raopeng said:I must apologize if this question sounds dump but if an isomorphism is established between two groups, is it true that their lie algebra is an isomorphism too? My idea is that since the tangent space is sent to tangent space also in the matrix space, both groups' lie algebra will be isomorphism also. For example we have: SU(2) × SU(2) ≈ SO(4) and su(2) × su(2) ≈ so(4).
Isomorphism is a mathematical concept that describes a one-to-one correspondence between two structures. In the context of groups and Lie algebras, it refers to a relationship between the abstract algebraic structure of a group and the linear algebraic structure of its associated Lie algebra. This means that for every operation performed on the group, there is a corresponding operation on the Lie algebra that preserves the same structure.
The isomorphism between groups and their Lie algebra is significant because it allows us to study the properties of a group through its associated Lie algebra, which is often easier to work with. Additionally, it helps us to connect the abstract concepts of group theory with the more concrete concepts of linear algebra, providing a powerful tool for understanding both fields.
To establish isomorphism between a group and its Lie algebra, we first need to define a mapping between the elements of the group and the elements of the Lie algebra. This mapping should preserve the group operation, meaning that if we apply the group operation to two elements, their corresponding elements in the Lie algebra should also satisfy the same operation. Once this mapping is established, we can then show that it is a bijective mapping, meaning that every element in the group has a unique corresponding element in the Lie algebra.
No, not all groups and their Lie algebras are isomorphic. In fact, there are many groups that do not have a corresponding Lie algebra, and vice versa. Additionally, even if a group and its Lie algebra are isomorphic, there may be multiple ways to establish this isomorphism, as there can be different mappings between the elements of the group and the elements of the Lie algebra that still preserve the group operation.
The isomorphism between groups and their Lie algebra has numerous applications in mathematics and physics. In mathematics, it is used to study the structure and properties of groups, as well as to classify different types of groups. In physics, it is used to study the symmetries of physical systems, such as in quantum mechanics and particle physics. It is also used in the development of Lie group and Lie algebra-based methods in differential geometry and mathematical physics.