The two-parameter deformation ofGL(2), its differential calculus, and Lie algebra

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 are known but explicit formulas useful for physical applications have been difficult to extract. One success here comes from “universal deformation formulas” (UDFs), expressions built from a Lie algebra which deform any algebra on which the Lie algebra operates as derivations. The most famous of these is the Moyal product, a special case of a class in which the Lie algebra is Abelian. Another comes from recognition that the Belavin–Drinfel’d solutions to the MCYBE are, in fact, infinitesimal deformations for which, in the case of the special linear groups, it is possible to give explicit formulas for the corresponding quantum Yang–Baxter equations. This review paper discusses, necessarily in brief, these and related topics, including “twisting” as a form of UDF and finding formulas for “preferred deformations” of Hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation.

The Yang-Baxter algebras (YBA) are introduced in a general framework stressing their power to exactly solve the lattice models associated to them. The algebraic Bethe Ansatz is developed as an eigenvector construction based on the YBA. The six-vertex model solution is given explicitly. The generalization of YB algebras to face language is considered. The algebraic BA for the SOS model of Andrews, Baxter and Forrester is described using these face YB algebras. It is explained how these lattice models yield both solvable massive QFT and conformal models in appropiated scaling (continuous) limits within the lattice light-cone approach. This approach permit to define and solve rigorously massive QFT as an appropiate continuum limit of gapless vertex models. The deep links between the YBA and Lie algebras are analyzed including the quantum groups that underly the trigonometric/hyperbolic YBA. Braid and quantum groups are derived from trigonometric/hyperbolic YBA in the limit of infinite spectral parameter. To conclude, some recent developments in the domain of integrable theories are summarized.

The q-deformed differential calculus is proposed and analysed in the framework of quantum plane. The q-deformed differential operator algebra is investigated and applied in the investigation of the quantum group SUq(2) and its representations. The generalization to n-dimensional differential calculus is made and shown to be a new solution to quantum plane by providing a new solution of the Yang-Baxter equation.

We propose that Baxter's Z-invariant six-vertex model at the rational gl(2) point on a planar but in general not rectangular lattice provides a way to study Yangian invariants. These are identified with eigenfunctions of certain monodromies of an auxiliary inhomogeneous spin chain. As a consequence they are special solutions to the eigenvalue problem of the associated transfer matrix. Excitingly, this allows to construct them using Bethe ansatz techniques. Conceptually, our construction generalizes to general (super) Lie algebras and general representations. Here we present the explicit form of sample invariants for totally symmetric, finite-dimensional representations of gl(n) in terms of oscillator algebras. In particular, we discuss invariants of three- and four-site monodromies that can be understood respectively as intertwiners of the bootstrap and Yang–Baxter equation. We state a set of functional relations significant for these representations of the Yangian and discuss their solutions in terms of Bethe roots. They arrange themselves into exact strings in the complex plane. In addition, it is shown that the sample invariants can be expressed analogously to Graßmannian integrals. This aspect is closely related to a recent on-shell formulation of scattering amplitudes in planar N=4 super Yang–Mills theory.

We construct an associative algebra with a decomposition into the direct sum of the underlying vector spaces of another associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant or the natural antisymmetric bilinear form is a Connes cocycle. The former is called a double construction of a Frobenius algebra and the latter is called a double construction of the Connes cocycle, which is interpreted in terms of dendriform algebras. Both of them are equivalent to a kind of bialgebras, namely, antisymmetric infinitesimal bialgebras and dendriform D-bialgebras, respectively. In the coboundary cases, our study leads to what we call associative Yang–Baxter equation in an associative algebra and D -equation in a dendriform algebra, respectively, which are analogues of the classical Yang–Baxter equation in a Lie algebra. We show that an antisymmetric solution of the associative Yang–Baxter equation corresponds to the antisymmetric part of a certain operator called \mathcal O -operator which gives a double construction of a Frobenius algebra, whereas a symmetric solution of the D -equation corresponds to the symmetric part of an \mathcal O -operator which gives a double construction of the Connes cocycle. By comparing antisymmetric infinitesimal bialgebras and dendriform D-bialgebras, we observe that there is a clear analogy between them. Due to the correspondences between certain symmetries and antisymmetries appearing in this analogy, we regard it as a kind of duality.

Infinite-dimensional Lie bialgebras via affinization of perm bialgebras and pre-Lie bialgebras

We investigate the algebras of the non-local charges and their generating functionals (the monodromy matrices) in classical and quantum non-linear σ models. In the case of the classical chiral σ models it turns out that there exists no definition of the Poisson bracket of two monodromy matrices satisfying antisymmetry and the Jacobi identity. Thus, the classical non-local charges do not generate a Lie algebra. In the case of the quantum O(N) non-linear σ model, we explicitly determine the conserved quantum monodromy operator from a factorization principle together withP,T, and O(N) invariance. We give closed expressions for its matrix elements between asymptotic states in terms of the known two-particleS-matrix. The quantumR-matrix of the model is found. The quantum non-local charges obey a quadratic Lie algebra governed by a Yang-Baxter equation.

The author presents a systematical study of left-covariant differential calculus on the quantum group GLp,q(2), a two-parameter deformation of the general linear group. In particular, the author explicitly constructs the most general bicovariant calculus. It depends on a new parameter s. The corresponding Lie algebra of left-coinvariant vector fields is a two-parameter deformation of the classical Lie algebra, p and q appear only in the combination r=pq. In the limit p,q to 1 one obtains non-standard bicovariant differential calculi on the classical Lie group.

A generalization of the results of Rasetti and Zanardi concerning avoiding\nerrors in quantum computers by using states preserved by evolution is\npresented. The concept of dynamical symmetry is generalized from the level of\nclassical Lie algebras and groups to the level of dynamical symmetry based on\nquantum Lie algebras and quantum groups (in the sense of Woronowicz). A natural\nconnection is proved between states preserved by representations of a quantum\ngroup and states preserved by evolution with dynamical symmetry of the\nappropriate universal enveloping algebra. Illustrative examples are discussed.\n

Consider a decomposition n = n 1 ⊕ ⋯ ⊕ n r \mathfrak {n}= \mathfrak {n}_1 \oplus \cdots \oplus \mathfrak {n}_r of the positive nilradical of a complex semisimple Lie algebra of rank r r , where each n k \mathfrak {n}_k is a module under an appropriate Levi factor. We show that this can be quantized as a finite-dimensional subspace n k q = n 1 q ⊕ ⋯ ⊕ n r q \mathfrak {n}^q_k = \mathfrak {n}^q_1 \oplus \cdots \oplus \mathfrak {n}^q_r of the positive part of the quantized enveloping algebra, where each n k q \mathfrak {n}^q_k is a module under the left adjoint action of a quantized Levi factor. Furthermore, we show that C ⊕ n q \mathbb {C}\oplus \mathfrak {n}^q is a left coideal, with the possible exception of components corresponding to some exceptional Lie algebras. Finally we use these quantizations to construct covariant first-order differential calculi on quantum flag manifolds, compatible in a certain sense with the decomposition above, which coincide with those introduced by Heckenberger-Kolb in the irreducible case.

For a Hopf algebra \({\mathcal{A}}\), we define the structures of differential complexes on two dual exterior Hopf algebras: (1) an exterior extension of \({\mathcal{A}}\) and (2) an exterior extension of the dual algebra \({\mathcal{A}}\)*. The Heisenberg double of these two exterior Hopf algebras defines the differential algebra for the Cartan differential calculus on \({\mathcal{A}}\). The first differential complex is an analogue of the de Rham complex. When \({\mathcal{A}}\)* is a universal enveloping algebra of a Lie (super)algebra, the second complex coincides with the standard complex. The differential is realized as an (anti)commutator with a BRST operator Q. We give a recursive relation that uniquely defines the operator Q. We construct the BRST and anti-BRST operators explicitly and formulate the Hodge decomposition theorem for the case of the quantum Lie algebra Uq(gl(N)).

The paper deals with non-commutative differential calculi. It contains the description of all homogeneous coordinate differential calculi on the quantum plane and of those on the two-generated Grassmannian algebra. In addition, the optimum algebra for the Pusz-Woronowicz calculus on the quantum hyperplane is found.

Yangians and their applications

We study covariant differential calculi on quantum symplectic and orthogonal planes as interesting examples of the Wess-Zumino formalism for general non-(anti-)commutative differential calculus on quantum planes. The aim of this exercise is to generate certain known R-matrix solutions of the quantum Yang-Baxter equation. This is achieved by treating the consistency conditions on the calculus in a manner different from that of Wess and Zumino.

As a good introduction to and example of the differential calculus on quantum vector spaces pioneered by Wess and Zumino (1990) the author considers two-parameter quantum deformation of the plane and its calculus. The main structures are derived anew without an R matrix and their generalizations to multiparameter deformation and to quantum superspace are briefly discussed in an arbitrary number of dimensions.