# Definition In mathematics, [[Lie Algebra|Lie algebras]] of [[Matrix Lie Group|matrix Lie groups]] are defined as: $ \mathfrak{g} \equiv \{X \in M_n(\mathbb{C})\, | \,e^{tX} \in G \,\,\forall\,t \in \mathbb{R}\} $ If we are working with [[Isometry Group|isometry groups]], the elements of these Lie algebras are *[[Hermitian Operator|anti-Hermitian matrices]]*. However, physicists define their Lie algebras as: $ \mathfrak{g}_{\text{physics}} \equiv \{X \in M_n(\mathbb{C})\, | \,e^{itX} \in G \,\,\forall\,t \in \mathbb{R}\} $ so that the elements of the Lie algebra are *[[Hermitian Operator|Hermitian matrices]]*. We then multiply these elements by $i$ to get anti-Hermitian matrices, which then exponentiate into the isometries. The most important thing to understand is that physicists *define* $X$ to be Hermitian, while mathematicians define it to be anti-Hermitian. This is where the factor of $i$ comes from. The reason for this discrepancy is that physicists use the elements of the [[Lie Algebra]] as observables, i.e., operators which correspond to physically measurable quantities. Clearly, these must have real eigenvalues, hence the need for them to be Hermitian. This definition is implicit in much of the physics literature. It really just amounts to a factor of $i$.