Berezin–Toeplitz Quantization and Invariant Symbolic Calculi

It is well known that one can often construct an invariant star-product by expanding the product of two Toeplitz operators asymptotically into a series of another Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin–Toeplitzquantization. We show that on bounded symmetric domains (Hermitian symmetric spaces of noncompact type), one can in fact obtain in a similar way any invariant star-productwhich is G-equivalent to the Berezin–Toeplitz star-product, by using, instead of Toeplitz operators, other suitable assignments fQ _f from compactly supported C functions f to bounded linear operators Q _f on the corresponding Hilbert spaces. (This procedureis referred to as prime quantization by some authors.) Along the way, we establish two technical results which are of interest in their own right, namely a controlled-growth parameter generalization of the classical theorem of Borel on the existence of a function with prescribed derivatives of all orders at a point, and the fact that any invariant bi-differential operator (Hochschild two-cochain) on a bounded symmetric domain automatically maps the Schwartz space into itself.     

Notes on the Elliptic Ruijsenaars Operators

We construct the spaces that the elliptic Ruijsenaars operators act on. It is shown that they are extensible to nonnegative selfadjoint operators on a space of square integrable functions, or preserve a finite dimensional vector space of entire functions. These facts are shown in terms of the R-operators which satisfy the Yang–lBaxter equation. The elliptic Ruijsenaars operators are considered as the elliptic analogues of the Macdonald operators or the difference analogues of the Lamé operators.     

Generating Operators of the Krasil'shchik–Schouten Bracket

It is proved that given a divergence operator on the structural sheaf of graded commutative algebras of a supermanifold, it is possible to construct a generating operator for the Krashil'shchik–Schouten bracket. This is a particular case of the construction of generating operators for a special class of bigraded Gerstenhaber algebras. Also, some comments on the generalization of these results to the context of n-graded Jacobi algebras are included.     

Generalized Hitchin Systems and the Knizhnik–zamolodchikov–bernard Equation on Ellipic Curves

The Knizhnik–Zamolodchikov–Bernard (KZB) equation on an elliptic curve with a marked point is derived by classical Hamiltonian reduction and further quantization. We consider classical Hamiltonian systems on a cotangent bundle to the loop group L(GL(N, C)) extended by the shift operators, to be related to the elliptic module. After reduction, we obtain a Hamiltonian system on a cotangent bundle to the moduli of holomorphic principle bundles and an elliptic module. It is a particular example of generalized Hitchin systems (GHS) which are defined as Hamiltonian systems on cotangent bundles to the moduli of holomorphic bundles and to the moduli of curves. They are extensions of the Hitchin systems by the inclusion the moduli of curves. In contrast with the Hitchin systems, the algebra of integrals are noncommutative on GHS. We discuss the quantization procedure in our example. The quantization of the quadratic integral leads to the KZB equation. We present an explicit form of higher quantum Hitchin integrals which, upon reducing from GHS phase space to the Hitchin phase space, gives a particular example of the Beilinson–Drinfeld commutative algebra of differential operators on the moduli of holomorphic bundles.     

Degenerations and Representations of Twisted Shibukawa–Ueno R-Operators

We study degenerations of the Belavin R-matrices via the infinite dimensional operators defined by Shibukawa–Ueno. We define a two-parameter family of generalizations of the Shibukawa–Ueno R-operators. These operators have finite dimensional representations which include Belavin's R-matrices in the elliptic case, a two-parameter family of twisted affinized Cremmer–Gervais R-matrices in the trigonometric case, and a two-parameter family of twisted (affinized) generalized Jordanian R-matrices in the rational case. We find finite dimensional representations which are compatible with the elliptic to trigonometric and rational degeneration. We further show that certain members of the elliptic family of operators have no finite dimensional representations. These R-operators unify and generalize earlier constructions of Felder and Pasquier, Ding and Hodges, and the authors, and illuminate the extent to which the Cremmer–Gervais R-matrices (and their rational forms) are degenerations of Belavin's R-matrix.     

Subprincipal Symbol for Toeplitz Operators

We establish some subprincipal estimates for Berezin–Toeplitz operators on symplectic compact manifolds. From this, we construct a family of subprincipal symbol maps and we prove that these maps are the only ones satisfying some expected conditions.     

Quantum Kac– Algebras and Vertex Representations

We introduce an affinization of the quantum Kac–Moody algebra associated to a symmetric generalized Cartan matrix. Based on the affinization, we construct a representation of the quantum Kac–Moody algebra by vertex operators from bosonic fields. We also obtain a combinatorial indentity about Hall–Littlewood polynomials.     

Lacunae of Hyperbolic Riesz Kernels and Commutative Rings of Partial Differential Operators

The purpose of this Letter is to demonstrate a close connection between the problem of describing supercomplete commutative rings of partial differential operators (in the sense of Krichever–Veselov) and the theory of lacunae for hyperbolic Riesz kernels. As an application, we give a simple and explicit construction of additional quantum integrals for the generalized Calogero–Moser problem associated with a finite reflection group W and W-invariant integer root multiplicities m. These quantum integrals are not W-invariant and cannot be constructed directly using the standard technique of Dunkl operators.     

From Agmon–Kannai Expansion to Korteweg–de Vries Hierarchy

We present a new method for computation of the Korteweg–de Vries hierarchy via heat invariants of the one-dimensional Schrödinger operator. As a result, new explicit formulas for the KdV hierarchy are obtained. Our method is based on an asymptotic expansion of resolvent kernels of elliptic operators due to S. Agmon and Y. Kannai.     

Noncommutative Dynamics of Random Operators

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra А on a transformation groupoid Γ = E × G where E is the total space of a principal fibre bundle over spacetime, and G a suitable group acting on Γ . We show that every a ∊ А defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita–Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra А which can be used to define a state dependent dynamics; i.e., the pair (А, ϕ), where ϕ is a state on А, is a “dynamic object.” Only if certain additional conditions are satisfied, the Connes–Nikodym–Radon theorem can be applied and the dependence on ϕ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair (А, ϕ) defines the so-called free probability calculus, as developed by Voiculescu and others, with the state ϕ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.