##
**Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras.**
*(English)*
Zbl 1117.33300

Kashiwara, Masaki (ed.) et al., Quantum many-body problems and representation theory. Tokyo: Mathematical Society of Japan (ISBN 4-931469-01-9/pbk). MSJ Mem. 1, 1-96 (1998).

This paper is a presentation of a series of lectures in which the author basically gave an exposition of results of his that were published in 1983–1984 and 1989–1995, as well as results of Aomoto, Matsuo, Heckman, Opdam, Varchenko, and a number of other authors.

In the introduction he gives the characteristic general direction of the investigations presented in the lectures, and also touches upon the methodological question of the relation between conceptual (imagined) mathematics and real (concrete) mathematics. The latter, in the author’s opinion, is represented by special functions, numbers and manipulations with them, leading to explicit relations and values for concrete data. The author claims that concrete mathematics is significantly “less” than conceptual mathematics. Different mathematical theories lead to the same special functions. And here one should recall PoincarĂ©’s appeal to look for new special functions, in which the essence of the matter lies. The analogy of different schemes of associating special functions to a conceptual mathematical theory leads to the statement of new interesting problems. For example, the representation theories of various Lie groups and Lie algebras, through the notion of spherical functions, characters of representations, Clebsch-Gordan coefficients, generating functions, lead to various interesting special functions.

The lectures are concentrated around the following themes:

(a) Compatibility conditions for the affine Knizhnik-Zamolodchikov (AKZ) equations and degenerate affine Hecke algebras.

Namely, the AKZ equations have the form \[ {{\partial\Phi}\over{\partial u_i}}=\Bigl(k\sum_{\alpha\in\Sigma_+}\nu_\alpha^i {{s_\alpha}\over{e^{(u,\alpha)}-1}}+x_i\Bigr)\Phi,\quad 1\leq i\leq n, \] where \(\Sigma\subset\mathbb{R}^n\) is a root system in \(\mathbb{R}^n\) and \(\Sigma_+\) the set of positive roots in \(\Sigma\), \(u=(u_1,\dots,u_n)\in\mathbb{C}^n\), \((\cdot,\cdot)\) is the standard scalar product. Further, \(W^{\text{a}}(\Sigma)\) is the affine Weyl group of the root system \(\Sigma\).

The AKZ equations are compatible (that is, they have a common solution) if and only if \(s_1=s_{\alpha_1},\dots,s_n=s_{\alpha_n}\) (where the \(\alpha_i\) are the simple roots) and \(x_1,\dots,x_n\) satisfy the relations of a degenerate affine Hecke algebra \([x_i,x_j]=0\) for all \(i,j\); \([s_i,x_j]=0\) for \(i\neq j\); \(s_ix_i-x_is_i=k\).

(b) The fundamental group of the domain of definition of the AKZ equations and affine Hecke algebras.

The relations of the fundamental group \[ \pi_1\Bigl(\Bigl(\mathbb{C}^n\setminus \bigcup_{\alpha\in\Sigma_+}\{u\in\mathbb{C}^n\colon e^{(\alpha,u)}-1=0\}\Bigr)/W^{\text{a}}(\Sigma)\Bigr), \] augmented by the quadratic relation \((T_i-t)(T_i+t^{-1})=0\), give a corepresentation of an affine Hecke algebra.

(c) Monodromy of the AKZ equations and isomorphism of affine and degenerate Hecke algebras.

The association of the monodromy of a special solution of an AKZ equation with a generator of an affine Hecke algebra gives the indicated isomorphism of algebras.

(d) The Dunkl operators and the isomorphism between the spaces of solutions of the AKZ equations and the spaces of eigenstates of multi-particle quantum problems.

The Dunkl operators have the form \[ D_i={\partial\over{\partial_i}} -k\Bigl (\sum_{\alpha\in\Sigma_+}\nu_\alpha^i{{s_\alpha} \over{e^{(u,\alpha)}-1}}+x_i\Bigr),\quad 1\leq i\leq n. \] These operators commute and we can substitute them into any \(W(\Sigma)\)-invariant polynomial. The resulting operator polynomials define a multi-particle integrable quantum system (as a rule the problem is equivalent to the Calogero-Sutherland system). Its eigenstates then form a space that is isomorphic to the space of solutions of the AKZ equations.

(e) Difference (quantum) analogues of the affine Knizhnik-Zamolodchikov equations and an isomorphism between their solution spaces and the spaces of eigenfunctions of the Macdonald eigenvalue problem.

(f) Double affine Hecke algebras and Macdonald polynomials.

(g) The classical \(r\)-matrices, the compatibility conditions for the AKZ equations and the construction of \(r\)-matrices using the factoring subalgebras in Kac-Moody algebras.

The affine KZ equations are written in terms of a \(W\)-invariant \(r\)-matrix \[ \partial_i\Phi(u)=\Bigl (\sum_{\alpha\in\Sigma_+}k_\alpha (e_i,\alpha)r_\alpha(u,\alpha)\Bigr)\Phi(u). \] The compatibility conditions for AKZ equations associated with a root system \(\Sigma\) are almost equivalent to the classical Yang-Baxter equations for this root system. Sufficient conditions for compatibility are indicated.

(h) Quantum elliptic multi-particle problems and dual affine KZ equations (an isomorphism between the solution spaces and the spaces of eigenstates of the corresponding multi-particle problem, given through elliptic Dunkl operators).

(i) Integral representations of solutions of KZ equations of various types.

(j) Discussion of the possibility of extending these results to the case of Knizhnik-Zamolodchikov-Bernard equations on elliptic curves and curves of higher genus.

All of these results are obtained for the AKZ equations and Hecke algebras associated with an arbitrary root system. The heart of the matter in many cases is first explained using meaningful examples corresponding to the simplest root system \(A_1\), and then presented in full generality. In many sections conjectures and unsolved problems are stated, encouraging interest in the development of the approach based on the theory of Hecke algebras. The main results are supplied with rather detailed proofs. The lectures unite facts scattered over many journal publications and are presented in a form that attracts the reader’s interest and enables an understanding of the technique and results of these theories.

For the entire collection see [Zbl 0919.00049].

In the introduction he gives the characteristic general direction of the investigations presented in the lectures, and also touches upon the methodological question of the relation between conceptual (imagined) mathematics and real (concrete) mathematics. The latter, in the author’s opinion, is represented by special functions, numbers and manipulations with them, leading to explicit relations and values for concrete data. The author claims that concrete mathematics is significantly “less” than conceptual mathematics. Different mathematical theories lead to the same special functions. And here one should recall PoincarĂ©’s appeal to look for new special functions, in which the essence of the matter lies. The analogy of different schemes of associating special functions to a conceptual mathematical theory leads to the statement of new interesting problems. For example, the representation theories of various Lie groups and Lie algebras, through the notion of spherical functions, characters of representations, Clebsch-Gordan coefficients, generating functions, lead to various interesting special functions.

The lectures are concentrated around the following themes:

(a) Compatibility conditions for the affine Knizhnik-Zamolodchikov (AKZ) equations and degenerate affine Hecke algebras.

Namely, the AKZ equations have the form \[ {{\partial\Phi}\over{\partial u_i}}=\Bigl(k\sum_{\alpha\in\Sigma_+}\nu_\alpha^i {{s_\alpha}\over{e^{(u,\alpha)}-1}}+x_i\Bigr)\Phi,\quad 1\leq i\leq n, \] where \(\Sigma\subset\mathbb{R}^n\) is a root system in \(\mathbb{R}^n\) and \(\Sigma_+\) the set of positive roots in \(\Sigma\), \(u=(u_1,\dots,u_n)\in\mathbb{C}^n\), \((\cdot,\cdot)\) is the standard scalar product. Further, \(W^{\text{a}}(\Sigma)\) is the affine Weyl group of the root system \(\Sigma\).

The AKZ equations are compatible (that is, they have a common solution) if and only if \(s_1=s_{\alpha_1},\dots,s_n=s_{\alpha_n}\) (where the \(\alpha_i\) are the simple roots) and \(x_1,\dots,x_n\) satisfy the relations of a degenerate affine Hecke algebra \([x_i,x_j]=0\) for all \(i,j\); \([s_i,x_j]=0\) for \(i\neq j\); \(s_ix_i-x_is_i=k\).

(b) The fundamental group of the domain of definition of the AKZ equations and affine Hecke algebras.

The relations of the fundamental group \[ \pi_1\Bigl(\Bigl(\mathbb{C}^n\setminus \bigcup_{\alpha\in\Sigma_+}\{u\in\mathbb{C}^n\colon e^{(\alpha,u)}-1=0\}\Bigr)/W^{\text{a}}(\Sigma)\Bigr), \] augmented by the quadratic relation \((T_i-t)(T_i+t^{-1})=0\), give a corepresentation of an affine Hecke algebra.

(c) Monodromy of the AKZ equations and isomorphism of affine and degenerate Hecke algebras.

The association of the monodromy of a special solution of an AKZ equation with a generator of an affine Hecke algebra gives the indicated isomorphism of algebras.

(d) The Dunkl operators and the isomorphism between the spaces of solutions of the AKZ equations and the spaces of eigenstates of multi-particle quantum problems.

The Dunkl operators have the form \[ D_i={\partial\over{\partial_i}} -k\Bigl (\sum_{\alpha\in\Sigma_+}\nu_\alpha^i{{s_\alpha} \over{e^{(u,\alpha)}-1}}+x_i\Bigr),\quad 1\leq i\leq n. \] These operators commute and we can substitute them into any \(W(\Sigma)\)-invariant polynomial. The resulting operator polynomials define a multi-particle integrable quantum system (as a rule the problem is equivalent to the Calogero-Sutherland system). Its eigenstates then form a space that is isomorphic to the space of solutions of the AKZ equations.

(e) Difference (quantum) analogues of the affine Knizhnik-Zamolodchikov equations and an isomorphism between their solution spaces and the spaces of eigenfunctions of the Macdonald eigenvalue problem.

(f) Double affine Hecke algebras and Macdonald polynomials.

(g) The classical \(r\)-matrices, the compatibility conditions for the AKZ equations and the construction of \(r\)-matrices using the factoring subalgebras in Kac-Moody algebras.

The affine KZ equations are written in terms of a \(W\)-invariant \(r\)-matrix \[ \partial_i\Phi(u)=\Bigl (\sum_{\alpha\in\Sigma_+}k_\alpha (e_i,\alpha)r_\alpha(u,\alpha)\Bigr)\Phi(u). \] The compatibility conditions for AKZ equations associated with a root system \(\Sigma\) are almost equivalent to the classical Yang-Baxter equations for this root system. Sufficient conditions for compatibility are indicated.

(h) Quantum elliptic multi-particle problems and dual affine KZ equations (an isomorphism between the solution spaces and the spaces of eigenstates of the corresponding multi-particle problem, given through elliptic Dunkl operators).

(i) Integral representations of solutions of KZ equations of various types.

(j) Discussion of the possibility of extending these results to the case of Knizhnik-Zamolodchikov-Bernard equations on elliptic curves and curves of higher genus.

All of these results are obtained for the AKZ equations and Hecke algebras associated with an arbitrary root system. The heart of the matter in many cases is first explained using meaningful examples corresponding to the simplest root system \(A_1\), and then presented in full generality. In many sections conjectures and unsolved problems are stated, encouraging interest in the development of the approach based on the theory of Hecke algebras. The main results are supplied with rather detailed proofs. The lectures unite facts scattered over many journal publications and are presented in a form that attracts the reader’s interest and enables an understanding of the technique and results of these theories.

For the entire collection see [Zbl 0919.00049].

Reviewer: V. Leksin (Kolomna) (MR1724948)

### MSC:

33C52 | Orthogonal polynomials and functions associated with root systems |

32G34 | Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation) |

20C08 | Hecke algebras and their representations |

05E05 | Symmetric functions and generalizations |

17B22 | Root systems |

17B38 | Yang-Baxter equations and Rota-Baxter operators |

17B65 | Infinite-dimensional Lie (super)algebras |

17B81 | Applications of Lie (super)algebras to physics, etc. |

81R12 | Groups and algebras in quantum theory and relations with integrable systems |