Integrals of motion from quantum toroidal algebras

Quantum Toroidal Algebras and Their Representations

Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac-Moody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic double affine Hecke algebras) to whom they are related via Schur-Weyl duality. In this review paper, we give a glimpse on some aspects of their very rich repre...

Quantum Toroidal Algebras and Their Vertex Representations

We construct the vertex representations of the quantum toroidal algebras Uq(sln+1,tor). In the classical case the vertex representations are not irreducible. However in the quantum case they are irreducible. For n=1, we construct a set of finitely many generators of Uq(sl2,tor).

Twisted Toroidal Lie Algebras

Using n finite order automorphisms on a simple complex Lie algebra we construct twisted n-toroidal Lie algebras. Thus obtaining Lie algebras wich have a rootspace decomposition. For the case n = 2 we list certain simple Lie algebras and their automorphisms, which produce twisted 2-toroidal algebras. In this way we obtain Lie algebras that are related to all Extended Affine Root Systems of K. Sa...

Notes: Toroidal Hecke algebras

0 Quantum Integrals of Motion for Variable Quadratic

We construct integrals of motion for several models of the quantum damped oscillators in a framework of a general approach to the time-dependent Schrödinger equation with variable quadratic Hamiltonians. An extension of the Lewis–Riesenfeld dynamical invariant is given. The time-evolution of the expectation values of the energy related positive operators is determined for the oscillators under ...