- #1
LagrangeEuler
- 717
- 20
Is it ##(\mathcal{R} without \{0\},\cdot)## Lie group?
LagrangeEuler said:I think you want to say that ##(\mathbb{R}\setminus\{0\},\cdot)## is ##GL(1,\mathbb{R})##. I never heard about algebra ##gl(1,\mathbb{R})##.
LagrangeEuler said:I always thought that advantage of Lie algebra is finite number of generators, while group has infinity elements. But in this case, if I understand you well, you have infinite number of generators. So I do not understand concept of Lie algebra any more. :(
LagrangeEuler said:When you take exponentials of real numbers you would not get negative numbers. So I'm not sure in this particular case how you will reproduce all members of group ##(\mathbb{R},\setminus\{0\})##, starting from Lie algebre ##\mathbb{R}##.
A Lie group without 0 is a mathematical concept that refers to a type of algebraic structure called a group, which also satisfies certain smoothness or continuity properties. This type of group does not include the element 0, or the identity element, in its structure.
A regular Lie group includes the element 0, or the identity element, in its structure. This means that the group's operation is closed under multiplication, and the identity element acts as the neutral element for this operation. A Lie group without 0, on the other hand, does not have an identity element and may not be closed under multiplication.
Examples of Lie groups without 0 include the multiplicative group of positive real numbers, the special unitary group, and the symplectic group. These groups do not have an identity element, as the identity element would be equal to 0, which is not included in their structure.
Studying Lie groups without 0 can provide insights into more general algebraic structures and their properties. It also allows for a deeper understanding of the behavior of groups without the constraints of an identity element. In addition, Lie groups without 0 have applications in various fields such as physics, geometry, and differential equations.
One of the main challenges in studying Lie groups without 0 is the complexity of their structure. These groups may not have an obvious geometric interpretation, making it more difficult to visualize and understand their properties. Additionally, the lack of an identity element can make some operations and proofs more challenging.