A Lie group is a type of mathematical group that is also a smooth manifold, meaning it is a continuous, non-intersecting surface. It is named after mathematician Sophus Lie and is used in many areas of mathematics, physics, and engineering.
A Lie group is considered real if its elements, or group elements, are defined using real numbers. This means that the group elements can be expressed as points on a real number line and that the group operations (such as multiplication) can be performed using real numbers.
A Lie group being real has several important implications. It means that the group elements and operations can be easily understood and visualized using real numbers, making it more accessible for practical applications. It also allows for the use of powerful mathematical tools, such as calculus and differential equations, in studying and analyzing the group.
Yes, a Lie group can be both real and complex. In fact, many important Lie groups, such as the special unitary group and the orthogonal group, are both real and complex. In these cases, the group elements are defined using both real and imaginary numbers, and the group operations involve both real and complex multiplication.
The concept of a real Lie group is related to many other areas of mathematics and science. It is used in differential geometry, topology, and algebraic geometry to study and classify different types of Lie groups. In physics, real Lie groups are used to describe symmetries and transformations in various systems, such as in quantum mechanics and general relativity.