Are there any applications of Lie algebras or Lie groups in cryptography? I am interested in the applications of algebra in general to cryptography. In particular, I am wondering if there are applications of Lie algebras and Lie groups to cryptography. 


Are Lie algebras or groups used in cryptography?
How could Lie algebras or groups be used in a cryptographic algorithm? 
Are there any, for example, cryptographic algorithms which use Lie algebras?
My knowledge of Lie algebras and groups is limited, so I hope I get at least some things right.


  
  Are Lie algebras or groups used in cryptography?
  


No. Except perhaps by some daredevils, unable to communicate with the rest of us.


  
  Are there any, for example, cryptographic algorithms which use Lie algebras?
  


No. Except perhaps some super exotic ones which nobody uses.

Let's focus on the more interesting question. 


  
  How could Lie algebras or groups be used in a cryptographic algorithm? 
  


I'd say the two main use cases of asymmetric crypto are key exchange and signatures. Given any group, we can build both. For key exchange we use Diffie-Hellman, and for signatures we can use for example Schnorr signatures.

A Lie group has a group structure. By the above, we can use them for key exchange and signatures. The answer to the following question is crucial: how hard is the discrete logarithm problem in a Lie group? Once that is answered, the following question is crucial: what is the size of the representation of private and public keys to obtain a 128-bit security level?
Once that is answered, the following question is crucial: are we more attractive than elliptic-curve groups or multiplicative groups of finite fields?

If the answer to the last question is no, then you probably don't want to use Lie groups. My 2-cents: they seem unlikely to be useful. For the discrete logarithm problem to be hard, we'd want our group to have as little extra structure as possible. By the definition of a Lie group, we require a lot of geometric structure which has no use for our protocols. This may make the dlog problem easier, and hence parameters larger.

For Lie algebras there is no direct use-case. When I googled this, I ended up here. So apparently someone has been able to Diffie-Hellman using some Lie algebra structure. Again, ask the questions from above. If you do worse than what's already out there, what is your use-case?

Of course, perhaps there are other ways to build protocols using the Lie algebra/group structure. It could be interesting, but it may be too little too late.