Customer Reviews

Theory of Continuous Groups (Dover Books on Mathematics)

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Short, and much more accessible than other modern treatments, getting quickly to the meat and bones. Loewner eschews "abstraction for the sake of abstraction", which lets you see what the content of continuous groups is about (representations, integration on groups) without your getting lost in a maze of formalism: you feel rooted.
To get through this little book quickly, you need to have a sense of what cosets do for you (i.e., not bothering to distinguish between elements that are equivalent to each other), and the notion in linear algebra of decomposing linear transformations orthogonally.
Loewner is especially useful if you've decided to dip into the three-volume work by Lie/Engels, or (better) the introductory version worked out by Scheffers of Lie's Vorlesungen ueber continuierliche Gruppen. Loewner's lectures in fact result from his attempt to get something out of the Lie/Engel treatise. The difficulty with trying to read Lie is that while he thought in terms of a group as an element in an r-dimensional sub-manifold of a transformation space, never-the-less he felt he had to couch his presentation in a form more congenial to the analysts of his day, which obscured his presentation. He added in small print what he was really trying to do -- structurally, and even with commutative diagrams! -- but first you have to wade through the analytic component-by-component treatment. What Loewner did was to extract the essence for you.
I recommend Loewner's precis, along with the more leisurely introduction by Lie/Scheffers.