Are SU(2) and SO(3) Groups Really Isomorphic?

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SUMMARY

SU(2) and SO(3) groups are not isomorphic; SU(2) serves as a double cover of SO(3). While their Lie algebras are isomorphic and they exhibit local isomorphism, the overall structures differ significantly. This distinction is crucial for understanding the relationship between these two mathematical entities in the context of group theory and topology.

PREREQUISITES
  • Understanding of group theory concepts
  • Familiarity with Lie algebras
  • Knowledge of double covering in topology
  • Basic principles of algebraic structures
NEXT STEPS
  • Study the properties of Lie algebras in depth
  • Explore the concept of double covers in topology
  • Investigate the applications of SU(2) in quantum mechanics
  • Learn about the geometric interpretations of SO(3) transformations
USEFUL FOR

Mathematicians, physicists, and students studying group theory, particularly those interested in the applications of SU(2) and SO(3) in theoretical physics and geometry.

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I have not seen why SU(2) and SO(3) groups are isomorphic?
 
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They aren't isomorphic. SU(2) is a double cover of SO(3).
 
Their Lie algebras are isomorphic, and they are locally isomorphic...but D H is right - they aren't isomorphic!
 

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