What Is a Lie Group Without an Identity Matrix Called?

In summary, the topic being discussed is the mathematical study of Lie groups without using the identity matrix as a member. This would result in a semigroup or a more general concept known as a "magma". A related concept is a torsor, described by John Baez as a group without its identity.
  • #1
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Is there a name for studying a Lie "group" that doesn't use the identity matrix as a member of the group?

I know it's not technically a group anymore, but is there any mathematical work pertaining to the general idea... and what is the terminology so that I can research it better?
 
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  • #2
If there is no identity then there are no inverses, so at best you'll be left with a semigroup.
 
  • #3
thanks! "Semigroup" led me to "magma", which is more the generalized concept I was thinking of. <3 Wiki.
 
  • #4

FAQ: What Is a Lie Group Without an Identity Matrix Called?

What is a Lie Group sans Identity Matrix?

A Lie Group sans Identity Matrix is a mathematical concept that describes a group of matrices that are closed under matrix multiplication and have a smooth structure. The term "sans Identity Matrix" means that the group does not include the identity matrix, which is a special matrix that does not change the values of other matrices when multiplied together.

How is a Lie Group sans Identity Matrix different from a regular Lie Group?

A regular Lie Group includes the identity matrix as one of its elements, while a Lie Group sans Identity Matrix does not. This means that the identity matrix is not considered a part of the group and does not have any special properties within the group.

What are some examples of Lie Groups sans Identity Matrix?

Some examples of Lie Groups sans Identity Matrix include the special orthogonal group, the special unitary group, and the special linear group. These groups are used in various areas of mathematics and physics to describe symmetries and transformations.

How are Lie Groups sans Identity Matrix used in scientific research?

Lie Groups sans Identity Matrix are used in scientific research to study symmetries and transformations in various systems, such as in quantum mechanics and general relativity. They are also used in data analysis and machine learning for dimension reduction and feature extraction.

Are there any practical applications of Lie Groups sans Identity Matrix?

Yes, there are several practical applications of Lie Groups sans Identity Matrix in fields such as computer graphics, robotics, and computer vision. They are also used in engineering and physics to model and analyze the behavior of complex systems.

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