Quantum groups: A survey of definitions, motivations and results

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 28 October 2012

1. Introduction

   1. 1. 1.1 Goal of this survey
      2. 1.2 References for quantum groups
      3. 1.3 Some missing topics and where to find them
      4. 1.4 Further references for the background topics
      5. 1.5 On reading these notes
      6. 1.6 Disclaimer
      7. 1.7 Acknowledgements

2. I. Hopf algebras and quasitriangular Hopf algebras

   1. 1. SRMCwMFFs

      1. 1.1 Definition of an algebra
      2. 1.2 Definition of a module
      3. 1.3 Motivation for SRMCwMFFs
      4. 1.4 Definition of SRMCwMFFs

   2. 2. Hopf algebras

      1. 2.1 Definition of Hopf algebras
      2. 2.2 Sweedler notation for the comultiplication
      3. 2.3 Hopf algebras give us SRMCwMFFs!
      4. 2.4 Group algebras are Hopf algebras
      5. 2.5 Enveloping algebras of Lie algebras are Hopf algebras
      6. 2.6 Definition of the adjoint action of a Hopf algebra on itself
      7. 2.7 Motivation for the definition of the adjoint action
      8. 2.8 Definition of an ad-invariant form on a Hopf algebra

   3. 3. Braided SRMCwMFFs

      1. 3.1 Motivation for braided SRMCwMFFs
      2. 3.2 Definition of braided SRMCwMFFs
      3. 3.3 Pictorial representation of braiding isomorphisms
      4. 3.4 What "natural isomorphism" means
      5. 3.5 The braid relation

   4. 4. Quasitriangular Hopf algebras

      1. 4.1 Motivation for quasitriangular Hopf algebras
      2. 4.2 Definition of quasitriangular Hopf algebras
      3. 4.3 Quasitriangular Hopf algebras give braided SRMCwMFFs

   5. 5. The quantum double

      1. 5.1 Motivation for the quantum double
      2. 5.2 Construction of the Hopf algebra $A^{*\text{coop}}$
      3. 5.3 Construction of the quantum double
      4. 5.4 If $A$ is an infinite dimensional Hopf algebra
      5. 5.5 An ad-invariant pairing on the quantum double

3. II. Lie algebras and enveloping algebras

   1. 1. Semisimple Lie algebras

      1. 1.1 Definition of a Lie algebra
      2. 1.2 Definition of a simple Lie algebra
      3. 1.3 Definition of the radical of a Lie algebra
      4. 1.4 Definition of simple modules for a Lie algebra
      5. 1.5 Definition of the adjoint representation of a Lie algebra
      6. 1.6 Definition of the Killing form
      7. 1.7 Characterizations of semisimple Lie algebras

   2. 2. Finite dimensional complex simple Lie algebras

      1. 2.1 Dynkin diagrams and Cartan matrices
      2. 2.2 Classification of finite dimensional complex simple Lie algebras
      3. 2.3 Triangular decomposition
      4. 2.4 Weights and weight spaces
      5. 2.5 Classification of simple modules
      6. 2.6 Roots and the root lattice
      7. 2.7 The inner product on ${𝔥}_{ℝ}^{*}$
      8. 2.8 The Weyl group corresponding to $𝔤$

   3. 3. Enveloping algebras

      1. 3.1 Motivation for the enveloping algebra
      2. 3.2 Definition of the enveloping algebra
      3. 3.3 A functorial way of realising the enveloping algebra
      4. 3.4 The enveloping algebra is a Hopf algebra
      5. 3.5 Modules for the enveloping algebra and the Lie algebra are the same!
      6. 3.6 The Lie algebra can be recovered from its enveloping algebra!
      7. 3.7 A basis for the enveloping algebra

   4. 4. The enveloping algebra of a complex simple Lie algebra

      1. 4.1 A presentation by generators and relations
      2. 4.2 Triangular decomposition
      3. 4.3 Grading on $𝔘{𝔫}^{-}$ and $𝔘{𝔫}^{+}$
      4. 4.4 Poincaré-Birkhoff-Witt bases of $𝔘{𝔫}^{-},$ $𝔘𝔥,$ and $𝔘{𝔫}^{+}$
      5. 4.5 The Casimir element in $𝔘𝔤$

4. III. Deformations of Hopf algebras

   1. 1. $h\text{-adic}$ completions

      1. 1.1 Motivation for $h\text{-adic}$ completions
      2. 1.2 The algebra $A\left[\left[h\right]\right],$ an example of an $h\text{-adic}$ completion
      3. 1.3 Definition of the $h\text{-adic}$ topology
      4. 1.4 Definitions of an $h\text{-adic}$ completion

   2. 2. Deformations

      1. 2.1 Motivation for deformations
      2. 2.2 Deformation as a Hopf algebra
      3. 2.3 Definition of equivalent deformations
      4. 2.4 Definition of the trivial deformation as a Hopf algebra
      5. 2.5 Deformation as an algebra
      6. 2.6 The trivial deformation as an algebra

5. IV. Perverse sheaves

   1. 1. The category $D_c^b\left(X\right)$

      1. 1.1 Complexes of sheaves
      2. 1.2 The category $K\left(X\right)$ and derived functions
      3. 1.3 Bounded complexes and constructible compelxes
      4. 1.4 Definition of the category $D_c^b\left(X\right)$

   2. 2. Functors

      1. 2.1 The direct image with compact support functor $f_{!}$
      2. 2.2 The inverse function $f^{*}$
      3. 2.3 The functor $f_{♭}$
      4. 2.4 The shift functor $\left[n\right]$
      5. 2.5 The Verdier duality functor $D$

   3. 3. Perverse sheaves

      1. 3.1 Definition of perverse sheaves
      2. 3.2 Intersection cohomology complexes

6. V. Quantum groups

   1. 1. Definition, uniqueness, existence

      1. 1.1 Making the Cartan matrix symmetric
      2. 1.2 The Poisson homomorphism $\delta$
      3. 1.3 The definition of the quantum group
      4. 1.4 Uniqueness of the quantum group
      5. 1.5 Definition of $q\text{-integers}$ and $q\text{-factorials}$
      6. 1.6 Presentation of the quantum group by generators and relations

   2. 2. The rational form of the quantum group

      1. 2.1 Definition of the rational form of the quantum group
      2. 2.2 Relating the rational form and the original form of the quantum group

   3. 3. Integral forms of the quantum group

      1. 3.1 Definition of integral forms
      2. 3.2 Motivation for integral forms
      3. 3.3 Definition of the non-restricted integral form of the quantum group
      4. 3.4 Definition of the restricted integral form of the quantum group

7. VI. Modules for quantum groups

   1. 1. Finite dimensional ${𝔘}_h𝔤\text{-modules}$

      1. 1.1 As algebras ${𝔘}_h𝔤\cong 𝔘𝔤\left[\left[h\right]\right]$
      2. 1.2 Definition of weight spaces in a ${𝔘}_h𝔤\text{-module}$
      3. 1.3 Classification of modules for ${𝔘}_h𝔤$

   2. 2. Finite dimensional $U_q𝔤\text{-modules}$

      1. 2.1 Construction of the Verma module $M\left(\lambda \right)$ and the simple module $L\left(\lambda \right)$
      2. 2.2 Twisting $L\left(\lambda \right)$ to get $L(\lambda ,\sigma )$
      3. 2.3 Classifications of the finite dimensional irreducible modules for $U_q𝔤$
      4. 2.4 Weight spaces for $U_q𝔤\text{-modules}$

8. VII. Properties of quantum groups

   1. 1. Triangular decomposition and grading

      1. 1.1 Triangular decomposition
      2. 1.2 The grading on ${𝔘}_h{𝔫}^{+}$ and ${𝔘}_h{𝔫}^{-}$

   2. 2. The inner product $⟨,⟩$

      1. 2.1 The pairing between ${𝔘}_h{𝔟}^{-}$ and ${𝔘}_h{𝔟}^{+}$
      2. 2.2 Extending the pairing to an ad-invariant pairing on ${𝔘}_h𝔤$
      3. 2.3 Duality between matrix coefficients for representations and $U_q𝔤$

   3. 3. The universal $ℛ\text{-matrix}$

      1. 3.1 Motivation for the $ℛ\text{-matrix}$
      2. 3.2 Existence and uniqueness of $ℛ$
      3. 3.3 Properties of the $ℛ\text{-matrix}$

   4. 4. An analogue of the Casimir element

      1. 4.1 Definition of the element $u$
      2. 4.2 Properties of the element $u$
      3. 4.3 Why the element $u$ is an analogue of the Casimir element

   5. 5. The element $T_{w_0}$

      1. 5.1 The automorphism $\varphi \circ \theta \circ S_h$
      2. 5.2 Definition of the element $T_{w_0}$
      3. 5.3 Properties of the element $T_{w_0}$

   6. 6. The Poincaré-Birkhoff-Witt

      1. 6.1 Root vectors in ${𝔘}_h𝔤$
      2. 6.2 Poincaré-Birkhoff-Witt bases of ${𝔘}_h{𝔫}^{-},$ ${𝔘}_h𝔥,$ and ${𝔘}_h{𝔫}^{+}$
      3. 6.3 The PBW-bases of ${𝔘}_h{𝔫}^{-}$ and ${𝔘}_h{𝔫}^{+}$ are dual bases with respect to $⟨,⟩$ (almost)

   7. 7. The quantum group is a quantum double (almost)

      1. 7.1 The identification of ${\left({𝔘}_h{𝔟}^{+}\right)}^{*\text{coop}}$ with ${𝔘}_h{𝔟}^{-}$
      2. 7.2 Recalling the quantum double
      3. 7.3 The relation between $D\left({\left({𝔘}_h{𝔟}^{+}\right)}^{*\text{coop}}\right)$ and ${𝔘}_h𝔤$
      4. 7.4 Using the $ℛ\text{-matrix}$ of $D\left({\left({𝔘}_h{𝔟}^{+}\right)}^{*\text{coop}}\right)$ to get the $ℛ\text{-matrix}$ of ${𝔘}_h𝔤$

   8. 8. The quantum Serre relations occur naturally

      1. 8.1 Definition of the algebras $U_h{𝔟}^{+}$ and $U_h{𝔟}^{-}$
      2. 8.2 The difference between the algebras $U_h{𝔟}^{\pm }$ and the algebras ${𝔘}_h{𝔟}^{\pm }$
      3. 8.3 A pairing between $U_h{𝔟}^{+}$ and $U_h{𝔟}^{-}$
      4. 8.4 The radical of $⟨,⟩$ is generated by the quantum Serre relations

9. VIII. Hall algebras and perverse sheaf algebras

   1. 1. Hall algebras

      1. 1.1 Quivers
      2. 1.2 Representations of a quiver
      3. 1.3 Definition of the Hall algebra
      4. 1.4 Connecting Hall algebras to the quantum group

   2. 2. An algebra of perverse sheaves

      1. 2.1 $\Gamma \text{-graded}$ vector spaces and the varieties $E_V$ with $G_V$ action
      2. 2.2 Definition of the categories ${𝒬}_V$ and ${𝒬}_T\otimes {𝒬}_W$
      3. 2.3 The Grothendieck group $𝒦$ associated to the categories ${𝒬}_V$
      4. 2.4 Definition of the multiplication in $𝒦$
      5. 2.5 Definition of the pseudo-comultiplication $r:𝒦\to 𝒦\otimes 𝒦$
      6. 2.6 The symmetric form on $𝒦$
      7. 2.7 Definition of the elements $L_{\overrightarrow{\nu }}\in 𝒦$
      8. 2.8 The connection between $𝒦$ and the quantum grouo
      9. 2.9 Dictionary between $𝒦$ and $U_q{𝔟}^{+}$
      10. 2.10 Definition of the constant $M^{\prime }(\tau ,\omega )$ which was used in (2.7)
      11. 2.11 Definition of the vector spaces ${\mathscr{H}}^{j+2\text{dim}(G\backslash \Omega )}\left(u_{!}(t_{♭}{s}^{*}{B}_1\otimes t_{♭}{s}^{*}{B}_2)\right)$ from (2.6)
      12. 2.12 Some remarks on Part II of Lusztig's book'

   3. 3. The connection between representations of quivers and perverse sheaves

      1. 3.1 Relating orbits and isomorphism classes of representations of $\Gamma$
      2. 3.2 Realizing the structure constants of the Hall algebra in terms of orbits
      3. 3.3 Rewriting the Hall algebra in terms of functions constant on orbits
      4. 3.4 The isomorphism between $𝒦$ and $K$

10. IX. Link invariants from quantum groups

    1. 1. 1.1 Knots, links and isotopy
       2. 1.2 Link invariants
       3. 1.3 Braids
       4. 1.4 Every link is the closure of a braid
       5. 1.5 Markov equivalence
       6. 1.6 Quantum dimensions and quantum traces
       7. 1.7 Quantum traces give us link invariants!

Notes and References

This is an excerpt from a paper entitled Quantum groups: A survey of definitions, motivations and results by Arun Ram. Research and writing supported in part by an Australian Research Council fellowship and a National Science Foundation grant DMS-9622985.

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