Generalization of orthogonal group to preserve different bilinear forms. Important because the Lorentz group is .
Definition of the indefinite orthogonal group by 
Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01Given a matrix with metric signature containing positive and negative entries, the indefinite orthogonal group is the set of all matrices that preserve the associated bilinear form, i.e.:Note that if , we just have the standard dot product, and that subcase corresponds to the following definition of the orthogonal group: Section "The orthogonal group is the group of all matrices that preserve the dot product".
As shown at all indefinite orthogonal groups of matrices of equal metric signature are isomorphic, due to the Sylvester's law of inertia, only the metric signature of matters. E.g., if we take two different matrices with the same metric signature such as:and:both produce isomorphic spaces. So it is customary to just always pick the matrix with only +1 and -1 as entries.
Indefinite special orthogonal group by Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01Like the special orthogonal group is to the orthogonal group, is the subset of with determinant equal to exactly 1.
Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of) , that has the same properties as the group.
Or in other words, associating to the more abstract notion of a group more concrete objects with which we are familiar (e.g. a matrix).
Each such matrix then represents one specific element of the group.
This is basically what everyone does (or should do!) when starting to study Lie groups: we start looking at matrix Lie groups, which are very concrete.
Or more precisely, mapping each group element to a linear map over some vector field (which can be represented by a matrix infinite dimension), in a way that respects the group operations:
As shown at Physics from Symmetry by Jakob Schwichtenberg (2015)
- page 51, a representation is not unique, we can even use matrices of different dimensions to represent the same group
- 3.6 classifies the representations of . There is only one possibility per dimension!
- 3.7 "The Lorentz Group O(1,3)" mentions that even for a "simple" group such as the Lorentz group, not all representations can be described in terms of matrices, and that we can construct such representations with the help of Lie group theory, and that they have fundamental physical application
Bibliography:
- www.youtube.com/watch?v=9rDzaKASMTM "RT1: Representation Theory Basics" by MathDoctorBob (2011). Too much theory, give me the motivation!
- www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609 The "Useless" Perspective That Transformed Mathematics by Quanta Magazine (2020). Maybe there is something in there amidst the "the reader might not know what a matrix is" stuff.
Classification of simple Lie groups by Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01A bit like the classification of simple finite groups, they also have a few sporadic groups! Not as spectacular since as usual continuous problems are simpler than discrete ones, but still, not bad.
An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011) by Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01This does not seem to go deep into the Standard Model as Physics from Symmetry by Jakob Schwichtenberg (2015), appears to focus more on more basic applications.
But because it is more basic, it does explain some things quite well.
Lie Groups, Physics, and Geometry by Robert Gilmore (2008) by Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01The author seems to have uploaded the entire book by chapters at: www.physics.drexel.edu/~bob/LieGroups.html
And the author is the cutest: www.physics.drexel.edu/~bob/Personal.html.
Overview:
- Chapter 3: gives a bunch of examples of important matrix Lie groups. These are done by imposing certain types of constraints on the general linear group, to obtain subgroups of the general linear group. Feels like the start of a classification
- Chapter 4: defines Lie algebra. Does some basic examples with them, but not much of deep interest, that is mostl left for Chapter 7
- Chapter 5: calculates the Lie algebra for all examples from chapter 3
- Chapter 6: don't know
- Chapter 7: describes how the exponential map links Lie algebras to Lie groups
Naive Lie theory by John Stillwell (2008) by Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01 Lie Algebras In Particle Physics by Howard Georgi (1999) by Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01 Pinned article: ourbigbook/introduction-to-the-ourbigbook-project
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