Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCharacteristic (Fedosov-)class of a twist constructed by Drinfeld
103
0
0.0
(
0
)
Authors
Jonas Schnitzer
Ask ChatGPT about the research
No Arabic abstract
In a seminal paper Drinfeld explained how to associate to every classical r-matrix for a Lie algebra $lie g$ a twisting element based on $mathcal{U}(lie g)[[hbar]]$, or equivalently a left invariant star product of the corresponding symplectic structure $omega$ on the 1-connected Lie group G of g. In a recent paper, the authors solve the same problem by means of Fedosov quantization. In this short note we provide a connection between the two constructions by computing the characteristic (Fedosov) class of the twist constructed by Drinfeld and proving that it is the trivial class given by $ frac{[omega]}{hbar}$.
rate research
Read More
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.
Fedosovs simple geometrical construction for deformation quantization of symplectic manifolds is generalized in three ways without introducing new variables: (1) The base manifold is allowed to be a supermanifold. (2) The star product does not have to be of Weyl/symmetric or Wick/normal type. (3) The initial geometric structures are allowed to depend on Plancks constant.
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.
We prove that the Drinfeld double of the category of sheaves on an orbifold is equivalent to the category of sheves on the corresponding inertia orbifold.
The Shapovalov determinant for a class of pointed Hopf algebras is calculated, including quantized enveloping algebras, Lusztigs small quantum groups, and quantized Lie superalgebras. Our main tools are root systems, Weyl groupoids, and Lusztig type isomorphisms. We elaborate powerful novel techniques for the algebras at roots of unity, and pass to the general case using a density argument. Key words: Hopf algebra, Nichols algebra, quantum group, representation
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
