Anthony Giaquinto

Bialgebra actions, twists, and universal deformation formulas

Abstract We use the concept of gauge transformations of quasi-Hopf algebras to study twists of algebraic structures based on actions of a bialgebra and relate this to the theory of universal deformation formulas. We establish new universal deformation formulas which are associated to central extensions of Heisenberg Lie algebras. These formulas are generalizations of the Moyal quantization formula.

Quantum groups and deformation quantization: Explicit approaches and implicit aspects

Deformation quantization, which gives a development of quantum mechanics independent of the operator algebra formulation, and quantum groups, which arose from the inverse scattering method and a study of Yang–Baxter equations, share a common idea abstracted earlier in algebraic deformation theory: that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations, i.e., “quantizations.” In deformation quantization the algebraic object is the algebra of “observables” (functions) on symplectic phase space, whose infinitesimal deformation is the Poisson bracket and global deformation a “star product,” in quantum groups it is a Hopf algebra, generally either of functions on a Lie group or (often its dual in the topological vector space sense, as we briefly explain) a completed universal enveloping algebra of a Lie algebra with, for infinitesimal, a matrix satisfying the modified classical Yang–Baxter equation (MCYBE). Frequently existence proofs a...

Boundary Solutions of the Quantum Yang–Baxter Equation and Solutions in Three Dimensions

Boundary solutions to the quantum Yang–Baxter (qYB) equation are defined to be those in the boundary of (but not in) the variety of solutions to the ‘modified’ qYB equation, the latter being analogous to the modified classical Yang–Baxter (cYB) equation. We construct, for a large class of solutions r to the modified cYB equation, explicit ‘boundary quantizations’, i.e., boundary solutions to the qYB equation of the form I + tr + t2r2 +⋯. In the last section we list and give quantizations for all classical r-matrices in sl(3) ∧ sl(3).

The Principal Element of a Frobenius Lie Algebra

We introduce the notion of the principal element of a Frobenius Lie algebra

Nonstandard Solutions of the Yang–Baxter Equation

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Higher Structures in Geometry and Physics

. The principal element corresponds to a choice of

Topics in Algebraic Deformation Theory

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