Vladimir K. Dobrev Invariant Differential Operators

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Vladimir K. Dobrev

Invariant Differential Operators

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in Mathematical Physics

Edited by

Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Dmitry Gitman, S ao Paulo, Brazil ̃

Alexander Lazarian, Madison, Wisconsin, USA Boris Smirnov, Moscow, Russia

Volume 39

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Vladimir K. Dobrev

Invariant

Differential Operators

Volume 2: Quantum Groups

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Mathematics Subject Classification 2010

16TXX, 17BXX, 17B37, 17B35, 17B62, 17B10, 17B81, 20G42, 33D80, 58B32, 81R50, 82D77, 16S30, 22E47, 22E15, 22E60, 81R05, 47A15, 47A46, 53A55, 70H33

Author

Vladimir K. Dobrev

Bulgarian Academy of Sciences Institute for Nuclear Research and Nuclear Energy

Tsarigradsko Chaussee 72 1784 Sofia

Bulgaria

dobrev@inrne.bas.bg

http://theo.inrne.bas.bg/_dobrev/

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Preface

This is volume 2 of our trilogy on invariant differential operators. In volume 1 we presented our canonical procedure for the construction of invariant differential operators and showed its application to the objects of the initial domain – noncompact semisimple Lie algebras and groups.

In volume 2 we show the application of our procedure to quantum groups. Sim- ilarly to the setting of volume 1 the main actors are in duality. Just as Lie algebras and Lie groups are in duality here the dual objects are the main two manifestations of quantum groups: quantum algebras and matrix quantum groups. Actually, quantum algebras typically are deformations of the universal enveloping algebras of semisimple Lie algebras. Analogously, matrix quantum groups typically are deformations of spaces of functions over semisimple Lie groups.

Chapter 1 presents first the necessary general background material on quantum algebras and some generalizations as Yangians. Then we present the necessary material onq-deformations of noncompact semisimple Lie algebras. Chapter 2 is devoted to highest weight modules over quantum algebras, mostly being considered Verma modules and singular vectors. The latter is given for the quantum algebras related to all semisimple Lie algebras. Chapter 3 considers positive energy representations of noncompact quantum algebras on the example ofq-deformed anti de Sitter algebra and q-deformed conformal algebra. In Chapter 4 we consider in detail the matrix quantum groups. Many important examples are considered together with the quantum algebras which are constructed using the duality properties. In many cases we consider the representations of quantum algebras that arise due to the duality. In Chapter 5 we consider systematically and construct induced infinite-dimensional representations of quantum algebras using as carrier spaces the corresponding dual matrix quantum groups. These representations are related to the Verma modules over the com- plexification of the quantum algebras, while the singular vectors produce invariant q-difference operators between the reducible induced infinite-dimensional represent- ations. This generalizes our considerations of volume 1 to the setting of quantum groups. These considerations are carried out for several interesting examples. In Chapter 6 we continue the same considerations for the invariantq-difference operators related to GL_q(n). Finally, in Chapter 7 we consider representations theq-deformed conformal algebra and the deformations of various representations and hierarchies of q-difference equations related in some sense to theq-Maxwell equations. Each chapter has a summary which explains briefly the contents and the most relevant literature. Besides, there are bibliography, author index, and subject index. Material from volume 1, Chapter N, formula n is cited as (I.N.n).

Note that initially we planned our monograph as a dilogy; however, later it turned out that the material on quantum groups deserves a whole volume, this volume.

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Volume 3 will cover applications to supersymmetry, the AdS/CFT correspondence, infinite-dimensional (super-)algebras including (super-)Virasoro algebras, and (q-) Schrödinger algebras.

Sofia, December 2016 Vladimir Dobrev

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1.5.5 q-Deformed Lorentz AlgebraU_q(so(3,1)) 26 1.5.6 q-Deformed Real Forms of so(5) 27 1.5.7 q-Deformed de Sitter Algebra so(4,1) 27 1.5.8 q-Deformed Anti de Sitter Algebra so(3,2) 28 1.5.9 q-Deformed AlgebrasU_q(sl(4,ℂ)) andU_q(su(2,2)) 28 1.5.10 q-deformed Poincaré and Weyl Algebras 32

2 Highest-Weight Modules over Quantum Algebras 34

2.1 Verma Modules, Singular Vectors, and Irreducible Subquotients 34 2.2 q-Fock Type Representations 37

2.3 Vertex Operators 39

2.4 Singular Vectors in Chevalley Basis 40 2.4.1 U_q(A_ℓ) 42

2.4.2 U_q(D_ℓ) 43 2.4.3 U_q(E_ℓ) 43 2.4.4 U_q(B_ℓ) 44 2.4.5 U_q(C_ℓ) 44 2.4.6 U_q(F₄) 45 2.4.7 U_q(G₂) 46

2.5 Singular Vectors in Poincaré–Birkhoff–Witt Basis 46

2.5.1 PBW Basis 46

2.5.2 Singular Vectors forU_q(A_ℓ) in PBW Basis 47 2.5.3 Singular Vectors forU_q(D_ℓ) in PBW Basis 51 2.6 Singular Vectors for Nonstraight Roots 56 2.6.1 Bernstein–Gel’fand–Gel’fand Resolution 56

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2.6.2 Case ofU_q(D_ℓ) in PBW Basis 61

2.6.3 Case ofU_q(D_ℓ) in the Simple Roots Basis 63 2.7 Representations at Roots of Unity 64 2.7.1 Generalities 64

2.7.2 The Example ofU_q(sl(2))at Roots of Unity 67 2.7.3 Classification in theU_q(sl(3,ℂ))Case 68 2.7.4 Cyclic Representations ofU_q(G) 76 2.8 Characters of Irreducible HWMs 79 2.8.1 Generalities 79

2.8.2 U_q(sl(3,ℂ)) 80

2.8.3 U_q(sl(3,ℂ))at Roots of Unity 81 2.8.4 Conjectures 83

3 Positive-Energy Representations of Noncompact Quantum Algebras 86

3.1 Preliminaries 86

3.2 Quantum Anti de Sitter Algebra 87 3.2.1 Representations 87

3.2.2 Roots of Unity Case 91 3.2.3 Character Formulae 95

3.3 Conformal Quantum Algebra 97 3.3.1 Generic Case 97

3.3.2 Roots of 1 Case 100 3.3.3 Massless Case 102 3.3.4 Character Formulae 104 4 Duality for Quantum Groups 107 4.1 Matrix Quantum Groups 107

4.1.1 Differential Calculus on Quantum Planes 111 4.2 Duality between Hopf Algebras 113

4.3 Matrix Quantum GroupGL_p,q(2) 113 4.4 Duality forGL_p,q(2) 115

4.5 Duality for Multiparameter QuantumGL(n) 119 4.5.1 Multiparameter Deformation ofGL(n) 120 4.5.2 Commutation Relations of the Dual Algebra 122 4.5.3 Hopf Algebra Structure of the Dual Algebra 125 4.5.4 Drinfeld–Jimbo Form of the Dual Algebra 128 4.5.5 Special Cases of Hopf Algebra Splitting 130 4.6 Duality for a Lorentz Quantum Group 131 4.6.1 Matrix Lorentz Quantum Group 131

4.6.2 Dual Algebras to the AlgebrasLqand̃Lq 133 4.6.3 Coalgebra Structure of the Dual Algebras 137

4.7 Duality for the Jordanian Matrix Quantum Group GL_g,h(2) 139 4.7.1 Jordanian Matrix Quantum Group GL_g,h(2) 140

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Contents IX

4.7.2 The Dual ofGL_g,h(2) 141

4.7.3 Algebra Structure of the Dual 143 4.7.4 Coalgebra Structure of the Dual 146 4.7.5 One-Parameter Cases 148

4.7.6 Application of a Nonlinear Map 149 4.8 Duality for Exotic Bialgebras 151 4.8.1 Exotic Bialgebras: General Setting 151 4.8.2 Exotic Bialgebras: Triangular Case 1 152 4.8.3 Exotic Bialgebras: Triangular Case 2 158 4.8.4 Exotic Bialgebras: Triangular Case 3 161 4.8.5 Higher-Order R-matrix Relations and Quantum

Planes 164

4.8.6 Exotic Bialgebras: Nontriangular CaseS03 167 4.8.7 Exotic Bialgebras: Nontriangular CaseS14 176 4.8.8 Exotic Bialgebras: Nontriangular CaseS14o 183 4.8.9 Exotic Bialgebras: Higher Dimensions 187

Conclusions and Outlook 198 5 Invariantq-Difference Operators 199 5.1 The Case ofGL_p,q(2) 199

5.1.1 Left and Right Action ofU_p,q(gl(2))onGL_p,q(2) 199 5.1.2 Induced Representations ofUp,qand Intertwining

Operators 203 5.1.3 The CaseU_q(sl(2)) 208 5.2 The Case ofGL_g,h(2) 209

5.2.1 Left and Right Action ofU_g,h(gl(2))onGL_g,h(2) 209 5.2.2 Induced Representations ofUg,hand Intertwining

Operators 212

5.2.3 Representations of the Jordanian AlgebraU_h(sl(2)) 218 5.2.4 Highest-Weight Modules overU_h(sl(2)) 219

5.2.5 Singular Vectors ofU_h(sl(2))Verma Modules 221

5.3 q-Difference Intertwining Operators for a Lorentz Quantum Algebra 223 5.3.1 A Matrix Lorentz Quantum Group 223

5.3.2 The Lorentz Quantum Algebra 224

5.3.3 Representations of the Lorentz Quantum Algebra 226 5.3.4 q-Difference Intertwining Operators 234

5.3.5 Classification of Reducible Representations 235 5.3.6 The Roots of Unity Case 236

5.4 Representations of the Generalized Lie Algebrasl(2)_q 238 5.4.1 Preliminaries 238

5.4.2 The Quantum Lie Algebrasl(2)_q 239 5.4.3 Highest-Weight Representations 240

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5.4.4 Highest-Weight Representations of the Restricted Algebra 244

5.4.5 Highest-Weight Representations at Roots of Unity 245 5.4.6 Highest-Weight Representations at Roots of Unity of the

Restricted Algebra 248

5.5 Representations ofU_q(so(3))of Integer Spin Only 250 5.5.1 Preliminaries 250

5.5.2 Matrix Quantum GroupSO_q(3)and the DualU_q(G) 250 5.5.3 Representations ofU_q(so(3)) 253

6 Invariantq-Difference Operators Related toGL_q(n) 261 6.1 Representations Related toGL_q(n) 261

6.1.1 Actions ofU_q(gl(n))andU_q(sl(n)) 263 6.1.2 Representation Spaces 266

6.1.3 Reducibility and Partial Equivalence 272 6.2 The Case ofU_q(sl(3)) 274

6.3 Polynomial Solutions ofq-Difference Equations in Commuting Variables 280

6.3.1 Procedure for the Construction of the Representations 281

6.3.2 Reducibility of the Representations and Invariant Subspaces 284

6.3.3 Newton Diagrams 290

6.4 Application of the Gelfand–(Weyl)–Zetlin Basis 292 6.4.1 Correspondence with the GWZ Basis 292

6.4.2 q-Hypergeometric Realization of the GWZ Basis 296 6.4.3 Explicit Orthogonality of the GWZ Basis 300 6.4.4 Normalized GWZ basis 303

6.4.5 Scalar Product and Normalized GWZ States 307 6.4.6 Summation Formulae 309

6.4.7 Weight Pyramid of theSU(3) UIRs 311

6.4.8 The Irregular Irreps in Terms of GWZ States 316 6.5 The Case ofU_q(sl(4)) 321

6.5.1 Elementary Representations 321 6.5.2 Intertwining Operators 324 7 q-Maxwell Equations Hierarchies 327 7.1 Maxwell Equations Hierarchy 327 7.2 Quantum Minkowski Space–Time 331 7.2.1 q-Minkowski Space–Time 331

7.2.2 Multiparameter Quantum Minkowski Space–Time 333 7.3 q-Maxwell Equations Hierarchy 335

7.4 q-d’Alembert Equations Hierarchy 340

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Contents XI

7.4.1 Solutions of theq-d’Alembert Equation 340 7.4.2 q-Plane-Wave Solutions 341

7.4.3 q-Plane-Wave Solutions for Non-Zero Spin 344

7.5 q-Plane-Wave Solutions of the Potentialq-Maxwell Hierarchy 347 7.6 q-Plane-Wave Solutions of the Fullq-Maxwell Equations 350 7.7 q-Weyl Gravity Equations Hierarchy 354

7.7.1 Linear Conformal Gravity 355

7.7.2 q-Plane-Wave Solutions ofq-Weyl Gravity 358 Bibliography 361

Author Index 391 Subject Index 392

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1 Quantum Groups and Quantum Algebras

Summary

We start with theq-deformationU_q(G)of the universal enveloping algebrasU(G)of simple Lie algeb- rasG called also quantum groups [251, 253] or quantum universal enveloping algebras [389, 521].

They arose in the study of quantum integrable systems, especially of the algebraic aspects of quantum inverse scattering method in papers by Faddeev, Kulish, Reshetikhin, Sklyanin and Takhta- jan [273, 274, 405, 408]. It was observed by Kulish–Reshetikhin [405] forG=sl(2,ℂ)and by Drinfeld [251, 253], Jimbo [360, 361] in general that the algebrasUq(G)have the structure of a Hopf algebra, cf. Abe [11]. This new algebraic structure was further studied in [441, 532, 588, 598]. Later, inspired by the Knizhnik-Zamolodchikov equations [395], Drinfeld has developed a theory of formal deformations and introduced a new notion of quasi-Hopf algebras [255]. The representations ofU_q(G)were considered first in [389, 405, 523, 532] for generic values of the deformation parameter. Actually all results from the representation theory ofGcarry over to the quantum group case. This is not so, however, if the deformation parameterqis a root of unity. Thus this case is very interesting from the mathematical point of view (see, e. g., [170–172, 175, 176, 442, 443]). Lately, quantum groups were intensively applied (with special emphasis on the case whenqis a root of unity) in rational conformal field theories [30–32, 304, 309, 319, 320, 482, 483, 524, 596].

We start this chapter with the general notions of Hopf algebras and quantum groups. Then we introduce quantum algebras first in Drinfeld’s definition and then in Jimbo’s definition. We present also the universal R-matrix and the Casimirs. We also give Drinfeld’ second realization of quantum affine algebras and Drinfeld’s realizations of Yangians. Then we discuss the q-deformations of noncompact Lie algebras. We propose a procedure forq-deformations of the real formsG of complex Lie (super) algebras associated with (generalized) Cartan matrices. Our procedure gives different q-deformations for the nonconjugate Cartan subalgebras ofG. We give several illustrations, for example,q-deformed Lorentz and conformal (super) algebras. Theq-deformed conformal algebra contains as a subalgebra aq-deformed Poincaré algebra and as Hopf subalgebras two conjugate 11- generatorq-deformed Weyl algebras. Theq-deformed Lorentz algebra is Hopf subalgebra of both Weyl algebras.

1.1 Hopf Algebras and Quantum Groups

LetFbe a field of characteristic 0; in fact, most of the time we shall work withF=ℂ orF=ℝ.

An associative algebraU overFwith unity 1_U is called abialgebra[11] if there exist two algebra homomorphisms calledcomultiplication(orcoproduct)$:

$:U →U ⊗U ,$(1_U) = 1_U ⊗1_U, (1.1) andcounit%:

%:U →F,%(1_U) = 1. (1.2)

The comultiplication$fulfills the axiom ofcoassociativity:

($⊗id)∘$= (id⊗$)∘$, (1.3)

DOI 10.1515/9783110427707-001

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where both sides are mapsU →U ⊗U ⊗U; the two homomorphisms fulfil:

(id⊗%)∘$=i₁, (%⊗id)∘$=i₂, (1.4) as mapsU →U ⊗F,U →F⊗U , wherei₁,i₂are the maps identifyingU withU ⊗F, F⊗U , respectively.

Next a bialgebra U is called a Hopf algebra [11] if there exists an algebra antihomomorphism𝛾calledantipode:

𝛾:U →U ,𝛾(1_U) = 1_U, (1.5)

such that the following axiom is fulfilled:

m∘(id⊗ 𝛾)∘$=i∘%, (1.6)

as mapsU →U , wheremis the usual product in the algebra:m(Y⊗Z) =YZ,Y,Z∈U andiis the natural embedding ofFintoU:i(c) =c1_U,c∈F.

The antipode plays the role of an inverse although there is no requirement that 𝛾²= id.

The operations of comultiplication, counit, and antipode are said to give the coalgebrastructure of a Hopf algebra.

Sometimes we shall use also the notation of Sweedler [570] for the coproduct ofa:

$_A(a) =a₍₁₎⊗a₍₂₎. (1.7)

One needs also theopposite comultiplication$^󸀠=0∘$, where0is the permutation in U ⊗U, that is,0(X⊗Y) =Y⊗X,X,Y∈U .

The comultiplication is said to becocommutativeif$^󸀠=$.

If the antipode has an inverse, then one uses also the notion ofopposite antipode:

𝛾^󸀠=𝛾^–1.

A Hopf algebraU is calledquasi-triangular Hopf algebraorquantum group[251, 253] if there exists an invertible elementR ∈ U ⊗U , calleduniversal R-matrix[251, 253], which intertwines$and$^󸀠:

R$(Y) =$^󸀠(Y)R,∀Y ∈U , (1.8)

and obeys also the relations:

($⊗id)R=R₁₃R₂₃,R=R_⋅₃, (1.9a) (id⊗$)R=R₁₃R₁₂,R=R₁_⋅, (1.9b) where the indices indicate the embeddings ofRintoU ⊗U ⊗U. For future use we write down:

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1.1 Hopf Algebras and Quantum Groups 3

R=R^󸀠_j⊗R^󸀠󸀠_j =∑

j

R^󸀠_j⊗R^󸀠󸀠_j. (1.10)

Then inU ⊗U ⊗U:

R₁₂=R^󸀠_j⊗R^󸀠󸀠_j ⊗1_U

and analogously forR₂₃,R₁₃. Further we shall denote 1_U ⊗1_U ⊗1_U just by 1_U. From the above it follows that:

(%⊗id)R= (id⊗%)R= 1_U. (1.11) [Proof: Apply%⊗id⊗id to both sides of (1.9b). On the LHS we have (using (1.4)):

(%⊗id⊗id)∘($⊗id)R= ((%⊗id)∘$)⊗id)R=

= (i₂⊗id) (R^󸀠_j⊗R^󸀠󸀠_j) =

= 1_U ⊗R^󸀠_j⊗R^󸀠󸀠_j =R₂₃ (1.12) On the RHS we have:

(%⊗id⊗id)R₁₃R₂₃= (%(R^󸀠_i)⊗1_U ⊗R^󸀠󸀠_i)R₂₃. (1.13) Comparing the first and third components of (1.12) and (1.13) we get (%⊗id)R= 1_Ufrom (1.11). Analogously it is proved (id⊗%)R= 1_U from (1.11).]

Using also (1.11) one has:

(𝛾 ⊗id)R=R^–1, (id⊗ 𝛾)R^–1=R. (1.14) Proof.For the first equality in (1.14) we consider:

R(𝛾 ⊗id)R=R^󸀠_j𝛾(R^󸀠_k)⊗R^󸀠󸀠_j R^󸀠󸀠_k =

= (m⊗id)∘(R^󸀠_j⊗ 𝛾(R^󸀠_k)⊗R^󸀠󸀠_j R^󸀠󸀠_k) =

= (m⊗id)∘(id⊗ 𝛾 ⊗id) (R^󸀠_j⊗R^󸀠_k⊗R^󸀠󸀠_j R^󸀠󸀠_k) =

= (m⊗id)∘(id⊗ 𝛾 ⊗id)R₁₃R₂₃=

= (m⊗id)∘(id⊗ 𝛾 ⊗id)∘($⊗id)R=

= ((m∘(id⊗ 𝛾)∘$)⊗id)R=

= (%⊗id)R= 1_U .◼ (1.15)

The termquantum group is used [253] also ifRis not inU ⊗ U but in some completion of it (cf. next subsection).

From (1.8) and one of (1.9) follows theYang–Baxter equation (YBE)forR:

R₁₂R₁₃R₂₃=R₂₃R₁₃R₁₂. (1.16)

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[Proof: Using (1.9b) we have:

R₁₂R₁₃R₂₃=R₁₂($⊗id)R=

= (R^󸀠_j⊗R^󸀠󸀠_j ⊗1_U) ($(R^󸀠_k)⊗R^󸀠󸀠_k) =

= ($^󸀠(R^󸀠_k)⊗R^󸀠󸀠_k) (R^󸀠_j⊗R^󸀠󸀠_j ⊗1_U) =

=R₂₃R₁₃R₁₂

where for the last equality one applies0to both sides of (1.9b).]

A quasi-triangular Hopf algebra is called triangular Hopf algebra if also the following holds:

0R^–1=R. (1.17)

The axiom of coassociativity (1.3) may be relaxed being replaced by:

($⊗id)∘$=I{(id⊗$)∘$}I^–1, (1.18) whereI∈U ⊗U ⊗U is invertible. The corresponding objects in which (1.18) holds are calledquasi-bialgebrasandquasi-Hopf algebras, respectively (cf. [255]).

1.2 Quantum Algebras

1.2.1 Drinfeld’s Definition

From now on (unless specified otherwise) we setF = ℂ. LetG be a complex simple Lie algebra; then theq-deformationU_q(G) of the universal enveloping algebrasU(G) is defined [251, 253] as the associative algebra overℂ with generatorsX_i^±,H_i,i = 1,. . .,ℓ= rankG and with commutation relations:

[H_i,H_j] = 0, [H_i,X_j^±] =±a_ijX_j^±, (1.19) [X⁺_i,X_j^–] =$_ij^q^Hi

/2 i –q^–_i^Hi^/2

q^1/2_i –q^–1/2_i =$_ij[H_i]_q

i, q_i=q⁽^!ⁱ^,^!ⁱ^)/2, andq-Serre relations:

n

∑

k=0

(–1)^k(n k)

q_i(X_i^±)^kX_j^±(X_i^±)^n–k= 0 , i≠j, (1.20) where (a_ij) = (2(!i,!j)/(!i,!i)) is theCartan matrixofG and (⋅,⋅) is the scalar product of the roots normalized so that for the short roots!we have (!,!) = 2 ,n= 1 –a_ij,

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(ⁿ_k)_q= ^[n]^q^!

[k]q![n–k]q!, [m]_q! = [m]_q[m– 1]_q. . .[1]_q, (1.21) [m]_q= ^q^m/2^–q^–m/2

q^1/2–q^–1/2 = ^sh(mh/2)

sh(h/2) = ^sin(⁰^m⁴⁾

sin(04) , q=e^h=e²⁰ⁱ⁴, h,4∈ ℂ, qâ_iîj=q⁽^!ⁱ^,^!^j⁾ =qâ_j^ji.

Remark 1.1.Expressions likeq^H/2 = e^hH/2 are made mathematically rigorous in the so-calledh-adic topology used in [251, 253] (q = e^h). [By standard notationF[[h]] is the ring of formal power series in the indeterminatehover the fieldF. EveryF[[h]]

moduleV(e. g.,U_q(G)) has theh-adic topology, which is characterized by requiring that{hⁿV|n≥0}is a base of neighbourhoods of 0 inVand that translations inVare continuous.] Physicists work formally with such exponents which is also justified as

explained below. ◊

Further we shall omit the subscriptqin theq-number[m]_qif no confusion can arise.

Note also that sometimes instead ofqone usesq^󸀠=q², so that [m]_q󸀠= ^q^m^–q^–m

q–q^–1 ≡[m]^󸀠_q. In [558] forG = sl(2) and in [251, 253, 360, 361] in general it was observed that the algebraU_q(G) is a Hopf algebra, the comultiplication, counit, and antipode being defined on the generators ofU_q(G) as follows:

$(H_i) =H_i⊗1 + 1⊗H_i, (1.22)

$(X_i^±) =X_i^±⊗q^H_iⁱ^/4+q^–H_i ⁱ^/4⊗X^±_i ,

%(H_i) =%(X^±_i) = 0 ,

𝛾(H_i) = –H_i, 𝛾(X_i^±) = –q^1/2_i^̂ X^±_i q^–_i^1/2^̂ = –q^±_i^1/2X^±_i, where1̂ ∈ H corresponds to1 = ¹

2∑_!∈B⁺!,B⁺ is the set of positive roots and1̂ =

1

2∑_!∈B⁺H_!.

The above definition is valid also whenG is an affine Kac-Moody algebra [251];

however, another realization, called Drinfeld’s second realization, was given in [254]

and will be presented below. It was also generalized to the complex Lie superalgebras with a symmetrizable Cartan matrix (cf., e. g., [385]).

The algebrasU_q(G) were calledquantum groups[251, 253] orquantum universal enveloping algebras[389, 521]. For shortness we shall call themquantum algebrasas it is now commonly accepted in the literature.

Forq →1 , (h →0) , we recover the standard commutation relations from (1.19) and q-Serre relations from (1.20) in terms of theChevalley generators H_i,X^±_i.

The elementsH_ispan the Cartan subalgebraH ofG, while the elementsX^±_i gen- erate the subalgebrasG^±. We shall use the standard triangular decompositioninto direct sums of vector subspacesG = H ⊕ ⊕

"∈BG_" = G⁺⊕H ⊕G^–,G^± = ⊕

"∈B^±G_", whereB=B⁺∪B^–is the root system ofG,B⁺,B^–, the sets of positive, negative, roots, respectively;BSwill denote the set of simple roots ofB. We recall thatH_jcorresponds

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to the simple roots!jofG, and if"^∨=∑jn_j!^∨j and"^∨≡2"/(","), then to"corresponds

H_"=∑jn_jH_j. The elements ofG which spanG_"(dimG_" = 1) are denoted byE_". These

Cartan–WeylgeneratorsH_",E_"[198, 360, 361, 575], may be normalized so that:

[E_",E_–_"] = [H_"]_q

", q_"≡q⁽^"^,^"^)/2 (1.23a)

[H_",E_±"_󸀠] =±("^∨,"^󸀠)E_±"_󸀠, ","^󸀠∈B⁺. (1.23b) To display more explicitly the Cartan–Weyl generators we need the notion of normal ordering [385]:

Definition 1.1.We say that the root systemBis in thenormal orderingif in the situation 𝛾=!+"∈B+, where!≠+",!,"∈B+, the roots are ordered as!<𝛾<". ◊ Then the Cartan–Weyl generators are constructed as follows: Let𝛾=!+",!<𝛾<", and [!;"] is a minimal segment including𝛾; that is, there do not exist roots!^󸀠,"^󸀠, such that !^󸀠 > !,"^󸀠 < "and!^󸀠+"^󸀠 = 𝛾. Then the root vectors E_±𝛾 are given as follows:

E_𝛾 = (ad_qE_!)E_"≡E_!E_"–q⁽^!^,^"^)/2E_"E_!, (1.24) E_–_𝛾 = (Ad_q–1E_")E_!=E_"E_!–q^–(^!^,^"^)/2E_!E_". (1.25) As an example we give the Cartan–Weyl generators forG = sl(n). LetX⁺_jk, X^–_kjbe the Cartan–Weyl generators corresponding to the roots!j,k+1, –!j,k+1, withj ≤ k; in particular,X⁺_jj=X⁺_j X^–_jj=X_j^–, correspond to the simple roots!j.

Here the normal ordering coincides with the lexicographic ordering. In the case of the root!_j,k+1we have two minimal segments since:

!j,k+1=!j+!j+1,k+1=!jk+!k, j<k,

!j,j+1=!j (1.26)

the orderings being:

!j<!j,k+1<!j+1,k+1, !jk<!j,k+1<!k, (1.27)

!j<!jk, !j+1,k+1<!k

Then instead of (I.2.46a,b) we have:

X⁺_jk= (ad_qX⁺_j)X_j+1,k⁺ ≡

≡X_j⁺X_j+1,k⁺ –q⁽^!^j^,^!^j+1,k+1^)/2X_j+1,k⁺ X_j⁺=

= (ad_qX⁺_j,k–1)X_k⁺≡

≡X_j,k–1⁺ X_k⁺–q⁽^!^k^,^!^jk^)/2X_k⁺X_j,k–1⁺ , j<k, (1.28a)

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X^–_kj= (Ad_q–1X_k^–)X_k–1,j^– ≡

≡X_k^–X_k–1,j^– –q^–(^!^k^,^!^jk^)/2X^–_k–1,jX^–_k =

= (Ad_q–1X_k,j+1^– )X^–_j ≡

≡X_k,j+1^– X_j^––q^–(^!^j^,^!^j+1,k+1^)/2X_j^–X_k,j+1^– , j<k. (1.28b) In the affine case, the Cartan–Weyl formulae are as above, though it is useful to write them down the analogues of (I.2.158a) and (I.2.161):

[E_k_d+!_̄ ,E_–(k_d+!)_̄ ] = [H_k_d+!_̄ ]_q

! = [H_!+kc]̂_q

!, (1.29a)

[Eⁱ

kd̄,E_{ℓ ̄}ⁱ

d] =$k,–ℓ[Hⁱ

kd̄]_q=$k,–ℓq^k^c/2^̂ –q^–k^c/2^̂

q^1/2–q^–1/2 . (1.29b) The action of$,%,𝛾on the Cartan–Weyl generators is obtained easily from (1.22) since

H_"andE_"are given algebraically in terms of the Chevalley generators. (Of course, if

!∉BSthe coalgebra operations$,𝛾look more complicated than (1.22).) The axioms in (1.1)–(1.6) are fulfilled by the explicit definition (1.22).

The opposite comultiplication and antipode [253, 361] introduced above define a Hopf algebraU_q(G)^󸀠, which is related toU_q(G) by:

U_q(G)^󸀠=U_q–1(G). (1.30)

1.2.2 Universal R-Matrix and Casimirs

ForG =sl(2) the universalR-matrix is given explicitly by [253]:

R=q^H^⊗^H/4∑

n≥0

(1 –q^–1)ⁿq^n(n–1)⁴

[n]! (q^H⁴X⁺)ⁿ⊗(q^–^H⁴X^–)ⁿ (1.31) whereH =H₁,X^± = X^±₁ ,r = 1. Note that thisR-matrix is not inU_q(sl(2))⊗U_q(sl(2)) , since it contains power series involving the generatorsX^±, but in some completion of it (in theh–adic topology used in [251, 253]). This is valid for theR-matrices of allU_q(G).

Hopf algebras with such anR-matrix are calledpseudo quasi-triangular Hopf algebras [253] oressentially quasi-triangular Hopf algebras[454].

Here we can point out the only serious inequivalence between the Drinfeld and Jimbo definitions. Namely, there is no element inŨ_q(G)⊗ ̃U_q(G) corresponding to the factorq^H^⊗^H/4. Nevertheless, the universalR-matrix can act on any tensor product of finite-dimensionalŨ_q(G)-modules.

ForG = sl(n) an explicit formula for Rwas given in [533]. Explicit multiplicative formulas forRwere given in [389, 424] for all complex simple Lie algebrasG and

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in [385] for all finite-dimensional superalgebras with symmetrizable Cartan matrices.

Then the universalR-matrix for the untwisted affine Lie algebras was given in [578].

Then this was obtained using the quantum Weyl group forA⁽¹⁾₁ in [426] and for general untwisted case in [167].

We recall results of [389, 447] where were given explicit multiplicative formulas forRfor anyU_q(G). For this they introducedq-version of the Weyl group forU_q(G). Let us recall that for!∈B,

s_!(D) =D–2(+,!)

(!,!)!, D∈H^∗ (1.32)

are the standard reflections inH^∗. TheWeyl group Wis generated by the reflections s_i ≡s_!

i, where!iis the simple root. Thus every elementw∈ Wcan be written as the product of simple reflections. It is said thatwis written in a reduced form if it is written with the minimal possible number of simple reflections; the number of reflections of a reduced form ofwis called the length ofw, denoted byℓ(w).

The elements of the q-Weyl group belong to the completion Ū_q(G) of U_q(G) [389]. They are defined by the action of the generating elements in the irreducible representations ofU_q(G).

In the case of sl(2,ℂ) the nontrivial element w of W is defined to act in the representation defined (e. g., [389]):

w|j,n>_q= (–1)^j–nq(n–j(j+1))/2|j, –n>_q . (1.33) It satisfies the relations [389]:

wX^±w^–1 = –q^±^1/2X^∓, wHw^–1= –H. (1.34) SinceŪ_q(G) is also a Hopf algebra we have [389]:

$(w) =R^–1w⊗w, %(w) = 1, 𝛾(w) =wq^H/2, (1.35) whereRis given by (1.31). Further let us introduce the element

u=∑

i

𝛾(a_i)b_i, (1.36)

wherea_i,b_iare the coordinates of the elementR:

R=∑

i

a_i⊗b_i. (1.37)

One may show that:

𝛾²(Y) =uYu^–1, (1.38)

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and

v=uq^–^r/2^̂ ∈ centre ofU_q(G), (1.39)

̂

ris used in (1.10). Let:be the unipotent central element, that is,:|j,n>_q= (–1)^2j|j,n>_q,:²= id. Then [389]

w²=v:=uq^–^r/2^̂ :. (1.40)

For arbitraryU_q(G) letL_Dbe an irreducible representation ofU_q(G). LetL_D=⊕j(W_D^j ⊗ L_j) be the decomposition ofL_Dinto irreducible (U_q(sl(2,ℂ)))_jsubmodules. Define the action ofw_iinL_Dasw_i=⊕_j(Id

W_D^j ⊗(w_i)_j), where (w_i)_jis the action ofwinL_jas in (1.33).

Further one has [389]:

w_iH_jw^–1_i =H_j–a_ijH_i, w_iX^±_iw^–1_i = –q^±^1/2X_i^∓. (1.41)

$w_i=R(i)^–1w_i⊗w_i, (1.42) whereR(i) =R(H_i,X_i^±|q_i),

(w_iw_j)^2–a^ij= 1, for i≠j, (w_i)²= 1, (1.43a) (w̃_iw̃_j)^2–a^ij= 1, for i≠j, (w̃_i)²= 1, (1.43b)

̃

w_i=w_iq^H

2 i/8

i . (1.43c)

Further lets₀=s_i

1. . .s_i

kbe the reduced form of the element ofWwith maximal length ℓ(s₀). It can be shown that the element

̃

w₀=w̃_i

1. . . ̃w_i

k (1.44)

is well defined and does not depend on the choice of decomposition ofs₀. Finally the result of [389] for the universalR-matrix is:

R = q^∑

n

i,j=1(B^–1)_ijH_i⊗H_j/4

(w̃₀⊗ ̃w₀)$(w̃₀)^–1,

(B_ij) = ((!i,!j)), (1.45a) or

R = q^∑ⁿ^i,j=1^(B^–1⁾^ij^Hⁱ^⊗^H^j^/4R(ĩ _k|s_i

1. . .s_i

k–1). . . . . . ̃R(i₂|s_i

1)R(ĩ ₁), (1.45b)

where

̃R(i_ℓ|s_i

1. . .s_i

ℓ–1) = (T_i^–1

1 ⊗T_i^–1

1). . .(T^–1_i

ℓ–1⊗T_i^–1

ℓ–1)R(ĩ _ℓ), (1.45c)

T_i(Y) =w̃^–1_i Yw̃_i. (1.45d)

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The same construction works for affine Lie algebras [389]. Earlier work in this case includes the explicit construction forA⁽¹⁾₁ in any representation [360, 405, 406, 558];

forA⁽¹⁾_n ,B⁽¹⁾_n ,C⁽¹⁾_n ,D⁽¹⁾_n in the vector representation [82, 362]; forB⁽¹⁾_n ,D⁽¹⁾_n in the spinor representation [500]; and forG⁽¹⁾₂ [415].

The centre of U_q(G), and for generic q the centre of Ũ_q(G), is generated by q-analogues of theCasimir operators[360, 361, 557]. ForG =sl(2) one has:

C₂= [(H+ 1)/2]²+X^–X⁺. (1.46) ForG =sl(n+ 1,ℂ) we shall need more explicit expressions for the Cartan–Weyl generators as in (1.28). Let!j,k+1 ∈ B⁺, 1≤ j≤ k≤ nbe a positive root given explicitly in terms of the simple roots!_j,j= 1,. . .,n(as in (1.26)) by:

!j,k+1=!j+!j+1+⋅ ⋅ ⋅+!k, j<k. (1.47) Then the corresponding root vectors elementsX_jk^±,j<kare defined inductively:

X_jk^± =±(q^1/4X^±_j X_j+1k^± –q^–1/4X_j+1k^± X_j^±), j<k. (1.48) Note that there is some inessential ambiguity in the definition (1.48), namely,X_jk^󸀠± = q^±ⁿX_jk^±for genericqis also a good choice. Particularly often are used the choicesn= 1/4 orn= –1/4. Thus, (1.48) differs by such normalization from (1.28). One can check (1.23) with

H_!

j,k+1=H_j+H_j+1+⋅ ⋅ ⋅+H_k, j<k. (1.49)

Now the Casimir operator is given by [478]:

C₂=K⁰( ∑

1≤i≤j≤n

K₁^–1. . .K_i–1^–1K_j+1. . .K_nX^–_ijX⁺_ijq(i+j–n–1)/2+

+

n

∑

j=0

K^–1₁ . . .K^–1_j K_j+1. . .K_nq^–j+n/2(q^1/2–q^–1/2)^–2), (1.50)

where

K⁰=K₁^a¹. . .K^a_nⁿ, a_i= (n+ 1 – 2i)/(n+ 1), K^±_i¹=q^±_i^Hⁱ^/2.

Forn= 1 this expression differs from (1.46) by an additive constant.

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1.2.3 Jimbo’s Definition

In some considerations it is useful to use a subalgebraŨ_q(G) ofU_q(G) generated by X^±_i and

K_i^±¹=q^±Hⁱ^/4

i , (1.51)

and then (1.19) is replaced by:

K_iK_i^–1=K_i^–1K_i= 1 , [K_i,K_j] = 0, K_iX_j^±K^–1_i =q^±^a^ij^/4

i X_j^±, (1.52a)

[X⁺_i ,X^–_j] =$ij

K_i²–K^–2_i q^1/2

i –q^–1/2

i

. (1.52b)

On the other hand one may forget (1.51) and defineŨ_q(G) with the generatorsX^±_i and K_i^±1and relations (1.20) and (1.52). In terms of these generators the coalgebra relations are:

$(K_i) =K_i⊗K_i, $(X_i^±) =X_i^±⊗K_i+K_i^–1⊗X^±_i (1.53a)

%(K_i) = 1 , %(X^±_i) = 0 , (1.53b) 𝛾(K_i) =K_i^–1, 𝛾(X^±_i) = –q^±_i^1/2X_i^±. (1.53c) This is actually how quantum groups are defined in [360, 361]. This definition has the advantage thatŨ_q(G) is an algebra in the strict sense of the notion. The algebraŨ_q(G) is also calledrational formofU_q(G), orJimbo quantum algebra.

Nevertheless, even if not used, relation (1.51) is present in a ‘hidden way’. That is why quantum algebras are called quantum groups a la Drinfeld-Jimbo in spite of the fact that the two definitions are not strictly equivalent. (In the mathematical literature (cf., e. g., Chari–Pressley [147]) one starts also by treatingq^±^1/2as formal variables.)

We shall point out now one of the inequivalences, the so-called twisting.

Let (31,. . .,3n) ∈ {±1}ⁿ. Then there exists an algebra homomorphism of Ũ_q(G) given by:

K_i󳨃→3iK_i, X⁺_i 󳨃→3iX_i⁺, X_i^–󳨃→X^–_i. (1.54) On the other hand, except from the identity automorphism3i = 1,∀i, there are no analogous automorphisms for U_q(G). We note that this inequivalence is not very important since these automorphisms (except for the identity one) do not respect the coalgebra structure ofŨ_q(G).