Keyed Hashing and verification Is there any standard algorithm where I can create a hash H(x,k) where x being the data and k being the private key. And verify on other end with H(x,l) where l being the public key?

How it can be achieved?
as SEJPM said you're looking for digital signature algorithms.
If you want names and standards, there's ECDSA based on elliptic curves and DSA based on the RSA philosophy.

https://en.wikipedia.org/wiki/Digital_signature

Good luck!