Alexander Mang

Alexander Mang

Mathematician

Research interests

My main interest is in operator-algebraic quantum groups, mostly compact/discrete quantum groups, in particular ones of combinatorial type, such as easy quantum groups or graph-theoretical quantum groups. Currently, I am learning more about functorial quantum field theories.

Publications

Abstract:
A resolution \(P\) of the counit of the Hopf \(\ast\)-algebra \(\mathcal{O}(U_n^+)\) of representative functions on van Daele and Wang's free unitary quantum group \(U_n^+\) in terms of free \(\mathcal{O}(U_n^+)\)-modules is computed for arbitrary \(n\). A different such resolution was recently found by Baraquin, Franz, Gerhold, Kula and Tobolski. While theirs has desirable properties which \(P\) lacks, \(P\) is still good enough to compute the (previously known) quantum group cohomology and comes instead with an important advantage: \(P\) can be arrived at without the clever combination of certain results potentially very particular to \(U_n^+\) that enabled the aforementioned authors to find their resolution. Especially, \(P\) relies neither on the resolution for \(O_n^+\) obtained by Collins, Härtel and Thom nor the one for \(SL_2(q)\) found by Hadfield and Krähmer. Rather, as shown in the present article, the recursion defining the Anick resolution of the counit of \(\mathcal{O}(U_n^+)\) can be solved in closed form. That suggests a potential strategy for determining the cohomologies of arbitrary easy quantum groups.
Keywords:
  • Anick resolution
  • Gröbner basis
  • free unitary quantum group
  • easy quantum group
  • compact quantum group
  • quantum group cohomology
Mathematics Subject Classification:
  • 20G42
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Abstract:
The first quantum group cohomology with trivial coefficients of the discrete dual of any unitary easy quantum group is computed. That includes those potential quantum groups whose associated categories of two-colored partitions have not yet been found.
Keywords:
  • Discrete quantum group
  • Quantum group cohomology
  • Trivial coefficients
  • Easy quantum group
  • Category of partitions
Mathematics Subject Classification:
  • 20G42 Primary
  • 05A18 Secondary
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with Moritz Weber
Abstract:
Within the framework of unitary easy quantum groups, we study an analogue of Brauer's Schur-Weyl approach to the representation theory of the orthogonal group. We consider concrete combinatorial categories whose morphisms are formed by partitions of finite sets into disjoint subsets of cardinality two; the points of these sets are colored black or white. These categories correspond to “half-liberated easy” interpolations between the unitary group and Wang's quantum counterpart. We complete the classification of all such categories demonstrating that the subcategories of a certain natural halfway point are equivalent to additive subsemigroups of the natural numbers; the categories above this halfway point have been classified in a preceding article. We achieve this using combinatorial means exclusively. Our work reveals that the half-liberation procedure is quite different from what was previously known from the orthogonal case.
Keywords:
  • Quantum group
  • Unitary easy quantum group
  • Unitary group
  • Half-liberation
  • Tensor category
  • Two-colored partition
  • Partition of a set
  • Category of partitions
  • Brauer algebra
Mathematics Subject Classification:
  • 05A18 Primary
  • 20G42 Secondary
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with Moritz Weber
Abstract:
Compact quantum groups can be studied by investigating their representation categories in analogy to the Schur–Weyl/Tannaka–Krein approach. For the special class of (unitary) “easy” quantum groups, these categories arise from a combinatorial structure: rows of two-colored points form the objects, partitions of two such rows the morphisms. Vertical/horizontal concatenation and reflection give composition, monoidal product and involution. Of the four possible classes \(\mathcal{O}\), \(\mathcal{B}\), \(\mathcal{S}\) and \(\mathcal{H}\) of such categories (inspired, respectively, by the classical orthogonal, bistochastic, symmetric and hyperoctahedral groups), we treat the first three—the non-hyperoctahedral ones. We introduce many new examples of such categories. They are defined in terms of subtle combinations of block size, coloring and non-crossing conditions. This article is part of an effort to classify all non-hyperoctahedral categories of two-colored partitions. It is purely combinatorial in nature. The quantum group aspects are left out.
Keywords:
  • Quantum group
  • Unitary easy quantum group
  • Unitary group
  • Half-liberation
  • Tensor category
  • Two-colored partition
  • Partition of a set
  • Category of partitions
  • Brauer algebra
Mathematics Subject Classification:
  • 05A18 Primary
  • 20G42 Secondary
View article:
Cite
with Moritz Weber
Abstract:
Within the framework of unitary easy quantum groups, we study an analogue of Brauer's Schur-Weyl approach to the representation theory of the orthogonal group. We consider concrete combinatorial categories whose morphisms are formed by partitions of finite sets into disjoint subsets of cardinality two; the points of these sets are colored black or white. These categories correspond to “half-liberated easy” interpolations between the unitary group and Wang's quantum counterpart. We complete the classification of all such categories demonstrating that the subcategories of a certain natural halfway point are equivalent to additive subsemigroups of the natural numbers; the categories above this halfway point have been classified in a preceding article. We achieve this using combinatorial means exclusively. Our work reveals that the half-liberation procedure is quite different from what was previously known from the orthogonal case.
Keywords:
  • Quantum group
  • Unitary easy quantum group
  • Unitary group
  • Half-liberation
  • Tensor category
  • Two-colored partition
  • Partition of a set
  • Category of partitions
  • Brauer algebra
Mathematics Subject Classification:
  • 05A18 Primary
  • 20G42 Secondary
Cite
with Moritz Weber
Abstract:
We classify certain categories of partitions of finite sets subject to specific rules on the coloring of points and the sizes of blocks. More precisely, we consider pair partitions such that each block contains exactly one white and one black point when rotated to one line; however, crossings are allowed. There are two families of such categories, the first of which is indexed by cyclic groups and is covered in the present article; the second family will be the content of a follow-up article. Via a Tannaka–Krein result, the categories in the two families correspond to easy quantum groups interpolating the classical unitary group \(U_n\) and Wang’s free unitary quantum group \(U^+_n\). In fact, they are all half-liberated in some sense and our results imply that there are many more half-liberation procedures than previously expected. However, we focus on a purely combinatorial approach leaving quantum group aspects aside.
Keywords:
  • Quantum group
  • Unitary easy quantum group
  • Unitary group
  • Half-liberation
  • Tensor category
  • Two-colored partition
  • Partition of a set
  • Category of partitions
Mathematics Subject Classification:
  • 05A18 Primary
  • 20G42 Secondary
View article:
Cite

Talks

Past

Area-dependent field theories and compact quantum groups at Hamburg University, Germany, November 14, 2023.
Classification and homological invariants of compact quantum groups of combinatorial type at Saarland University, Germany, March 9, 2023.
Introduction to fusion‐2‐categories, part 4 at
Research Seminar Algebra and Mathematical Physics
, Hamburg University, Germany, January 24, 2023.
Compact quantum groups of combinatorial type at
Research Seminar Algebra and Mathematical Physics
, Hamburg University, Germany, November 8, 2022.
Quantizing groups through combinatorics at
Seminar Algebra – Geometry – Combinatorics
, TU Dresden, Germany, May 19, 2022.
Easy quantum groups — what they are, why they are easy and why they are difficult at
Operator Algebra Seminar
, KU Leuven, Belgium, April 22, 2022.
Half-liberating the unitary groups at
Operator Algebras Seminar
, Oslo University, Norway, March 30, 2022.
Series of Talks on L2‐Betti-Numbers, No. 2 — L2-Theory and Operator Algebras at
Free Probability Research Seminar
, Saarland University, Germany, February 24, 2021.
Unitary easy quantum group cohomology at
Cohomological properties of easy quantum groups
, Będlewo Conference Center, Poland, November 2–8, 2021.
Hochschild cohomology for easy quantum groups with Moritz Weber at
Non-commutative algebra, probability and analysis in action
, Greifswald University, Germany, September 20–21, 2021.
Half-Liberation of the Unitary Group auf
Cohomology of Quantum Groups and Quantum Automorphism Groups of Finite Graphs
, Saarland University, Germany, October 8–12, 2018.
Easy Quantum Groups between \(U_n\) and \(U_n^+\) at
Young Researchers’ Symposium, 3rd Joint Annual Meeting of the German Mathematical Society and the Society for Mathematics Didactics, Students’ Section,
Paderborn University, Germany, March 4–5, 2018.

Biography

2022-2024 Postdoc (wissenschaftlicher Mitarbeiter) at Hamburg University
2023 PhD in mathematics (Dr. rer. nat.) from Saarland University, supervisor: Moritz Weber, thesis: Classification and homological invariants of compact quantum groups of combinatorial type, defense: March 9, 2023
2019 Master in mathematics (M. Sc.) from Saarland University, supervisor: Moritz Weber, thesis: Classification of All Non-Hyperoctahedral Categories of Two-Colored Partitions
2017 Bachelor in mathematics (B. Sc.) from Saarland University, supervisor: Moritz Weber, thesis: Neutral Pair Partition Categories. All Categories of Two-Colored Pair Partitions Quantizing the Unitary Group

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