Representation of a Lie group
Ok, so imagine you have a group of friends, and all of you like to dance together in a certain way. You all have different moves, but when you dance together, you make a nice pattern.
Now, imagine you want to teach a new friend how to dance with your group. You could try to teach them every single move that each person does, but that might take a really long time and be confusing for your new friend.
Instead, you could teach your new friend some basic moves that represent the whole group's style of dancing. These moves could be combined in different ways to make the larger patterns that your group does.
In math, there are also groups, called Lie groups. These groups have certain moves or transformations that they can do. And just like with your dance group, we can represent the whole group's transformations with some basic moves or transformations.
These basic moves are called a representation of the Lie group. They capture the essence of the group's transformations, and we can use them to describe how the group acts on different objects or spaces.
Now, imagine you want to teach a new friend how to dance with your group. You could try to teach them every single move that each person does, but that might take a really long time and be confusing for your new friend.
Instead, you could teach your new friend some basic moves that represent the whole group's style of dancing. These moves could be combined in different ways to make the larger patterns that your group does.
In math, there are also groups, called Lie groups. These groups have certain moves or transformations that they can do. And just like with your dance group, we can represent the whole group's transformations with some basic moves or transformations.
These basic moves are called a representation of the Lie group. They capture the essence of the group's transformations, and we can use them to describe how the group acts on different objects or spaces.