Prof. Dr. Moritz Weber

Daniel Gromada

Reading seminar on quantum groups and Hopf algebras

(Winter term 2018/2019)


Time and Place

Wednesday, 14-16, SR6 (room 217, building E2 4)

  • 17 Oct - Introduction/motivation/outlook
  • 24 Oct - Definitions (compact quantum group, compact matrix quantum group, Hopf algebra), Examples (free orthogonal quantum group, free symmetric quantum group, quantum sl(2), SL_q(2),...), basic concepts and facts (commutative vs noncommutative algebras, duality)
    [Reading: Klimyk-Schmüdgen 1.1-1.2.7 (not 1.2.5), 3.1.1-3.1.3, 4.1.1-4.1.2; Neshveyev-Tuset 1.1; Indian notes Moritz: Def.2.4-Ex.2.6, Ex.2.9-Ex.2.16]
  • 7 Nov - cont. of definitions, examples, basic concepts
  • 14 Nov - Quantum symmetry (compact quantum spaces and their symmetries, the quantum plane, actions), examples (O_n^+ and quantum spheres, S_n^+ and n points, quantum automorphism groups of graphs, the symmetries of the quantum plane)
    [Reading: Kassel IV.1, IV.3, IV.5, IV.6, IV.7; Indian notes Moritz: 2.1, 2.3, 2.7, 2.8, 2.9; action of O_n^+ on the free sphere: BanicaGoswami or SpeicherWeber; quantum automorphism groups of graphs and their actions on graph C*-algebras: SchmidtWeber, see also H_n^+ as a quantum symmetry group: BanicaBichonCollins and Bichon - don't try to read all of these articles in detail, just try to understand the main idea of quantum symmetry here...]
  • 21 Nov - Haar state and representations of compact quantum groups (existence of the Haar state, definitions of representations, decompositions into irreducibles, fusion rules)
    [Reading: Indian notes Moritz: 3.1-3.7, 4.1-4.14; Neshveyev-Tuset: 1.2, 1.3]
    [Talks for 21 and 28 Nov: Haar1 (M.Kl., 3.1-3.7 without 3.2; also sketch 4.12-4.14 assuming that 4.10 is true), Haar2 (A.We., 3.2), Rep1 (L.Ju., 4.1-4.3), Rep2 (J.Sch., 4.4-4.7), Rep3 (A.Ma., 4.8-4.11), RepEx (M.St., Fusion rules of O_n^+ and maybe mentioning those of S_n^+ and U_n^+, see [2,3,4] in the list of references of Moritz' Indian notes and also 7.4.2 there, see also 6.4.12 in Timmermann for O_n^+, see also the last lines of Ex. 5.9 and Ex. 5.10 in FreslonWeber for an overview on O_n^+ and S_n^+)]
  • 28 Nov - cont. of Haar state and representations of CQG
    [probably talks Rep2, Rep3, RepEx]
  • 12 Dec - Representations of U_q(sl2) and SL_q(2)
    [Reading. Representation theory of sl2: Kassel V.4, V.5; representation theory of Uq(sl2): Kassel VI.3, VII.2, VII.7 (alternatively K.–S. 3.2.1, 3.2.2, 3.4); duality: Kassel V.7, VII.4, VII.5 (see also K.–S. 1.2.5, 1.3.5, 4.2.1)]
  • 16 Jan - Intertwiner spaces.
    [Reading. Tensor categories: Kassel sec. XI.1.1, def. XI.1.2, sec. XI.2.1 only strict tensor categories, sec. XI.3.1 only briefly introducing representation category of a group; Categories of partitions: Indian notes sec. 5 up to Prop. 5.5 see also arXiv:1710.06199, arXiv:1509.00988; R-matrices: Klimyk–Schmüdgen 8.1.1, 8.1.2]
  • 23 Jan - Easy quantum groups.
    [Reading: Indian notes. Tannaka–Krein duality: 4.16 – 4.18 (see also Malacarne), Easy quantum groups: rest of Sec. 5, Classification of categories of partitions: Sec. 6 (with focus on 6.1 – 6.6)]
  • 6 Feb - the quantum future of the world (or something like this)


Symmetry is a central concept in mathematics and science; many groundbreaking discoveries relied on the study
of symmetry. Mathematically, symmetries are understood in the first place as an invariance under an action of a
group. However, the developments of modern mathematics and quantum physics revealed the need to go beyond
groups in order to capture a new understanding of symmetry. This was the birth of quantum groups in the 1980's,
the pioneers being amongst others Drinfeld and Jimbo for algebraic approaches and Woronowicz for an analytic
or topological one.

In this seminar, we will address the following questions and topics: The seminar is held in English. This seminar is for students having some basic background in functional analysis.

Announcement of the seminar


Last update: 14 November 2018   Moritz Weber Impressum