I'm a graduate aerospace engineering who's really interested in the topic of numerical methods and in particular for structure preserving schemes. Hopefully at September I will start a phd in this subject, as a consequence of my work of thesis. However, I do feel my mathematical background is kinda poor, coming from an aerospace engineering master degree. Indeed, I'm already excited to follow some of the courses I will in my phd courses about advanced linear algebra and functional analysis. During the summer I have some spare time, and I would love to study some of the books I encountered during my work of thesis. In particular "Geometric Numerical Integration" by Hairer et al. As soon as I started reading this book I encountered some mathematical topics that I never encountered during my degree courses: Lie algebra, Lie groups (groups in general) and manifold. I would like to know what are the books you suggest to get into these topics.
I saw that "Lie Groups, Lie Algebras, and Representations" by Brian C. Hall is a really good book as a starting point for this topic, but I know there's a more general introduction to lie groups which is related to manifolds. So which books do you suggest to introduce me to these topics ? Also, I would like to refresh some knowledge about the algebra courses I did in the past, but I don't really know which book I should refer to. Thank you so much for your help.