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Algebra (Shibukawa) Lab
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The lab studies the Yang–Baxter equation, which underlies why lattice models are solvable.
AboutAbout the lab
Toward the equation that makes solvability possible
Prof. Shibukawa studies the Yang–Baxter equation and its solutions, within the field of integrable systems.
In statistical mechanics, the (quantum) Yang–Baxter equation is the equation that guarantees a lattice model is solvable.
The braid relation, which is equivalent to this equation, also makes sense as an equation on morphisms in a tensor category.
The lab considers solutions in various tensor categories, seeking to grasp connections with representation theory, Hopf algebras, and solitons from a unified perspective.
Key terms
- Yang–Baxter equation
- An equation that guarantees that a lattice model in statistical mechanics is solvable. It is also closely related to representation theory and quantum groups
- Integrable system
- An equation equipped with a mechanism that makes it solvable, or the system that such an equation describes
- Representation theory
- A field that represents elements of algebra in concrete forms such as matrices and studies their structure
ResearchResearch themes
The lab studies the Yang–Baxter equation, which guarantees the mechanism by which lattice models are solvable, and its solutions. By considering solutions in different categories, it looks for connections to other fields.
*Illustrative imageBraid Relation on Tensor Categories
The Yang–Baxter equation is originally an equation about linear maps. Because the equivalent braid relation also makes sense for morphisms in a tensor category, solutions can be considered in various tensor categories.
*Illustrative imageDynamical Yang–Baxter Maps
These are solutions of the Yang–Baxter equation in a category obtained by slightly modifying the category of all sets. Through this research, the lab seeks to understand how the equation relates to representation theory, Hopf algebras, solitons, and more.
MembersFaculty
- ProfessorYoichi Shibukawa
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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: “Braid 4perm” Claudio Rocchini / CC BY 2.5 (cropped)
- Braid Relation on Tensor Categories: “Figure eight knot complement” François Guéritaud, Saul Schleimer, and Henry Segerman. / CC BY-SA 3.0 (cropped)
- Dynamical Yang–Baxter Maps: “KdV simulation” Mikko Laine / CC BY-SA 3.0 (cropped)
- Related lab (幾何系
（古畑仁） ): “Canal-helix” Ag2gaeh / CC BY-SA 3.0 (cropped) - Related lab (数理科学系
（眞崎聡） ): “Baltic Sea Wave (Cien Water)” Marek Nowocień / CC BY 2.5 (cropped) - Related lab (数理科学系
（坂井哲） ): “3Dpercolation” David Adams / CC BY 3.0 (cropped) - Related lab (数理科学系
（黒田紘敏） ): “Newly fallen snow crystals accumulating on a window sill” W.carter / CC0 (cropped)
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