##  [Lie Group](/lie-group) 

  ##  [Lie Group](https://quantummech.quantumdictionary.io/lie-group-0) 

  

 [![Quantum Mechanics Dictionary](/sites/default/files/styles/large/public/2026-01/Quantum%20Mechanics.png.webp?itok=tC_FGCW0)](/topic-specific-dictionaries/modern-physics/quantum-mechanics)

- Modern Physics -

**Quantum Mechanics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A symmetry concept defining transformations that leave key properties of a quantum system invariant. It governs conserved quantities, degeneracies, and selection rules through the action of generators and representations. It does not imply exact invariance when symmetry-breaking terms or boundary effects are present in the Hamiltonian.



 

 

 

 

 





 

 



 ##  [Lie Group](https://algebra.quantumdictionary.io/lie-group-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A group that is also a finite-dimensional smooth manifold in which the group operations (multiplication and inversion) are smooth maps; thus it combines algebraic group structure with differential-geometric structure.