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 → 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.Essentially, such a pseudogroup is the famous RTT algebra (see [FRT]) associated with a given braiding, i.e. an invertible operator R : V ⊗2 → V ⊗2 , satisfying the Quantum Yang-Baxter EquationHereafter V is a vector space over the ground field K (= R or C) and I is the identity operator. In [W] a general scheme of defining differential forms and the de Rham complex on a matrix pseudogroup was suggested. Also, the author considered analogs of vector fields, introducing them by duality. In some subsequent publications (see f.e. [IP, FP]) the algebra generated by such fields in the case related to the QG U q (sl(m)) was identified as a (modified) Reflection Equation (RE) algebra 1 (see below).Recently, we have (partially 2 ) generalized this construction replacing the RTT algebra by other quantum matrix algebras, in particular, by an RE algebra. Namely, taking a copy of the RE algebra (denoted M) as a quantum function algebra, we treated another copy of this algebra (denoted L) as an analog of the one-sided or adjoint differential operators. Below, we deal with the total algebra (denoted B(L, M)) where we assume that M is equipped with a left (rightinvariant) action of L.Besides, the braiding R coming in the definition of the algebra B(L, M) is taken to be a Hecke symmetry. This means that it is subject to the equationwhere q ∈ K is assumed to be generic. In particular, such a Hecke symmetry comes from the QG U q (sl(m)). This Hecke symmetry and all related objects will be called standard. The RTT and RE algebras associated with a standard Hecke symmetry are deformations of the algebra Sym(gl(m)). With the use of other Hecke symmetries we can get analogous deformations of the super-algebra Sym(gl(m|n)).Note that any RE algebra associated with a Hecke symmetry has another basis in which the permutation relations between basic elements are quadratic-linear. So, the RE algebra in this basis becomes more similar to the enveloping algebra U (gl(m|n)). We call this quadraticlinear algebra the modified RE (mRE) algebra 3 . Namely in this form the RE algebra comes in constructing quantum analogs of vector fields.Besides, in [GPS2] we have con...

Noncommutative Geometry and Dynamical Models on U(u(2))Background

“…Emphasize that the action of the partial derivatives is not subject to the classical Leibniz rule. By contrary, these derivatives meet a new version of the Leibniz rule which was found by Meljanac and Skoda [10] (see also [6] where we have given another but equivalent form of this Leibniz rule). In order to describe this form of the Leibniz rule, consider the following coproduct in the algebra D ( ) 1 1 .…”

“…A couple of years ago we suggested a new type of differential calculus [6,7] on the algebras U(gl(m)), the super-algebras U(gl(m|n)), and their braided analogs the so-called modified Reflection Equation algebra L(R) related to a Hecke symmetry R. The calculus is based on so-called permutation relations between elements of the NC algebra in question and partial derivatives on it.…”

“…First, recall that according to Gurevich D et al, [6,7] the Weyl algebra W(U(gl(m) h )) is a unital associative algebra generated by …”

Derivatives in noncommutative calculus and deformation property of quantum algebras

“…First, consider the Lie subalgebra su(2) ⊂ u(2) . As follows from [GPS2] for any polynomial of the form f (x, y, z) = f 1 (x) f 2 (y) f 3 (z) the action of the partial derivative ∂ x is defined by…”

“…2. This algebra can be equipped with a braided bi-algebra structure (see [GPS2] for a definition). This structure is determined by the usual counit and the coproduct such that for = 1 it has the form…”