A Lie [group](Groups.md) is a type of [infinite Group](Infinite%20group.md) with group operations given by differentiable functions. A Lie Group is a [[differentiable manifold]] with the property that make it a group. That is, a smooth manifold $G,$ is a Lie group if it is equipped with a product $G\times G \rightarrow G$ In addition, the inverse map is also smooth. A Lie Group is also a [continuous group](Continuous%20groups.md) since smooth maps are also [continuous](Maps.md#Continuous%20maps). ^60524f # Lie subgroups # Matrix and non-matrix Lie groups A Lie Group may be a [matrix Lie group](Matrix%20Lie%20group.md) or a [non-matrix Lie groups](Non-matrix%20Lie%20groups.md). # Lie group dimensionality Note that the dimensionality of a Lie Group is defined as $\mathrm{dim}/\mathbb{R}$ or $\mathrm{dim}/\mathbb{C}$, which for us will be relevant when looking at the dimensionality of [matrix Lie group](Matrix%20Lie%20group.md)s. A Lie Group elements are [matrix exponentials](Matrix%20exponentials.md), which give the [Lie group representation](Lie%20group%20representations.md). #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups