Following the findings in the field of deformable algebra, can deformation of an n-dimensional vector space contribute to changing its initial dimension?
How will this contribute to finding the appropriate deformations for infinite-dimensional algebra on the basis that algebra is a vector space with an additional structure, with the hope that this will lead to new physical systems with an infinite number of symmetries, and thus complementarity in the hope of enhancing one of what the theory of strings aspires to ?
This is one of these cases where the answer is: it depends on what you mean by deformation of a vector space.
What can certainly happen is e.g. that you have a family of finite dimensional elliptic operator say parametrised by a parameter t \in (-1,1) whose kernel for K_t for t != 0 has dimension n, whereas for t = 0 its kernel has (finite) dimension n + k > n. For algebraic families in algebraic geometry one can have similar jumping phenomena e.g. classically, when considering the linear systems of a "special" divisor on a curve of genus g, if the divisor has degree with 0
We can say, that the concept of deforming an n-dimensional vector space can indeed change its initial dimension, and this idea extends to exploring deformations in infinite-dimensional algebras, particularly in the context of theoretical physics. These deformations can impact symmetries in physical systems, offering alternative descriptions that may complement the insights from theories like string theory. Overall, studying deformed algebras is an ongoing area of research with the potential to uncover new mathematical structures and enhance our understanding of fundamental physics.
Thank you very much for your interaction (Salem Saleh and Rogier Brussée ).
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics