##
**A guide to quantum groups.**
*(English)*
Zbl 0839.17010

Cambridge: Cambridge University Press. xv, 651 p. (1995).

The theory of quantum groups is scarcely more than a dozen years old, but already several books have been devoted to it, and there is also a separate preprint server with many hundreds of items available. As one might guess from the name, quantum groups arose originally in the process of abstracting some of the mathematical data involved in the quantum inverse scattering problem, as studied particularly by L. D. Faddeev and the school around him in Leningrad (now St. Petersburg again). In essence they are associative algebras whose defining relations are expressed in terms of a matrix of coefficients called a “quantum \(R\)-matrix”, arising from some integrable system. Later it was noticed that such algebras are Hopf algebras and that they are related to the other Hopf algebra associated with a Lie algebra (namely, the universal enveloping algebra) (which is why the Mathematics Subject Classification for Quantum groups is “17B37’) and the machine began to turn. Connections with many other fields, such as the construction of knot invariants and the representation theory of algebraic groups in characteristic \(p\) followed.

In the present book the authors give a fairly comprehensive overview of the field and some of its applications. It is well-written and contains many useful examples and bibliographic notes. There is a 70-page bibliography and indexes of notation and terminology. Some more detailed notes on the chapters follow.

Chapter 1 introduces the subject from the standpoint of the connection between classical and quantum mechanics. The authors discuss Poisson-Lie groups and the associated infinitesimal version (Lie bialgebras). From there they discuss deformations of Poisson structures and quantization using the Moyal bracket. This chapter is principally motivation. Note that only the Moyal deformation is discussed, although other treatments are possible.

Chapter 2 begins the deeper understanding of Lie bialgebras associated to a Poisson-Lie group. If \(\mathfrak g\) is a Lie algebra (over a field of characteristic zero), an element \(r \in {\mathfrak g} \otimes {\mathfrak g}\) defines a Lie bialgebra structure arising from the coboundary \(\delta(X) = [X,r]\) if and only if the symmetric part of \(r\) is a \(\mathfrak g\)-invariant element of \({\mathfrak g} \otimes {\mathfrak g}\) and the expression \[ [[r,r]] = [r_{12}, r_{13}] + [r_{12},r_{23}] + [r_{13},r_{23}] \] is a \(\mathfrak g\)-invariant element of \({\mathfrak g} \otimes {\mathfrak g} \otimes {\mathfrak g}\). When the last condition is trivially satisfied, that is, when \([[r,r]] = 0\), this is called the classical Yang-Baxter equation (CYBE), which arose explicitly in this form as an integrability condition for classical Hamiltonian systems, although it is also a special case of the Schouten bracket in differential geometry. It is important to note that if \(\mathfrak g\) is a finite-dimensional complex simple Lie algebra, then all Lie bialgebra structrures on \(\mathfrak g\) arise in this way.

Chapter 3, then, considers the parametrization and explicit description of solutions of the CYBE. It is interesting to notice that different methods are required, depending on whether the sought-after solutions are skew-symmetric or not. Relating this back to Poisson-Lie groups, the special case when \(\mathfrak g\) is an algebra of Lie-algebra-valued functions on some space (e.g., a domain in \(\mathbb{C}\)). Solutions in this last case (in the e.g.) are elliptic functions, trigonometric functions or rational functions \(^*\)under some hypothesis). This is important for the later study of so-called Knizhnik-Zamolodchikov equations in conformal field theory.

Chapter 4 begins the examination of these objects from the point of view of Hopf algebras. Remember that to a Lie group we have associated both a Lie algebra and a Lie bialgebra. The importance here is in the special class of “quasitriangular Hopf algebras”, which is defined with respect to a certain property of the comultiplication of a Hopf algebra \(A\). So as not to burden this review with too many diagrams, we give one of the main properties of quasitriangular Hopf algebras as a “quasi-definition”. A quasitriangular Hopf algebra \(A\) contains a distinguished invertible element \({\mathcal R} \in A \otimes A\), called the universal \(R\)-matrix, from which the isomorphism \(V \otimes W \to W \otimes V\) for the tensor product of two representation spaces of \(A\) is constructed. The element \(\mathcal R\) satisfies the so-called quantum Yang-Baxter equation \[ {\mathcal R}_{12} {\mathcal R}_{13} {\mathcal R}_{23} = {\mathcal R}_{23} {\mathcal R}_{13} {\mathcal R}_{12}. \] The isomorphism between \(V \otimes W\) and \(W \otimes V\) shows that the category of representations of a quasitriangular Hopf algebra is a quasitensor (or braided monoidal) category. (This differs from a tensor category in that the commutativity isomorphisms are no longer involutive). Chapter 4 also contains some reference material on Hopf algebras.

Monoidal categories are obtained from abstracting the tensor product operation (on some objects). Motivated by the categories of representations of Hopf algebras and various geometric examples (directed tangles and ribbon tangles) related to knot theory, the authors discuss in Chapter 5 these abstract notions. In particular, the authors treat combinatorial data that arise from the decomposition of the tensor product in a semisimple abelian quasitensor category into simple objects of the tensor product of any two simple objects. Such data arise as so-called fusion rules in conformal field theories. The final section of this chapter is devoted to the work of Reshetikhin and Turaev on the invariants of ribbon tangles.

Chapter 6 is devoted to the quantization of Lie bialgebras. The authors specialize from general deformations of Hopf algebras to deformations of enveloping algebras, called “quantized universal enveloping algebras” (or QUE algebras), and to deformations of the algebra of functions on a Lie group. Recall that before a Poisson-Lie group structure or a Lie bialgebra structure was used to quantize a Lie group or Lie algebra, so to incorporate this into a universal enveloping algebra one needs to define an appropriate coalgebra structure. This chapter also contains a construction of the standard quantization of the Lie algebra \({\mathfrak s}{\mathfrak l}_2(\mathbb{C})\) with the standard Lie bialgebra structure, showing in particular that it is quasitriangular as a Hopf algebra. This construction is then extended to the quantization of any finite-dimensional complex simple Lie algebra. As motivation, the origins of the quantum Yang-Baxter equation in two-dimensional lattice models in statistical mechanics is indicated. For a (finite-dimensional complex simple) Lie algebra \(\mathfrak g\) this quantized universal enveloping algebra is denoted \(U_h({\mathfrak g})\), where \(h\) is the deformation “parameter” (coming from a power series algebra over a Hopf algebra).

Chapters 7 and 13 are devoted to quantized function algebras. Chapter 7 starts with the quantization \({\mathcal F}_h(+SL_2(\mathbb{C}))\) of the algebra of polynomial functions on \(\text{SL}_2(\mathbb{C})\) and show that it is the dual Hopf algebra to the quantization of the universal enveloping algebra \(U({\mathfrak s}{\mathfrak l}_2(\mathbb{C}))\) constructed in Chapter 6. Generalizing this from \(\text{SL}_2(\mathbb{C})\) leads to the quantization of the algebra of polynomials on an arbitrary complex simple Lie group \(G\). The authors also discuss multi-parameter deformations and a deformation of the de Rham complex on \(\text{GL}_{n+1}(\mathbb{C})\) which is compatible with the above deformation of the algebra of functions. Chapter 13 is concerned with harmonic analysis on quantized algebras of functions. In particular, there is a natural \(*\)-structure on the algebra of representative functions on a complex simple Lie group \(G\). This \(*\)-structure allows the authors to discuss unitary representations. Notions of compactness of a quantum group and the tentative development of a representation theory for compact quantum groups are discussed in this chapter. Chapter 13 concludes with a brief discussion of \(q\)-special functions.

Chapter 8 is concerned with the construction of a basis of \(U_h({\mathfrak g})\) analogous to the PoincarĂ©-Birkhoff-Witt basis of \(U({\mathfrak g})\). This entails the extension of the Weyl group action from the classical situation of an action of \(W\) on a Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\) and then to an action of a covering of \(W\) on \(U({\mathfrak g})\) to the “quantum situation” of an action on the subalgebra \(U({\mathfrak h})[[h]]\) of \(U_h({\mathfrak g})\). But this action on \(U({\mathfrak h})[[h]]\) does not extend fully as in the classical case, but only to an action on \(U_h({\mathfrak g})\) of an infinite cover of \(W\), the braid group \({\mathcal B}_{\mathfrak g}\) of \(\mathfrak g\). The relationship between the braid group action and the coalgebra structure of \(U_h(\mathfrak g)\) (incompatible!) and the quasitriangularity of \(U_h({\mathfrak g})\) are also discussed.

The next chapter concerns the topic of specializations of \(U_h({\mathfrak g})\). Although one cannot do this in a direct way (since \(U_h({\mathfrak g})\) is a formal object), one can form a new algebra \(U_q({\mathfrak g})\) defined over the field of rational functions of an indeterminate \(q\), and an “integral” form, defined over \(Z[q,q^{-1}]\). The authors point out the similarities and differences between \(U_h({\mathfrak g})\) and \(U_q({\mathfrak g})\), pointing out in particular if \(\dim({\mathfrak g}) < \infty\), then, although \(U_h({\mathfrak g})\) is isomorphic as an algebra to \(U({\mathfrak g})[[h]]\), it is not true that \(U_q({\mathfrak g}) \otimes_{Q(q)} \mathbb{C}(q)\) is isomorphic to \(U({\mathfrak g})\otimes_\mathbb{C} \mathbb{C}(q)\), and that the universal \(R\)-matrix does not translate explicitly to \(U_q\), though it does act on \(U_q\)-modules. They also discuss the non-restricted and restricted integral forms of \(U_q({\mathfrak g})\). The non-restricted specialization arises by replacing \(q\) by \(\varepsilon\) in the defining relations. When \(\varepsilon\) is a primitive \(\ell\)th root of unity \((\ell > 1)\), \(U_\varepsilon({\mathfrak g})\) is finite-dimensional over its center, and thus its representation theory reduces essentially to the representation theory of a finite-dimensional algebra. The authors show that this center \(Z_\varepsilon\) is the algebra of regular functions on a complex algebraic variety and that this variety is a finite ramified covering of the big cell in the group \(G\) with Lie algebra \(\mathfrak g\). The definition of the restricted specialization resembles the well-known construction of a Chevalley-Kostant basis in the classical case.

After constructing these specializations, in Chapters 10 and 11 the authors present results, due mostly to G. Lusztig, on the representations of QUE algebras. Chapter 10 considers the generic case when \(\varepsilon\) is not a root of unity. Among the results here are the quantum analogue of the Frobenius-Schur duality theorem and the role of the Hecke algebra in the representation theory. In Chapter 11 the authors discuss the root-of-unity case, including the conjectural dimension formula, the Kazhdan-Lusztig character formula (proved in Chapter 16 as an extension of the Kohno-Drinfel’d theorem showing that the category of finite-dimensional representations of \(U_h({\mathfrak g})\) is equivalent, as a quasitensor category, to the category of finite-dimensional representations of \(\mathfrak g\) equipped with a quasitensor structure defined via the Knizhnik-Zamolodchikov equation) and tensor products of representations of the restricted specialization \(U^{\text{res}}_\varepsilon ({\mathfrak g})\).

Chapter 12 is concerned with infinite-dimensional quantum groups and their representations, namely the Yangians (deformations of the Lie algebra of maps \(\mathbb{C} \to {\mathfrak g})\) and quantizations of affine (Kac-Moody) algebras. The finite-dimensional representations of these algebras are used to compute the rational and trigonometric solutions of the quantum Yang-Baxter equation.

Chapter 14 applies the general theory to the construction of canonical bases of any (finite-dimensional) representation of a Lie algebra \(\mathfrak g\). Classically this is approached by various methods, but most of them are restricted to the cases when \(\mathfrak g\) is of classical type of just to the case of \({\mathfrak s}{\mathfrak l}_m(\mathbb{C})\). The approaches using quantum groups eliminate these problems and in fact are applicable when \(\mathfrak g\) is an arbitrary symmetrizable Kac-Moody algebra. In this chapter the authors discuss Kashiwara’s construction of crystal bases, and Lusztig’s construction of canonical bases (which turns out to have the same result). Most of the material here is only sketched.

The remaining chapter, Chapter 15, concerns the use of representations of quantum groups by Turaev and others to construct the Jones polynomial, the HOMFLY polynomial, and the Kauffman polynomial from knot theory.

In summary, then, although this book makes considerable demands on the part of the reader, it should be useful for anyone who wants to understand “what the fuss is all about”.

In the present book the authors give a fairly comprehensive overview of the field and some of its applications. It is well-written and contains many useful examples and bibliographic notes. There is a 70-page bibliography and indexes of notation and terminology. Some more detailed notes on the chapters follow.

Chapter 1 introduces the subject from the standpoint of the connection between classical and quantum mechanics. The authors discuss Poisson-Lie groups and the associated infinitesimal version (Lie bialgebras). From there they discuss deformations of Poisson structures and quantization using the Moyal bracket. This chapter is principally motivation. Note that only the Moyal deformation is discussed, although other treatments are possible.

Chapter 2 begins the deeper understanding of Lie bialgebras associated to a Poisson-Lie group. If \(\mathfrak g\) is a Lie algebra (over a field of characteristic zero), an element \(r \in {\mathfrak g} \otimes {\mathfrak g}\) defines a Lie bialgebra structure arising from the coboundary \(\delta(X) = [X,r]\) if and only if the symmetric part of \(r\) is a \(\mathfrak g\)-invariant element of \({\mathfrak g} \otimes {\mathfrak g}\) and the expression \[ [[r,r]] = [r_{12}, r_{13}] + [r_{12},r_{23}] + [r_{13},r_{23}] \] is a \(\mathfrak g\)-invariant element of \({\mathfrak g} \otimes {\mathfrak g} \otimes {\mathfrak g}\). When the last condition is trivially satisfied, that is, when \([[r,r]] = 0\), this is called the classical Yang-Baxter equation (CYBE), which arose explicitly in this form as an integrability condition for classical Hamiltonian systems, although it is also a special case of the Schouten bracket in differential geometry. It is important to note that if \(\mathfrak g\) is a finite-dimensional complex simple Lie algebra, then all Lie bialgebra structrures on \(\mathfrak g\) arise in this way.

Chapter 3, then, considers the parametrization and explicit description of solutions of the CYBE. It is interesting to notice that different methods are required, depending on whether the sought-after solutions are skew-symmetric or not. Relating this back to Poisson-Lie groups, the special case when \(\mathfrak g\) is an algebra of Lie-algebra-valued functions on some space (e.g., a domain in \(\mathbb{C}\)). Solutions in this last case (in the e.g.) are elliptic functions, trigonometric functions or rational functions \(^*\)under some hypothesis). This is important for the later study of so-called Knizhnik-Zamolodchikov equations in conformal field theory.

Chapter 4 begins the examination of these objects from the point of view of Hopf algebras. Remember that to a Lie group we have associated both a Lie algebra and a Lie bialgebra. The importance here is in the special class of “quasitriangular Hopf algebras”, which is defined with respect to a certain property of the comultiplication of a Hopf algebra \(A\). So as not to burden this review with too many diagrams, we give one of the main properties of quasitriangular Hopf algebras as a “quasi-definition”. A quasitriangular Hopf algebra \(A\) contains a distinguished invertible element \({\mathcal R} \in A \otimes A\), called the universal \(R\)-matrix, from which the isomorphism \(V \otimes W \to W \otimes V\) for the tensor product of two representation spaces of \(A\) is constructed. The element \(\mathcal R\) satisfies the so-called quantum Yang-Baxter equation \[ {\mathcal R}_{12} {\mathcal R}_{13} {\mathcal R}_{23} = {\mathcal R}_{23} {\mathcal R}_{13} {\mathcal R}_{12}. \] The isomorphism between \(V \otimes W\) and \(W \otimes V\) shows that the category of representations of a quasitriangular Hopf algebra is a quasitensor (or braided monoidal) category. (This differs from a tensor category in that the commutativity isomorphisms are no longer involutive). Chapter 4 also contains some reference material on Hopf algebras.

Monoidal categories are obtained from abstracting the tensor product operation (on some objects). Motivated by the categories of representations of Hopf algebras and various geometric examples (directed tangles and ribbon tangles) related to knot theory, the authors discuss in Chapter 5 these abstract notions. In particular, the authors treat combinatorial data that arise from the decomposition of the tensor product in a semisimple abelian quasitensor category into simple objects of the tensor product of any two simple objects. Such data arise as so-called fusion rules in conformal field theories. The final section of this chapter is devoted to the work of Reshetikhin and Turaev on the invariants of ribbon tangles.

Chapter 6 is devoted to the quantization of Lie bialgebras. The authors specialize from general deformations of Hopf algebras to deformations of enveloping algebras, called “quantized universal enveloping algebras” (or QUE algebras), and to deformations of the algebra of functions on a Lie group. Recall that before a Poisson-Lie group structure or a Lie bialgebra structure was used to quantize a Lie group or Lie algebra, so to incorporate this into a universal enveloping algebra one needs to define an appropriate coalgebra structure. This chapter also contains a construction of the standard quantization of the Lie algebra \({\mathfrak s}{\mathfrak l}_2(\mathbb{C})\) with the standard Lie bialgebra structure, showing in particular that it is quasitriangular as a Hopf algebra. This construction is then extended to the quantization of any finite-dimensional complex simple Lie algebra. As motivation, the origins of the quantum Yang-Baxter equation in two-dimensional lattice models in statistical mechanics is indicated. For a (finite-dimensional complex simple) Lie algebra \(\mathfrak g\) this quantized universal enveloping algebra is denoted \(U_h({\mathfrak g})\), where \(h\) is the deformation “parameter” (coming from a power series algebra over a Hopf algebra).

Chapters 7 and 13 are devoted to quantized function algebras. Chapter 7 starts with the quantization \({\mathcal F}_h(+SL_2(\mathbb{C}))\) of the algebra of polynomial functions on \(\text{SL}_2(\mathbb{C})\) and show that it is the dual Hopf algebra to the quantization of the universal enveloping algebra \(U({\mathfrak s}{\mathfrak l}_2(\mathbb{C}))\) constructed in Chapter 6. Generalizing this from \(\text{SL}_2(\mathbb{C})\) leads to the quantization of the algebra of polynomials on an arbitrary complex simple Lie group \(G\). The authors also discuss multi-parameter deformations and a deformation of the de Rham complex on \(\text{GL}_{n+1}(\mathbb{C})\) which is compatible with the above deformation of the algebra of functions. Chapter 13 is concerned with harmonic analysis on quantized algebras of functions. In particular, there is a natural \(*\)-structure on the algebra of representative functions on a complex simple Lie group \(G\). This \(*\)-structure allows the authors to discuss unitary representations. Notions of compactness of a quantum group and the tentative development of a representation theory for compact quantum groups are discussed in this chapter. Chapter 13 concludes with a brief discussion of \(q\)-special functions.

Chapter 8 is concerned with the construction of a basis of \(U_h({\mathfrak g})\) analogous to the PoincarĂ©-Birkhoff-Witt basis of \(U({\mathfrak g})\). This entails the extension of the Weyl group action from the classical situation of an action of \(W\) on a Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\) and then to an action of a covering of \(W\) on \(U({\mathfrak g})\) to the “quantum situation” of an action on the subalgebra \(U({\mathfrak h})[[h]]\) of \(U_h({\mathfrak g})\). But this action on \(U({\mathfrak h})[[h]]\) does not extend fully as in the classical case, but only to an action on \(U_h({\mathfrak g})\) of an infinite cover of \(W\), the braid group \({\mathcal B}_{\mathfrak g}\) of \(\mathfrak g\). The relationship between the braid group action and the coalgebra structure of \(U_h(\mathfrak g)\) (incompatible!) and the quasitriangularity of \(U_h({\mathfrak g})\) are also discussed.

The next chapter concerns the topic of specializations of \(U_h({\mathfrak g})\). Although one cannot do this in a direct way (since \(U_h({\mathfrak g})\) is a formal object), one can form a new algebra \(U_q({\mathfrak g})\) defined over the field of rational functions of an indeterminate \(q\), and an “integral” form, defined over \(Z[q,q^{-1}]\). The authors point out the similarities and differences between \(U_h({\mathfrak g})\) and \(U_q({\mathfrak g})\), pointing out in particular if \(\dim({\mathfrak g}) < \infty\), then, although \(U_h({\mathfrak g})\) is isomorphic as an algebra to \(U({\mathfrak g})[[h]]\), it is not true that \(U_q({\mathfrak g}) \otimes_{Q(q)} \mathbb{C}(q)\) is isomorphic to \(U({\mathfrak g})\otimes_\mathbb{C} \mathbb{C}(q)\), and that the universal \(R\)-matrix does not translate explicitly to \(U_q\), though it does act on \(U_q\)-modules. They also discuss the non-restricted and restricted integral forms of \(U_q({\mathfrak g})\). The non-restricted specialization arises by replacing \(q\) by \(\varepsilon\) in the defining relations. When \(\varepsilon\) is a primitive \(\ell\)th root of unity \((\ell > 1)\), \(U_\varepsilon({\mathfrak g})\) is finite-dimensional over its center, and thus its representation theory reduces essentially to the representation theory of a finite-dimensional algebra. The authors show that this center \(Z_\varepsilon\) is the algebra of regular functions on a complex algebraic variety and that this variety is a finite ramified covering of the big cell in the group \(G\) with Lie algebra \(\mathfrak g\). The definition of the restricted specialization resembles the well-known construction of a Chevalley-Kostant basis in the classical case.

After constructing these specializations, in Chapters 10 and 11 the authors present results, due mostly to G. Lusztig, on the representations of QUE algebras. Chapter 10 considers the generic case when \(\varepsilon\) is not a root of unity. Among the results here are the quantum analogue of the Frobenius-Schur duality theorem and the role of the Hecke algebra in the representation theory. In Chapter 11 the authors discuss the root-of-unity case, including the conjectural dimension formula, the Kazhdan-Lusztig character formula (proved in Chapter 16 as an extension of the Kohno-Drinfel’d theorem showing that the category of finite-dimensional representations of \(U_h({\mathfrak g})\) is equivalent, as a quasitensor category, to the category of finite-dimensional representations of \(\mathfrak g\) equipped with a quasitensor structure defined via the Knizhnik-Zamolodchikov equation) and tensor products of representations of the restricted specialization \(U^{\text{res}}_\varepsilon ({\mathfrak g})\).

Chapter 12 is concerned with infinite-dimensional quantum groups and their representations, namely the Yangians (deformations of the Lie algebra of maps \(\mathbb{C} \to {\mathfrak g})\) and quantizations of affine (Kac-Moody) algebras. The finite-dimensional representations of these algebras are used to compute the rational and trigonometric solutions of the quantum Yang-Baxter equation.

Chapter 14 applies the general theory to the construction of canonical bases of any (finite-dimensional) representation of a Lie algebra \(\mathfrak g\). Classically this is approached by various methods, but most of them are restricted to the cases when \(\mathfrak g\) is of classical type of just to the case of \({\mathfrak s}{\mathfrak l}_m(\mathbb{C})\). The approaches using quantum groups eliminate these problems and in fact are applicable when \(\mathfrak g\) is an arbitrary symmetrizable Kac-Moody algebra. In this chapter the authors discuss Kashiwara’s construction of crystal bases, and Lusztig’s construction of canonical bases (which turns out to have the same result). Most of the material here is only sketched.

The remaining chapter, Chapter 15, concerns the use of representations of quantum groups by Turaev and others to construct the Jones polynomial, the HOMFLY polynomial, and the Kauffman polynomial from knot theory.

In summary, then, although this book makes considerable demands on the part of the reader, it should be useful for anyone who wants to understand “what the fuss is all about”.

Reviewer: J.S.Joel (Kelly)

### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |