##
**Lectures on algebraic quantum groups.**
*(English)*
Zbl 1027.17010

Quantum groups can be studied from different angles, and many of these deserve the publication of books accounting their main achievements. This book is mainly concerned with the representation theory of quantum algebras from a ring-theoretical point of view.

More precisely, two main types of algebras are studied: “quantized coordinate algebras” like the quantized coordinate algebra of an algebraic semisimple group, the algebra of functions on quantum linear spaces, or on quantum matrices, and so on; and quantized enveloping algebras of Lie algebras. There are books focussing on quantized coordinate algebras of semisimple algebraic groups; [L. I. Korogodskij and Y. S. Soibelman, Algebras of functions on quantum groups. Part I. Mathematical Surveys and Monographs, 56. American Mathematical Society, Providence, RI (1998; Zbl 0923.17017)] has a more geometrical and analytical flavor; and [A. Joseph, Quantum groups and their primitive ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin (1995; Zbl 0808.17004)] is in the spirit of algebraic Lie theory. The origin of research in this area is the seminal work of Soibelman, Vaksman and Levendorskii on representations of the quantum algebra of functions on a compact quantum group. The guiding principle is that the representation theory of a quantized algebra is governed by the Poisson geometry of its classical limit, in particular by the foliation by symplectic leaves (Soibelman et al. attribute this idea to Drinfeld). This principle generalizes the celebrated orbit method of Kirillov. It fits perfectly in the case of the “standard” compact quantum group, and less accurately in the multiparametric case. An account is given in [Korogodskij and Soibelman, loc. cit.]; see also the references therein. The results of Soibelman et al. inspired Hodges and Levasseur, who stated a conjecture with an analogous claim for the spectrum of the quantized coordinate algebra of an algebraic semisimple group at a generic parameter, and proved it in the case of \(sl(n)\) [T. Hodges and T. Levasseur, J. Algebra 168, 455-468 (1994; Zbl 0814.17012)]. The conjecture was subsequently proved by A. Joseph in [J. Algebra 169, 441-511 (1994; Zbl 0814.17013)]; in his book he gives a full account of these results, as well as of many other results from his previous work on the structure of the quantized enveloping algebra, and much more. See also [T. Hodges, T. Levasseur and M. Toro, Adv. Math. 126, 52-92 (1997; Zbl 0878.17009)]. Finally, all this work motivated also the study of the quantized coordinate algebra of an algebraic semisimple group at a root of one [C. De Concini and V. Lyubashenko, Adv. Math. 108, 205-262 (1994; Zbl 0846.17008), C. De Concini and C. Procesi, Aust. Math. Soc. Lect. Ser. 9, 127-160 (1997; Zbl 0901.17005)]. Remarkably, the Poisson geometry of the classical limit still plays an important rôle in the representation theory of this algebra.

There are also books on quantized enveloping algebras of Lie algebras: the difficult but magnificent [G. Lusztig, Introduction to quantum groups, Progress in Mathematics 110, Boston, Birkhäuser (1993; Zbl 0788.17010)] and [J. C. Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics 6. Providence, RI, American Mathematical Society (1996; Zbl 0842.17012)]. There are other books on quantum groups with emphasis on other points of view, from Hopf algebra theory to physics. While the book of Joseph is an advanced research monograph, the book under review is aimed at beginners and is based on an advanced course given by the authors at the Centro di Ricerca Matematica, in Barcelona. Proofs of the main results are generally not given, but instead there is a general overview of the developments in the area.

The book is divided into three parts. The first part contains preliminary material, including the definition of various “quantum algebras”; several appendices supplement this part with brief introductions to material needed later. The second part provides a general framework to study the spectrum of a quantum algebra “at a generic \(q\)”. Namely, let us assume that the base field is \(\mathbb C\). Let \(H = (\mathbb C^{\times})^{\theta}\) be a torus and let \(X(H) \simeq \mathbb Z^{\theta}\) be the group of its characters. Let \(A\) be an algebra provided with a rational action of \(H\) by algebra automorphisms, or equivalently with a \(X(H)\)-grading. An \(H\)-ideal is an ideal stable under the action of \(H\), or equivalently, a homogeneous ideal. If \(I\) is an arbitrary ideal of \(A\), the largest \(H\)-ideal contained in \(I\) is denoted \((I:H)\). An \(H\)-ideal \(J\) is \(H\)-prime if whenever the product of two \(H\)-ideals is contained in \(J\), one of them is also contained in \(J\). If \(P\) is prime, then \((P:H)\) is \(H\)-prime. Let \(H\)-\(\text{Spec }A = \{H\)-prime ideals of \(A\} \). If \(J\) is an \(H\)-prime ideal, then set \(\text{Spec }_J A = \{P \in \text{Spec } A \mid (P:H) = J\}\). The \(H\) stratification of \(\text{Spec } A\) is the disjoint union \(\text{coprod}_{J \in H\text{-Spec } A} \text{Spec}_J A\). Assume that \(A\) is noetherian. Then any \(H\)-prime ideal is prime, and \(\text{Spec}_J A\) is homeomorphic to the prime spectrum of some Laurent polynomial algebra. Furthermore, if \(A\) is a suitable Ore extension then \(H\)-\(\text{Spec } A\) is finite, so that there are a finite number of \(H\)-strata in \(\text{Spec } A\). Thus, a fairly close approximation to \({\text{Spec}} A\) in this case would be to describe \(H\)-\(\text{Spec } A\), i.e. the space of graded prime ideals. This second part contains also finer results on the interrelation between prime and \(H\)-prime ideals. For instance, if \(H\)-\(\text{Spec } A\) is finite and \(A\) “satisfies the Nullstellensatz” (see Definition II.7.14) then the Dixmier-Moeglin equivalence holds (primitive, locally closed and rational prime ideals are the same, and coincide with prime ideals maximal in their strata).

The subject of the third part is “quantum algebras at a root of 1”, the main examples being quantized enveloping algebras and quantum algebras of functions. In the first chapter of this part, some basic theory of finite-dimensional modules over an affine PI algebra is developed. The next two chapters are accounts of the representation theory of \(U_{\varepsilon}(sl(2))\) and \(\mathcal O_{\varepsilon}(SL(2))\), respectively. Here and below \(\varepsilon\) is a root of 1. In the next chapter, the notion of Hopf PI triple is presented, along with some basic properties. A Hopf PI triple is a pair \((H, Z_0)\) where \(H\) is an affine Hopf algebra and \(Z_0\) is a central Hopf subalgebra such that \(H\) is a prime ring and a finitely generated \(Z_0\)-module. After a quick introduction to Poisson geometry and quantization in Chapter 5, the next two chapters are expositions of the representation theory of \(U_{\varepsilon}(g)\) and \(\mathcal O_{\varepsilon}(G)\), respectively; here \(g\) is a simple Lie algebra and \(G\) is its corresponding simply-connected algebraic group. The last chapters contain discussions on homological properties, the Azumaya locus and finally an interesting collection of open problems and their actual state.

More precisely, two main types of algebras are studied: “quantized coordinate algebras” like the quantized coordinate algebra of an algebraic semisimple group, the algebra of functions on quantum linear spaces, or on quantum matrices, and so on; and quantized enveloping algebras of Lie algebras. There are books focussing on quantized coordinate algebras of semisimple algebraic groups; [L. I. Korogodskij and Y. S. Soibelman, Algebras of functions on quantum groups. Part I. Mathematical Surveys and Monographs, 56. American Mathematical Society, Providence, RI (1998; Zbl 0923.17017)] has a more geometrical and analytical flavor; and [A. Joseph, Quantum groups and their primitive ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin (1995; Zbl 0808.17004)] is in the spirit of algebraic Lie theory. The origin of research in this area is the seminal work of Soibelman, Vaksman and Levendorskii on representations of the quantum algebra of functions on a compact quantum group. The guiding principle is that the representation theory of a quantized algebra is governed by the Poisson geometry of its classical limit, in particular by the foliation by symplectic leaves (Soibelman et al. attribute this idea to Drinfeld). This principle generalizes the celebrated orbit method of Kirillov. It fits perfectly in the case of the “standard” compact quantum group, and less accurately in the multiparametric case. An account is given in [Korogodskij and Soibelman, loc. cit.]; see also the references therein. The results of Soibelman et al. inspired Hodges and Levasseur, who stated a conjecture with an analogous claim for the spectrum of the quantized coordinate algebra of an algebraic semisimple group at a generic parameter, and proved it in the case of \(sl(n)\) [T. Hodges and T. Levasseur, J. Algebra 168, 455-468 (1994; Zbl 0814.17012)]. The conjecture was subsequently proved by A. Joseph in [J. Algebra 169, 441-511 (1994; Zbl 0814.17013)]; in his book he gives a full account of these results, as well as of many other results from his previous work on the structure of the quantized enveloping algebra, and much more. See also [T. Hodges, T. Levasseur and M. Toro, Adv. Math. 126, 52-92 (1997; Zbl 0878.17009)]. Finally, all this work motivated also the study of the quantized coordinate algebra of an algebraic semisimple group at a root of one [C. De Concini and V. Lyubashenko, Adv. Math. 108, 205-262 (1994; Zbl 0846.17008), C. De Concini and C. Procesi, Aust. Math. Soc. Lect. Ser. 9, 127-160 (1997; Zbl 0901.17005)]. Remarkably, the Poisson geometry of the classical limit still plays an important rôle in the representation theory of this algebra.

There are also books on quantized enveloping algebras of Lie algebras: the difficult but magnificent [G. Lusztig, Introduction to quantum groups, Progress in Mathematics 110, Boston, Birkhäuser (1993; Zbl 0788.17010)] and [J. C. Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics 6. Providence, RI, American Mathematical Society (1996; Zbl 0842.17012)]. There are other books on quantum groups with emphasis on other points of view, from Hopf algebra theory to physics. While the book of Joseph is an advanced research monograph, the book under review is aimed at beginners and is based on an advanced course given by the authors at the Centro di Ricerca Matematica, in Barcelona. Proofs of the main results are generally not given, but instead there is a general overview of the developments in the area.

The book is divided into three parts. The first part contains preliminary material, including the definition of various “quantum algebras”; several appendices supplement this part with brief introductions to material needed later. The second part provides a general framework to study the spectrum of a quantum algebra “at a generic \(q\)”. Namely, let us assume that the base field is \(\mathbb C\). Let \(H = (\mathbb C^{\times})^{\theta}\) be a torus and let \(X(H) \simeq \mathbb Z^{\theta}\) be the group of its characters. Let \(A\) be an algebra provided with a rational action of \(H\) by algebra automorphisms, or equivalently with a \(X(H)\)-grading. An \(H\)-ideal is an ideal stable under the action of \(H\), or equivalently, a homogeneous ideal. If \(I\) is an arbitrary ideal of \(A\), the largest \(H\)-ideal contained in \(I\) is denoted \((I:H)\). An \(H\)-ideal \(J\) is \(H\)-prime if whenever the product of two \(H\)-ideals is contained in \(J\), one of them is also contained in \(J\). If \(P\) is prime, then \((P:H)\) is \(H\)-prime. Let \(H\)-\(\text{Spec }A = \{H\)-prime ideals of \(A\} \). If \(J\) is an \(H\)-prime ideal, then set \(\text{Spec }_J A = \{P \in \text{Spec } A \mid (P:H) = J\}\). The \(H\) stratification of \(\text{Spec } A\) is the disjoint union \(\text{coprod}_{J \in H\text{-Spec } A} \text{Spec}_J A\). Assume that \(A\) is noetherian. Then any \(H\)-prime ideal is prime, and \(\text{Spec}_J A\) is homeomorphic to the prime spectrum of some Laurent polynomial algebra. Furthermore, if \(A\) is a suitable Ore extension then \(H\)-\(\text{Spec } A\) is finite, so that there are a finite number of \(H\)-strata in \(\text{Spec } A\). Thus, a fairly close approximation to \({\text{Spec}} A\) in this case would be to describe \(H\)-\(\text{Spec } A\), i.e. the space of graded prime ideals. This second part contains also finer results on the interrelation between prime and \(H\)-prime ideals. For instance, if \(H\)-\(\text{Spec } A\) is finite and \(A\) “satisfies the Nullstellensatz” (see Definition II.7.14) then the Dixmier-Moeglin equivalence holds (primitive, locally closed and rational prime ideals are the same, and coincide with prime ideals maximal in their strata).

The subject of the third part is “quantum algebras at a root of 1”, the main examples being quantized enveloping algebras and quantum algebras of functions. In the first chapter of this part, some basic theory of finite-dimensional modules over an affine PI algebra is developed. The next two chapters are accounts of the representation theory of \(U_{\varepsilon}(sl(2))\) and \(\mathcal O_{\varepsilon}(SL(2))\), respectively. Here and below \(\varepsilon\) is a root of 1. In the next chapter, the notion of Hopf PI triple is presented, along with some basic properties. A Hopf PI triple is a pair \((H, Z_0)\) where \(H\) is an affine Hopf algebra and \(Z_0\) is a central Hopf subalgebra such that \(H\) is a prime ring and a finitely generated \(Z_0\)-module. After a quick introduction to Poisson geometry and quantization in Chapter 5, the next two chapters are expositions of the representation theory of \(U_{\varepsilon}(g)\) and \(\mathcal O_{\varepsilon}(G)\), respectively; here \(g\) is a simple Lie algebra and \(G\) is its corresponding simply-connected algebraic group. The last chapters contain discussions on homological properties, the Azumaya locus and finally an interesting collection of open problems and their actual state.

### MSC:

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

16W35 | Ring-theoretic aspects of quantum groups (MSC2000) |

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