Braided Weyl algebras and differential calculus on U(u(2))

Abstract

On any Reflection Equation algebra corresponding to a skew-invertible Hecke symmetry (i.e. a special type solution of the Quantum Yang-Baxter Equation) we define analogs of the partial derivatives. Together with elements of the initial Reflection Equation algebra they generate a "braided analog" of the Weyl algebra. When $q\to 1$, the braided Weyl algebra corresponding to the Quantum Group $U_q(sl(2))$ goes to the Weyl algebra defined on the algebra $\Sym((u(2))$ or that $U(u(2))$ depending on the way of passing to the limit. Thus, we define partial derivatives on the algebra $U(u(2))$, find their "eigenfunctions", and introduce an analog of the Laplace operator on this algebra. Also, we define the "radial part" of this operator, express it in terms of "quantum eigenvalues", and sketch an analog of the de Rham complex on the algebra $U(u(2))$. Eventual applications of our approach are discussed.

LaTex, 18 pages. Accepted in Journal of Geometry and Physics

References in corpus (3)

• Braided Differential Operators on Quantum Algebras
• Cayley-Hamilton theorem for quantum matrix algebras of GL(m|n) type
• Braided affine geometry and q-analogs of wave operators

Cited by in corpus (6)

• Matrix Capelli identities related to Reflection Equation algebra
• The $q$-immanants and higher quantum Capelli identities
• Doubles of associative algebras and their applications
• Generalized Heisenberg algebra, realizations of the $\mathfrak{gl}(n)$ algebra and applications
• Derivatives in noncommutative calculus and deformation property of quantum algebras
• On the quantum argument shift method