–H̃!h!/4⊗S^±_!, !∈B⁺_s∩BS.

Further as before the set of the compact roots inBmay be identified with the root system ofM^sℂ. Thus formulae (1.109b),(1.112), and (1.113) give also the deformation U_q(M^s). Since the centre ofM is compact (it is in the Cartan subalgebraH^mwhich is compact), then the deformationU_q(Z^m) is given as after (1.112). Thus the Hopf algebra U_q(M) is given. Otherwise, the considerations for the factorsN,Ñ go as forN0,Ñ 0. Thus our scheme provides a different q-deformation for each conjugacy class of Cartan subalgebras.

1.5.2.2 q-Deformations for Arbitrary Parabolic Subalgebras and Reductive Lie Algebras

Until now our data are the nonconjugate Cartan subalgebrasH = Hk⊕A and the related with Bruhat decompositions (1.106). In these decompositions special role for theq-deformations is played by the minimal parabolic subalgebrasP0P̃0. A stand-ard parabolic subalgebra is any subalgebraP^󸀠ofG such thatP₀⊆P^󸀠. The number of standard parabolic subalgebras, includingP0andG, is 2^r,r= dimA. They are all of the formP^󸀠 = M^󸀠⊕A^󸀠⊕N^󸀠,M^󸀠 ⊇ M,A^󸀠 ⊆ A,N^󸀠 ⊆ N;M^󸀠is the cent-ralizer ofA^󸀠inG (modA^󸀠);N^󸀠(resp.Ñ^󸀠 = (N^󸀠) is comprised from the negative (resp. positive) root spaces of the restricted root systemB^󸀠_Rof (G,A^󸀠). One also has the corresponding Bruhat decompositions:

G =Ñ ^󸀠⊕A^󸀠⊕M^󸀠⊕N^󸀠. (1.114)

1.5q-Deformations of Noncompact Lie Algebras 25

Note thatM^󸀠is a noncompact reductive Lie algebra which has a noncompact Cartan subalgebraH^󸀠m≅Hk⊕Hn, whereHnis noncompact andA ≅Hn⊕A^󸀠. This Cartan subalgebraH^󸀠mofM^󸀠will be chosen for the purposes of theq-deformation.

Thus we need to extend our scheme to noncompact reductive Lie algebras. Let

̂G =G ⊕Z = ̂K ⊕ ̂Qbe a real reductive Lie algebra, whereG is the semisimple part ofĜ;Z is the centre ofĜ;K̂ ,Q̂ are the +1, –1 eigenspaces of the Cartan involution

̂(; ̂A^󸀠 = A^󸀠⊕Zpis the analogue ofA^󸀠;Zp = Z ∩ ̂Q. The root system of the pair (Ĝ;Â^󸀠) coincides withB^󸀠_R, and the subalgebrasÑ^󸀠andN^󸀠are inherited fromG. The decomposition (1.114) then is:

̂G = ̃N^󸀠⊕ ̂A^󸀠⊕ ̂M^󸀠⊕N^󸀠, (1.115) where M̂ ^󸀠 = M^󸀠s ⊕ ̂Z^󸀠m, Ẑ^󸀠m = Z^󸀠m ⊕ Z ∩ ̂K. As in the compact reductive case we choose a deformation which preserves the splitting ofĜ, that is,U_q(Ĝ) = U_q(G)⊗U_q(Z), and even further into simple Lie subalgebras and one-dimensional central subalgebras.

Remark 1.5.A general property of the deformations U_q(G) obtained by the above procedure is thatU_q(M0),U_q(P̃0),U_q(P0) are Hopf subalgebras ofU_q(G). ◊

1.5.3 Example so(p,r)

LetG = so(p,r), withp ≥ r ≥ 2 orp > r = 1 with generators:M_AB = –M_BA,A,B = 1,. . .,p+r,'AB= diag (–⋅ ⋅ ⋅– +⋅ ⋅ ⋅+), (ptimes minus,rtimes plus) which obey:

[M_AB,M_CD] =i('BCM_AD–'ACM_BD–'BDM_AC+'ADM_BC)

Besides the “physical” generatorM_ABwe shall also use the “mathematical” generator Y_AB= –iM_AB. One has:K ≅so(p)⊕so(r) ifr≥2 andK ≅so(p) ifr= 1. The generators ofK areM_ABwith 1≤A<B≤pandp+ 1≤A<B≤p+r. The split rank is equal to r;M0≅so(p–r), ifp–r≥2, andM0= 0 ifp–r= 0, 1, dimÑ = dimN =r(p– 1).

Furthermore the dimensions of the roots in the root systemBofso(p+r,ℂ) and inBR

depending on the parity ofp+rare given by:

roots p+reven p+rodd

|B^±r| r(r– 1) r²

|B^±_i| (p–r)(p–r– 2)/4 (p–r– 1)²/4

|B^±_c| r(p–r) r(p–r– 1)

|B^±_R| r² r(r+ 1)

Note that the algebraso(2n+ 1, 1) has only one conjugacy class of Cartan subalgebras.

Thus in these cases ourq-deformation is unique. The algebraso(2n, 1) has two con-jugacy classes of Cartan subalgebras, and in these cases there are twoq-deformations which we ilustrate below forn= 1.

1.5.4 Example so(2,1)

Using notation from aboveA,B = 1, 2, 0, (– – +);Y₁₂is the generator ofK, and we may chooseY₂₀for the generator ofA;M0= 0. Thus we can choose eitherY₂₀orY₁₂ as a generator ofH andH^ℂ. LetB^± = {±!}be the root system ofG^ℂ = sl(2,ℂ). If H^ℂis generated byY₂₀(andH = H0 = A), then!is a real root, and this deform-ation, denoted byU_q⁰(so(2, 1)), is given by formulae (1.110) and (1.22) overℝ. IfH^ℂ is generated byY₁₂, then!is a singular compact root, and the deformation, denoted, U¹_q(so(2, 1)), is given by formula (1.113) withh_!∈ ℝ.