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Mapping spaces and automorphism groups of toric noncommutative spaces
Gwendolyn E Barnes1,2,3, Alexander Schenkel4,5, Richard J Szabo1,2,3
11Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS UK.
Letters in Mathematical Physics
|February 4, 2020
Summary
We introduce a sheaf theory method for toric noncommutative geometry, enabling the formalization of mapping spaces between toric noncommutative spaces. This approach simplifies the description of the automorphism group
Area of Science:
- Noncommutative Geometry
- Algebraic Geometry
- Category Theory
Background:
- Toric noncommutative spaces are generalizations of toric varieties with applications in mathematical physics and algebraic geometry.
- Mapping spaces and automorphism groups are fundamental objects in the study of geometric and algebraic structures.
- Existing methods for studying these structures in noncommutative settings can be complex and lack comprehensive formalization.
Purpose of the Study:
- To develop a novel sheaf theory approach for toric noncommutative geometry.
- To formalize the concept of mapping spaces between toric noncommutative spaces.
- To analyze the internalized automorphism group of toric noncommutative spaces and its associated Lie algebra.
Main Methods:
- Development of a sheaf-theoretic framework tailored for toric noncommutative spaces.
- Application of sheaf cohomology and related tools to study mapping spaces.
- Investigation of the structure of the automorphism group using categorical and algebraic methods.
Main Results:
- A rigorous formalization of mapping spaces between toric noncommutative spaces is established.
- The 'internalized' automorphism group of a toric noncommutative space is studied.
- An elementary description of the Lie algebra of the automorphism group is derived in terms of braided derivations.
Conclusions:
- The developed sheaf theory approach provides a powerful and unified framework for toric noncommutative geometry.
- This work offers new insights into the structure of mapping spaces and automorphism groups in noncommutative settings.
- The elementary description of the Lie algebra simplifies further analysis and applications of these geometric objects.
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