Quantum groups and deformation quantization: Explicit approaches and implicit aspects

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.

Lie algebras and quantum groups are not usually studied by an undergraduate. However, in the study of these structures, there are interesting questions that are easily accessible to an upper-level undergraduate. Here we look at the expansion of a nested set of brackets that appears in relations presented in a paper of Lum on toroidal algebras. We illuminate certain terms that must be in the expansion, providing a partial answer for the closed form. Lie algebras and quantum groups are not topics that you are apt to hear undergraduates math majors discussing in their spare time. However, there are a surprising number of nontrivial questions in this area that are undergraduate appropriate. In this paper, we will give a brief overview of the broad mathematical setting, and then discuss an accessible problem that involves expanding a nested set of brackets. Lie algebras, their universal enveloping algebras and quantum groups are a fundamental part of representation theory that have many applications within mathematics and mathematical physics. Lie algebras and Lie groups were originally discovered by Sophus Lie in the late nineteenth century [Borel 2001]. Given a Lie algebra, we associate a unique associative algebra called the universal enveloping algebra. In 1985, Jimbo and Drinfeld discovered q-analogues of these universal enveloping algebras called “quantum groups”, which have been a recent area of study (see [Lusztig 1993]). In order to find the quantum analogue of a Lie algebra it is often desirable to understand the defining relationships of the Lie algebra inside of its universal enveloping algebra. The motivation for this project came from a paper by Lum in which he gives a nice presentation of a toroidal Lie algebra that could be useful in understanding this Lie algebra’s quantum group [Lum 1998]. All of these relations utilize a nested set of brackets called t.k/. For simplicity, we have modified t.k/ by a scalar. In this paper we seek to understand the expansion of this object.

After a presentation of the context and a brief reminder of deformation quantization, we indicate how the introduction of natural topological vector space topologi es on Hopf algebras associated with Poisson Lie groups, Lie bialgebras and their doubles explains their dualities a nd provides a comprehensive framework. Relations with deformation quantization and applications to the deformation quantization of symmetric spaces are described.

For pt. I see ibid., vol.23, p.4185 (1990). The SUq(2) algebra is realized by means of both the Poisson brackets in classical mechanics and commutators in quantum mechanics in a system with q-deformed oscillators of two different types. The structures of the Hopf algebra and the quantum Yang-Baxter equation are also discussed on a quantum level. A set of j-representations of the quantum algebra SUq(2) is constructed based on the 'type-II' q-oscillators. Multi-deformations of the oscillators of the two types and multi-deformed algebras expressed in Poisson brackets as well as in Lie brackets are proposed.

Hopf algebras have important connections to quantum theory, Lie algebras, knot and braid theory, operator algebras and other areas of physics and mathematics. They have been intensely studied in the past; in particular, the solution of a number of conjectures of Kaplansky from the 1970s has led to progress on the classification of semisimple Hopf algebras and on the structure of pointed Hopf algebras. Among the topics covered are results toward the classification of finite-dimensional Hopf algebras (semisimple and non-semisimple), as well as what is known about the extension theory of Hopf algebras. Some papers consider Hopf versions of classical topics, such as the Brauer group, while others are closer to work in quantum groups. The book also explores the connections and applications of Hopf algebras to other fields.

We study a general ansatz for an odd supersymmetric version of the Kronecker elliptic function, which satisfies the genus one Fay identity. The obtained result is used for construction of the odd supersymmetric analogue for the classical and quantum elliptic R-matrices. They are shown to satisfy the classical Yang–Baxter equation and the associative Yang–Baxter equation. The quantum Yang–Baxter equation is discussed as well. It acquires additional term in the case of supersymmetric R-matrices.

Faddeev, Reshetikhin and Takhtadzhyan showed, some time ago [1], how to construct a nice algebraic structure out of a given matrix R ∈ End(C n ⊗ C n ) satisfying the Yang-Baxter equation. They referred to these algebraic structures as the algebras of “Functions over the Quantum Group”: they are in fact bialgebras which may have an antipode, i.e. may be Hopf algebras, when some extra conditions are satisfied.

Quantum group and quantum symmetry

Integrable massive QFT and conformal invariant models follow from lattice integrable models in suitable scaling limits. There are Yang-Baxter algebras (YBA) associated with all these two-dimensional models. These YBA allow one to construct the exact solution (spectrum, S-matrix, form-factors,…) for this class of theories. Braid groups and quantum groups are derived as limiting cases of YBA when θ (spectral parameter) goes to ±∞.KeywordsHopf AlgebraQuantum GroupBraid GroupScaling LimitConformal WeightThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Part 1 Mathematical aspects of quantum groups theory and non-commutative geometry: Hopf algebra and Poisson structure of Lie groups and algebras quantum groups, algebras and their duality non-commutative spaces and quantum groups invariant differential calculi elements of quantum groups representations theory tensor products of representations q-tensors, q-vectors, q-scalars. Part 2 Deformation of harmonic oscillators: q-deformation of single-harmonic oscillator different forms of commutation relations representations q real and roots of unity cases algebraic maps from non-deformed to deformed oscillators path integral quantization of q-oscillator. Part 3 Q-deformation of space-time symmetries: relativistic space-time symmetries Poincare group as a classical deformation of Galilei group its representations multiparametric q-deformation of linear groups, twisted groups and algebras q-Poincare group as a q-subgroup of q-conformal one and its induced representations. Part 4 Non-commutative geometry and unification models: unified models, the problem of natural introduction of Higgs fields unified models of Higgs fields in the frame of non-commutative geometry and others.

International audience

The Poincar-Birkhoff-Witt Theorem deals with the structure and universal property of the universal enveloping algebra U (L) of a Lie algebra L , e.g., over R or C .K. H. Hofmann and L. Kramer (HK) [On weakly complete group algebras of Compact Groups, J. Lie Theory 30 (2020) 407-426] recently introduced the weakly complete universal enveloping algebra U(g) of a profinite-dimensional topological Lie algebra g.Here it is shown that the classical universal enveloping algebra U (|g|) of the abstract Lie algebra underlying g is a dense subalgebra of U(g) , algebraically generated by g U(g) .It is further shown that, inspite of U being a left adjoint functor, it nevertheless preserves projective limits in the form U(lim i g/i) = lim i U(g/i) , for profinite-dimensional Lie algebras g represented as projective limits of their finite-dimensional quotients.The required theory is presented in an appendix which is of independent interest.In a natural way, a weakly complete enveloping algebra U(g) is a weakly complete symmetric Hopf algebra with a Lie subalgebra P(U(g)) of primitive elements containing g (indeed properly if g = {0}), and with a nontrivial multiplicative pro-Lie group G(U(g)) of grouplike units, having P(U(g)) as its Lie algebra -in contrast with the classical Poincar-Birhoff-Witt environment of U (L) , thus providing a new aspect of Lie's Third Fundamental Theorem: Indeed a canonical pro-Lie subgroup * (g) of G(U(g)) is identified whose Lie algebra is naturally isomorphic to g.The structure of U(g) is described in detail for dim g = 1 .The primitive and grouplike components and their mutual relationship are evaluated precisely.In (HK), cited above, and in the work of R. Dahmen and K. H. Hofmann [The pro-Lie group aspect of weakly complete algebras and weakly complete group Hopf algebras, J. Lie Theory 29 (2019) 413-455] the real weakly complete group Hopf algebra R[G] of a compact group G was described.In particular, the set P(R[G])) of primitive elements of R[G] was identified as the Lie algebra g of G .It is now shown that for any compact group G with Lie algebra g there is a natural morphism of weakly complete symmetric Hopf algebras g : U(g) R[G] , implementing the identity on g and inducing a morphism of pro-Lie groups * (G) G(R[G]) = G : yet another aspect of Sophus Lie's Third Fundamental Theorem !

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)).

We find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild cohomology with one‐dimensional coefficients for all these Hopf algebras. In fact, the resolutions can be endowed with a Yetter–Drinfeld structure. General results of Bichon then allow us to compute also the corresponding bialgebra cohomologies. Finding the resolution rests on two pillars. We take as a starting point the resolution for the free orthogonal quantum group presented by Collins, Härtel, and Thom or its algebraic generalization to quantum symmetry groups of bilinear forms due to Bichon. Then, we make use of the fact that the free unitary quantum groups and some of its non‐Kac versions can be realized as a glued free product of a (non‐Kac) free orthogonal quantum group with , the finite group of order 2. To obtain the resolution also for more general universal cosovereign Hopf algebras, we extend Gromada's proof from compact quantum groups to the framework of matrix Hopf algebras. As a by‐product of this approach, we also obtain a projective resolution for the freely modified bistochastic quantum groups. Only a special subclass of free unitary quantum groups and universal cosovereign Hopf algebras decompose as a glued free product in the described way. In order to verify that the sequence we found is a free resolution in general (as long as the parameter matrix is generic, two conditions which are automatically fulfilled in the free unitary quantum group case), we use the theory of Hopf bi‐Galois objects and Bichon's results on monoidal equivalences between the categories of Yetter–Drinfeld modules over universal cosovereign Hopf algebras for different parameter matrices.