Which Clifford Algebras Admit Multiplicative Inverses?

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In summary, Clifford algebras with inverse, also known as Clifford algebras with involution, are a type of algebraic structure that has many applications in physics, mathematics, and engineering. They are a generalization of Clifford algebras and have an additional structure called an involution, which allows for the existence of inverses for each element. Resources such as books and online materials are available for further study. They have also been applied to real-world problems in fields such as computer graphics, robotics, and quantum mechanics.
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Hello,
Let's assume we have a Clifford algebra of one of the following two kinds: [tex]\mathcal{C}\ell_{n,0}[/tex] or [tex]\mathcal{C}\ell_{0,n}[/tex].

Is it possible to say which algebras will admit a multiplicative inverse for each multivector element?
 
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[tex]Cl_{0,1}=\bf{C}[/tex]
[tex]Cl_{0,2}=\bf{H}[/tex]

See the table on p. 11 in Atiyah, Bott, Shapiro "http://www.dleex.com/read/7878" "
 
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FAQ: Which Clifford Algebras Admit Multiplicative Inverses?

What are Clifford algebras with inverse?

Clifford algebras with inverse, also known as Clifford algebras with involution, are a type of algebraic structure that is built upon a vector space and equipped with a multiplication operation. This multiplication operation satisfies certain properties, including the existence of an inverse for each element in the algebra.

What is the significance of Clifford algebras with inverse?

Clifford algebras with inverse have many applications in physics, mathematics, and engineering. They can be used to represent and study geometric objects, rotations in space, and electromagnetic fields. They also have connections to other areas of mathematics, such as topology, differential geometry, and representation theory.

How are Clifford algebras with inverse related to Clifford algebras?

Clifford algebras with inverse are a generalization of Clifford algebras, which are also built upon a vector space and equipped with a multiplication operation. The main difference is that Clifford algebras with inverse have an additional structure, known as an involution, which allows for the existence of inverses for each element.

How can I learn more about Clifford algebras with inverse?

There are many resources available to learn more about Clifford algebras with inverse. Some recommended books include "Clifford Algebras and their Applications in Mathematical Physics" by Rafal Ablamowicz and Bertfried Fauser, and "Geometric Algebra for Physicists" by Chris J. L. Doran and Anthony N. Lasenby. Online resources, such as lecture notes and video lectures, are also available for further study.

Can Clifford algebras with inverse be applied to real-world problems?

Yes, Clifford algebras with inverse have many applications in real-world problems. For example, they have been used in computer graphics and computer vision for 3D object recognition and manipulation. They have also been applied in robotics and control theory for modeling and control of complex systems. Additionally, they have been used in quantum mechanics to study entanglement and quantum teleportation.

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