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Ring Theory and Representation Theory Lab
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The lab works to uncover the structure of the module categories and derived categories of algebras.
AboutAbout the lab
Understanding algebras through their collections of modules
Prof. Tsukamoto studies the representation theory of algebras.
An algebra is a ring that also has the structure of a vector space.
Rather than studying an algebra directly, the approach of representation theory is to study the structure of the entire collection of modules that reflects its properties (the module category).
Established in the 1970s, this is a relatively new field that continues to develop in close connection with commutative ring theory, algebraic geometry, and the representation theory of Lie algebras and quantum groups.
Key terms
- algebra
- A ring that has the structure of a vector space. In addition to addition and multiplication, it allows scalar multiplication by numbers
- module category
- The collection of all modules, which reflects the properties of an algebra. In the representation theory of algebras, this structure is what is studied
- tilting theory
- A theory that studies the structure of the module category and derived category of an algebra using tilting modules and related objects as clues
ResearchResearch themes
One of the main goals of the representation theory of algebras is to uncover the structure of module categories and their derived categories. The lab is particularly interested in two themes: quasi-hereditary algebras and extriangulated categories (ET categories).
*Illustrative imageQuasi-hereditary algebras
This is a class of algebras introduced to study the representation theory of Lie algebras and algebraic groups. The lab studies strongly quasi-hereditary algebras, a class with good homological properties, using category-theoretic methods, and has recently been examining them from the viewpoint of tilting theory.
*Illustrative imageExtension-closed subcategories and ET categories
Extension-closed subcategories do not necessarily have the structure of an abelian or triangulated category, so ET categories were introduced as a framework to treat them in a unified way. In a co-authored paper, the professor focused on the fact that the extension kernel of a t-structure has the structure of an ET category, and generalized the method of constructing a new t-structure via a torsion pair (HRS tilting).
MembersFaculty
- Associate ProfessorMayu Tsukamoto
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Combined group axiom commutative diagram” Struoc / CC BY-SA 4.0 (cropped)
- Quasi-hereditary algebras: “Affine Dynkin diagrams” self / Public domain (cropped)
- Extension-closed subcategories and ET categories: “Complex ses diagram” Mbw314 / Public domain (cropped)
- Related lab (Ishii Lab): “Surface of degree 7” Jean Constant / CC BY-SA 4.0 (cropped)
- Related lab (Analytic and Algebraic Number Theory Lab): “ZetaFunction” Gaussianer / Public domain (cropped)
- Related lab (Combinatorics, Theoretical Computer Science, Algorithm Theory, and Optimization Theory Lab (Hiroshi Hirai)): “Imperialist-competitive-algorithm-flowchart-bg” Original author: Icasite Author of the derivative work: Spiritia / CC BY-SA 3.0 (cropped)
- Related lab (Algebraic Geometry, Representation Theory, Infinite-Dimensional Integrable Systems, and Mathematical Physics Lab (Shintaro Yanagida)): “Truncated icosahedron cherry model by George W. Hart” George W. Hart / CC BY-SA 4.0 (cropped)
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