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Register a new userTwisted bialgebroids versus bialgebroids from a Drinfeld twist
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Bialgebroids (resp. Hopf algebroids) are bialgebras (Hopf algebras) over noncommutative rings. Drinfeld twist techniques are particularly useful in the (deformation) quantization of Lie algebras as well as underlying module algebras (=quantum spaces). Smash product construction combines these two into the new algebra which, in fact, does not depend on the twist. However, we can turn it into bialgebroid in the twist dependent way. Alternatively, one can use Drinfeld twist techniques in a category of bialgebroids. We show that both techniques indicated in the title: twisting of a bialgebroid or constructing a bialgebroid from the twisted bialgebra give rise to the same result in the case of normalized cocycle twist. This can be useful for better description of a quantum deformed phase space. We argue that within this bialgebroid framework one can justify the use of deformed coordinates (i.e. spacetime noncommutativity) which are frequently postulated in order to explain quantum gravity effects.
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We introduce Lie-Nijenhuis bialgebroids as Lie bialgebroids endowed with an additional derivation-like object. They give a complete infinitesimal description of Poisson-Nijenhuis groupoids, and key examples include Poisson-Nijenhuis manifolds, holomorphic Lie bialgebroids and flat Lie bialgebra bundles. To achieve our goal we develop a theory of generalized derivations and their duality, extending the well-established theory of derivations on vector bundles.
Two one-parameter families of twists providing kappa-Minkowski * -product deformed spacetime are considered: Abelian and Jordanian. We compare the derivation of quantum Minkowski space from two perspectives. The first one is the Hopf module algebra point of view, which is strictly related with Drinfelds twisting tensor technique. The other one relies on an appropriate extension of deformed realizations of nondeformed Lorentz algebra by the quantum Minkowski algebra. This extension turns out to be de Sitter Lie algebra. We show the way both approaches are related. The second path allows us to calculate deformed dispersion relations for toy models ensuing from different twist parameters. In the Abelian case one recovers kappa-Poincare dispersion relations having numerous applications in doubly special relativity. Jordanian twists provide a new type of dispersion relations which in the minimal case (related to Weyl-Poincare algebra) takes an energy-dependent linear mass deformation form.
In this thesis we give obstructions for Drinfeld twist deformation quantization on several classes of symplectic manifolds. Motivated from this quantization procedure, we further construct a noncommutative Cartan calculus on any braided commutative algebra, as well as an equivariant Levi-Civita covariant derivative for any non-degenerate equivariant metric. This generalizes and unifies the Cartan calculus on a smooth manifold and the Cartan calculus on twist star product algebras. We prove that the Drinfeld functor leads to equivalence classes in braided commutative geometry and commutes with submanifold algebra projection.
We consider closed XXX spin chains with broken total spin $U(1)$ symmetry within the framework of the modified algebraic Bethe ansatz. We study multiple actions of the modified monodromy matrix entries on the modified Bethe vectors. The obtained formulas of the multiple actions allow us to calculate the scalar products of the modified Bethe vectors. We find an analog of Izergin-Korepin formula for the scalar products. This formula involves modified Izergin determinants and can be expressed as sums over partitions of the Bethe parameters.
We consider abelian twisted loop Toda equations associated with the complex general linear groups. The Dodd--Bullough--Mikhailov equation is a simplest particular case of the equations under consideration. We construct new soliton solutions of these Toda equations using the method of rational dressing.
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