Elements of a Lie algebra can (should!) be seen a continuous analogue to the generating set of a group in finite groups.
For continuous groups however, we can't have a finite generating set in the strict sense, as a finite set won't ever cover every possible point.
But the generator of a Lie algebra can be finite.
And just like in finite groups, where you can specify the full group by specifying only the relationships between generating elements, in the Lie algebra you can almost specify the full group by specifying the relationships between the elements of a generator of the Lie algebra.
The reason why the algebra works out well for continuous stuff is that by definition an algebra over a field is a vector space with some extra structure, and we know very well how to make infinitesimal elements in a vector space: just multiply its vectors by a constant that cana be arbitrarily small.
Lie Groups, Physics, and Geometry by Robert Gilmore (2008) 7.2 "The covering problem" gives some amazing intuition on the subject as usual.
The product of a exponential of the compact algebra with that of the non-compact algebra recovers a simple Lie from its algebra by 
Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16Furthermore, the non-compact part is always isomorphic to , only the non-compact part can have more interesting structure.
The most important example is perhaps and , both of which have the same Lie algebra, but are not isomorphic.
Because the Lie bracket has to be a bilinear map, all we need to do to specify it uniquely is to specify how it acts on every pair of some basis of the Lie algebra.
Then, together with the Baker-Campbell-Hausdorff formula and the Lie group-Lie algebra correspondence, this forms an exceptionally compact description of a Lie group.
The group of all transformations that preserve some bilinear form, notable examples:
- orthogonal group preserves the inner product
- unitary group preserves a Hermitian form
- Lorentz group preserves the Minkowski inner product
What happens to the definition of the orthogonal group if we choose other types of symmetric bilinear forms by Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16We looking at the definition the orthogonal group is the group of all matrices that preserve the dot product, we notice that the dot product is one example of positive definite symmetric bilinear form, which in turn can also be represented by a matrix as shown at: Section "Matrix representation of a symmetric bilinear form".
By looking at this more general point of view, we could ask ourselves what happens to the group if instead of the dot product we took a more general bilinear form, e.g.:The answers to those questions are given by the Sylvester's law of inertia at Section "All indefinite orthogonal groups of matrices of equal metric signature are isomorphic".
- : another positive definite symmetric bilinear form such as ?
- what if we drop the positive definite requirement, e.g. ?
The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose by Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16Let's show that this definition is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.
Note that:and for that to be true for all possible and then we must have:i.e. the matrix inverse is equal to the transpose.
These matricese are called the orthogonal matrices.
TODO is there any more intuitive way to think about this?
Elements of the orthogonal group have determinant plus or minus one by Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16This is crap, became slow very fast. The battery is IMPOSSIBLE to remove!!! youtu.be/kO-RwIQ_i1w?t=162 Battery was 4.163V when thing wouldn't turn on anymore. But topbattery.co.uk/product/original-battery-for-tablet-lenovo-yoga-tab-3-yt3-x50fyt3-x50m-sl20076-2/ says it is 3.6V. What?
Group of the unitary matrices.
Complex analogue of the orthogonal group.
One notable difference from the orthogonal group however is that the unitary group is connected "because" its determinant is not fixed to two disconnected values 1/-1, but rather goes around in a continuous unit circle. is the unit circle.
Pinned article: Introduction to the OurBigBook Project
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Intro to OurBigBook
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Figure 3. Visual Studio Code extension installation.
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