# Definition Assuming the [[Group Representation|representation space]] is equipped with an [[inner product]] which is preserved by the operators of a group $G$, we have a [[group representation]] of the form: $ \Pi: G \rightarrow \text{Isom}(V) $ and a corresponding [[Lie algebra]] representation $ \Pi: \mathfrak{g} \rightarrow \mathfrak{isom}(V) $ In this case, the representation is called *unitary*. See also: [[Theorem on the Unitarity of Representations]]. # Physics It is natural to require symmetry generators (i.e., elements of the [[Lie algebra]]) in quantum mechanics to be [[Hermitian Operator|anti-Hermitian]], (i.e., be elements of $\mathfrak{isom}(V)$), because dividing them by $i$ yields [[Hermitian Operator|Hermitian operators]] which are [[Diagonalizable Matrix|diagonalizable]] to have real [[Eigenvalue|eigenvalues]], as required of observables. Note that in the [[physicists' definition of Lie algebras]], elements of $\mathfrak{isom}(V)$ will be Hermitian from the get-go.