Universal quantum knot invariants

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.

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.

In a recent article Turaev and Viro [TV] have constructed nontrivial “quantum” invariants of closed compact 3-manifolds M 3 in the form of state sums (called partition functions in statistical physics and vacuum functionals in quantum field theory) associated with the quantized universal enveloping algebra Uq(sl(2, C)) when q is a complex root of unity of a certain degree 2r > 4. The state sum is first defined for a given triangulation X of M and then shown to be independent of the triangulation X of M thus giving rise to a well defined invariant of M, which thus depends on q. This result may be viewed as a rigorous mathematical construction of what is called a topological quantum field theory. In fact, in the language of physicists, a triangulation corresponds to the introduction of a high-energy cut-off. Now topological quantum field theories have trivial dynamics, in other words they only deal with the vacuum sector, are scale invariant and more generally independent of any metrics. Invariance under subdivision is just the statement that the associated renormalization group transformation is trivial. This result suggests that the familiar techniques from algebraic topology should become useful to construct and discuss other topological quantum field theories. Moreover, similar combinatorial techniques could become helpful in discussing particle structures of low dimensional quantum field theories, whose vacuum sector is given in terms of a topological quantum field theory.

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 !

Note on the cohomology of color Hopf and Lie algebras

Let [Formula: see text] be a groupoid and assume that [Formula: see text] is a family of algebras with identity. First, we introduce the notion of a weak Hopf (groupoid)[Formula: see text]-coalgebra by that if, for each pair [Formula: see text], there is given a unital homomorphism [Formula: see text] satisfying certain properties, generalizing the notion of Hopf group-coalgebras as introduced by Turaev from groups to groupoids and Hopf algebra structures to weak Hopf algebra structures. Then one considers now the direct sum [Formula: see text] of these algebras. It is an algebra, without identity, except when [Formula: see text] is a finite groupoid, but the product is non-degenerate. The maps [Formula: see text] can be used to define a coproduct [Formula: see text] on [Formula: see text] and the conditions imposed on these maps give that [Formula: see text] is a weak multiplier Hopf algebra. It is [Formula: see text]-cograded as explained in this paper. We study these so-called groupoid-cograded weak multiplier Hopf algebras. They are, as explained above, more general than the weak Hopf group-coalgebras (introduced by Van Daele and Wang), generalizing the Turaev’s Hopf group-coalgebras. Moreover, our point of view makes it possible to use results and techniques from the theory of weak multiplier Hopf algebras in the study of weak Hopf groupoid-coalgebras (and generalizations).

Hopf algebras of set systems

Caltech High Energy Physics (HEP) has a broad program in both experimental and theoretical physics. We are known for our creativity and leadership. The future is uncertain and we strive to be involved in all the major areas of experimental and theoretical HEP physics so no matter where the important discoveries occur we are well positioned to play an important role. An outstanding group of postdoctoral scholars, graduate students, staff scientists, and technical and administrative personnel support our efforts in experimental and theoretical physics. The PI’s on this grant are involved in the following program of experimental and theoretical activities: I) EXPERIMENTAL PHYSICS Our CMS group, led by Harvey Newman and Maria Spiropulu, has played a key role in the discovery and interpretation of the Higgs boson and in searches for new physics. They have important hardware responsibilities in both ECAL and HCAL and are also involved in the upgrades needed for the High Luminosity LHC. Newman's group also develops and operates Grid-based computing, networking, and collaborative systems for CMS and the US HEP community. The charged lepton (Mu2e) and quark BaBar flavor physics group is led by David Hitlin and Frank Porter. On Mu2e they have been instrumental in the design of the calorimeter. Construction responsibilities include one third of the crystals and associated readout as well as the calibration system. They also will have responsibility for a major part of the online system software. Although data taking ceased in 2008 the Caltech BaBar group is active on several new forefront analyses. The neutrino group is led by Ryan Patterson. They are central to NOvA's core oscillation physics program, to calibration, and to detector readiness being responsible for the production and installation of 12,000 APD arrays. They have key roles in neutrino appearance and disappearance analysis in MINOS and MINOS+. Sunil Golwala leads the dark matter direct detection effort. Areas of activity include: CDMS II data analysis, contributions to SuperCDMS Soudan operations and analysis, R&D towards SuperCDMS SNOLAB, development of a novel screener for radiocontamination (the BetaCage), and development of new WIMP detector concepts. Ren-Yuan Zhu leads the HEP crystal laboratory for the advanced detector R&D effort. The crystal lab is involved in development of novel scintillating crystals and has proposed several crystal based detector concepts for future HEP experiments at the energy and intensity frontiers. Its current research effort is concentrated on development of fast crystal scintillators with good radiation hardness and low cost. II) THEORETICAL PHYSICS The main theme of Sergei Gukov's current research is the relation between the geometry of quantum group invariants and their categorification, on the one hand, and the physics of supersymmetric gauge theory and string theory, on the other. Anton Kapustin's research spans a variety of topics in non-perturbative Quantum Field Theory (QFT). His main areas of interest are supersymmetric gauge theories, non-perturbative dualities in QFT, disorder operators, Topological Quantum Field Theory, and non-relativistic QFT. He is also interested in the foundations and possible generalizations of Quantum Mechanics. Hirosi Ooguri's current research has two main components. One is to find exact results in Calabi-Yau compactification of string theory. Another is to explore applications of the AdS/CFT correspondence. He also plans to continue his project with Caltech postdoctoral fellows on BPS spectra of supersymmetric gauge theories in diverse dimensions. John Preskill works on quantum information science. This field may lead to important future technologies, and also lead to new understanding of issues in fundamental physics John Schwarz has been exploring a number of topics in superstring theory/M-theory, supersymmetric gauge theory, and their AdS/CFT relationships. Much of the motivation for these studies is the desire to gain a deeper understanding of superstring theory and M-theory. The research interests of Mark Wise span particle physics, cosmology and nuclear physics. His recent work has centered on extensions of the standard model where baryon number and lepton number are gauged and extensions of the standard model that have novel sources of baryon number violation and new sources of charged lepton flavor violation

At present there can be no doubt to say that the quantum group theory has been one of the most important, modern and rapidly developing directions of mathematics and mathematical physics in the end of the twentieth century. Although initially (in the early 1980’s) the quantum group theory was formulated for solving problems in the theory of integrable systems and statistical physics, later the surprising connection of this theory with many branches of mathematics, and the theoretical and mathematical physics was discovered. Today the quantum group theory is connected with such mathematical fields as special functions (especially, with (q-orthogonal polynomials and basic hypergeometric series), the theory of difference and differential equations, combinatorial analysis and representation theory, matrix and operator algebras, noncommutative geometry, knot theory, topology, category theory and so on. From point of view of the mathematical physics there exists interconnection of the quantum group theory with the quantum inverse scattering method, conformai and quantum fields theory and so on. It is expected that the quantum groups will provide deeper understanding of concept of symmetry in physics.KeywordsQuantum GroupSimple RootBasic Hypergeometric SeriesReduction ChainGeneral Projection OperatorThese 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.

In this article we construct three explicit natural subgroups of the Brauer-Picard group of the category of representations of a finite-dimensional Hopf algebra. In examples the Brauer Picard group decomposes into an ordered product of these subgroups, somewhat similar to a Bruhat decomposition. Our construction returns for any Hopf algebra three types of braided autoequivalences and correspondingly three families of invertible bimodule categories. This gives examples of so-called (2-)Morita equivalences and defects in topological field theories. We have a closer look at the case of quantum groups and Nichols algebras and give interesting applications. Finally, we briefly discuss the three families of group-theoretic extensions.

Differential Hopf algebras arise in several contexts in algebraic topology. The Bockstein spectral sequence of an //-space is one example that has been investigated by many authors [3; 1; 7; 8]. Borel [3] and Araki [1] proved algebraic theorems about the structure of differential Hopf algebras of special kinds. These special theorems enabled them to determine the odd torsion in the cohomology of the exceptional Lie groups. If X and Tare //-spaces and/:X-> Tis a fibre map which is multiplicative, then the spectral sequence of / is a spectral sequence of Hopf algebras. This situation was first discussed by J. C. Moore [17], and later by the author [5; 6]. The techniques of [5] were later extended by the author (in unpublished work) to prove theorems about the homology and cohomology suspensions, i.e., when X is the space of paths of the //-space Y. The proofs rested upon a general theorem about the structure of this spectral sequence. Some of these suspension theorems had been proved by Moore using a different spectral sequence of Hopf algebras [12]. In this paper we make a study of differential Hopf algebras, and prove general theorems on the structure of their homology. These theorems generalize the results of Borel and Araki. Applied to the case of multiplicative fibre maps, we obtain a general theorem about the structure of the spectral sequence (even in the nonacyclic case), which, in particular, yields simple proofs of the suspension theorems mentioned above. Applied to the Bockstein spectral sequence, we get information on torsion in //-spaces. This study of differential Hopf algebras depends on two spectral sequences which may be defined in different circumstances. If one of them is defined, then the terms of that spectral sequence satisfy the conditions necessary for the other to be defined. Thus we get a spectral sequence for the term of the other spectral sequence(2), and this spectral sequence has a very simple form which makes it easy to calculate the form of its homology. Thus the structure of the limit

Monopoles and topological field theory

A fundamental problem in the theory of Hopf algebras is the classification and construction of finite-dimensional (minimal) triangular Hopf algebras (A,R) introduced by Drinfeld. Only recently Etingof and the author completely solved this problem for semisimple A over algebraically closed fields of characteristics 0 and p>>dim(A) (any p if one assumes that A is also cosemisimple). In this paper we take the first step towards solving this problem for finite-dimensional pointed Hopf algebras over an algebraically closed field k of characteristic 0. We first prove that the fourth power of the antipode of any triangular pointed Hopf algebra A is the identity. We do that by focusing on minimal triangular pointed Hopf algebras (A,R) (every triangular Hopf algebra contains a minimal triangular sub Hopf algebra) and proving that the group algebra of the group of grouplike elements of A (which must be abelian) admits a minimal triangular structure and consequently that A has the structure of a biproduct. We also generalize our result on the order of the antipode to any finite-dimensional quasitriangular Hopf algebra A whose Drinfeld element u acts as a scalar in any irreducible representation of A (e.g. when A^* is pointed). Second, we describe a method of construction of finite-dimensional pointed Hopf algebras which admit a minimal triangular structure, and classify all their minimal triangular structures. We conclude the paper by proving that any minimal triangular Hopf algebra which is generated as an algebra by grouplike elements and skew primitive elements is isomorphic to a minimal triangular Hopf algebra constructed using our method.

In this book we shall review the systematic approaches and applications of a theory which investigates situations where Feynman path integrals of physical systems can be evaluated exactly leading to a complete understanding of the quantum physics. These mathematical formalisms are in large part motivated by the symmetries present in integrable systems and topological quantum field theories which make these latter examples exactly solvable problems. Besides providing conceptual understandings of the solvability features of these special classes of problems, this framework yields geometric approaches to evaluating the quantum spectrum of generic quantum mechanical and quantum field theoretical partition functions. The techniques that we shall present here in fact motivate an approach to studying generic physical problems by relating their properties to those of integrable and topological field theories. In doing so, we shall therefore also review some of the more modern quantum field theoretical and mathematical ideas which have been at the forefront of theoretical physics over the past two decades.KeywordsPartition FunctionPath IntegralEquivariant CohomologyCoadjoint OrbitSupersymmetric Quantum MechanicThese 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.

We study algebraic and homological properties of two classes of infinite-dimensional Hopf algebras over an algebraically closed field k k of characteristic 0 0 . The first class consists of those Hopf k k -algebras that are connected graded as algebras, and the second class are those Hopf k k -algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel’fand–Kirillov dimension. It is shown that if the Hopf algebra H H is a connected graded Hopf algebra of finite Gel’fand–Kirillov dimension n n , then H H is a noetherian domain which is Cohen–Macaulay, Artin–Schelter regular, and Auslander regular of global dimension n n . It has S 2 = I d H S^2 = \mathrm {Id}_H , and it is Calabi–Yau. Detailed information is also provided about the Hilbert series of H H . Our results leave open the possibility that the first class of algebras is (properly) contained in the second. For this second class, the Hopf k k -algebras of finite Gel’fand–Kirillov dimension n n with connected coalgebra, the underlying coalgebra is shown to be Artin–Schelter regular of global dimension n n . Both these classes of Hopf algebras share many features in common with enveloping algebras of finite-dimensional Lie algebras. For example, an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra. Nevertheless, we construct, as one of our main results, an example of a Hopf k k -algebra H H of Gel’fand–Kirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to U ( g ) U(\mathfrak {g}) for any Lie algebra g \mathfrak {g} .