Solving differential equations was apparently Lie's original motivation for developing Lie groups. It is therefore likely one of the most understandable ways to approach it.
It appears that Lie's goal was to understand when can a differential equation have an explicitly written solution, much like Galois theory had done for algebraic equations. Both approaches use symmetry as the key tool.
- www.researchgate.net/profile/Michael_Frewer/publication/269465435_Lie-Groups_as_a_Tool_for_Solving_Differential_Equations/links/548cbf250cf214269f20e267/Lie-Groups-as-a-Tool-for-Solving-Differential-Equations.pdf Lie-Groups as a Tool for Solving Differential Equations by Michael Frewer. Slides with good examples.
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.
Bibliography:
Lie Groups, Physics, and Geometry by Robert Gilmore (2008) 7.2 "The covering problem" gives some amazing intuition on the subject as usual.
A single exponential map is not enough to recover a simple Lie group from its algebra by 
Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16 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.
Cool data embedded in the Bitcoin blockchain Len Sassaman tribute by Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16Tribute to computer security researcher Len Sassaman, who killed himself on 2011-07-03, starting with an ASCII art portrait followed by text.
Because it comes so early in the blockchain, and because it is the first ASCII art on the blochain as far as we can see, and because is so well done, this is by far the most visible ASCII art of the Bitcoin blockchain.
Present at tx 930a2114cdaa86e1fac46d15c74e81c09eee1d4150ff9d48e76cb0697d8e1d72. It does not show well on Bitcoin Inscription Indexer however with rationale described at: github.com/cirosantilli/bitcoin-inscription-indexer/tree/3f53e152ec9bb0d070dbcb8f9249d92f89effa70#smart-newline-joining
But it can be seen at on bitcoinStrings.com at: bitcoinstrings.com/blk00003.txt.
Transaction: www.blockchain.com/btc/tx/930a2114cdaa86e1fac46d15c74e81c09eee1d4150ff9d48e76cb0697d8e1d72 from 2011-07-30, a few weeks after the suicide.
Discussion: bitcoin.stackexchange.com/questions/3370/in-which-block-was-len-sassaman-memorialised/101276#101276
Created by famous computer security researcher Dan Kaminsky and Travis Goodspeed, presumably this other security researcher, evidence:
- signature on the tribute
- the art is highlighted at Video 1. "Black OPS of TCP/IP by Dan Kaminsky (2011)", which happened very few days after the art was uploaded to the blockchain, thus making it exceedingly unlikely that someone else could have done it
"Bernanke" is a reference to Ben Bernanke, who was one of the economists in power in the US Government during the financial crisis of 2007-2008, and much criticized by some, as shown for example in the documentary Inside Job (2010). As hinted in the Genesis block message, the United States Government bailed out many big banks that were going to go bankrupt with taxpayer money, even though it was precisly those banks that had started the crisis through their reckless investment, thus violating principles of the free market and business accountability. This was one of the motivations for the creation Bitcoin, which could reduce government power over economic policy.
It is worth mentioning that there do exist some slightly earlier "artistic" inscriptions in the form Punycode inscription in the Namecoin blockchain, but as far as we've seen, the are all trivial compared to
BitLen in terms of artistic value/size.---BEGIN TRIBUTE---
#./BitLen
:::::::::::::::::::
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=-=-=-=-=-=-=-=-=-=
LEN "rabbi" SASSAMA
1980-2011
Len was our friend.
A brilliant mind,
a kind soul, and
a devious schemer;
husband to Meredith
brother to Calvin,
son to Jim and
Dana Hartshorn,
coauthor and
cofounder and
Shmoo and so much
more. We dedicate
this silly hack to
Len, who would have
found it absolutely
hilarious.
--Dan Kaminsky,
Travis Goodspeed
P.S. My apologies,
BitCoin people. He
also would have
LOL'd at BitCoin's
new dependency upon
ASCII BERNANKE
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----END TRIBUTE----Code 1.
Len Sassaman tribute
. 
Official portrait of Ben Bernanke (2008)
Source. Reference image from Wikipedia for the ASCII art.Black OPS of TCP/IP by Dan Kaminsky (2011)
Source. Presented at the BlackHat 2011 conference. Dan unveils the Len memorial at the given timestamp around 8:41. The presentation was done on 2011-08-03 or 04, so very few days after the upload to the blockchain.From the JSON transaction we understand the encoding format:So it is really encoded one line at a time in the
"out":[
{
"spent":false,
"tx_index":0,
"type":0,
"addr":"1CqKQ2EqUscMkeYRFMmgepNGtfKynXzKW7",
"value":1000000,
"n":0,
"script":"76a91481ccb4ee682bc1da3bda70176b7ccc616a6ba9da88ac"
},
{
"spent":false,
"tx_index":0,
"type":0,
"addr":"157sXa7duStAvq3dPLWe7J449sgh47eHzw",
"value":1000000,
"n":1,
"script":"76a9142d2d2d424547494e20545249425554452d2d2d2088ac"
},
...
{
"spent":false,
"tx_index":0,
"type":0,
"addr":"157sXYpjvAyEJ6TdVFaVzmoETAQnHB6FGU",
"value":1000000,
"n":77,
"script":"76a9142d2d2d2d454e4420545249425554452d2d2d2d2088ac"
}script of the transaction outputs.The most important example is perhaps and , both of which have the same Lie algebra, but are not isomorphic.
Every Lie algebra has a unique single corresponding simply connected Lie group by Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16This simply connected is called the universal covering group.
Most commonly refers to: exponential map.
Like everything else in Lie group theory, you should first look at the matrix version of this operation: the matrix exponential.
The exponential map links small transformations around the origin (infinitely small) back to larger finite transformations, and small transformations around the origin are something we can deal with a Lie algebra, so this map links the two worlds.
The idea is that we can decompose a finite transformation into infinitely arbitrarily small around the origin, and proceed just like the product definition of the exponential function.
The definition of the exponential map is simply the same as that of the regular exponential function as given at Taylor expansion definition of the exponential function, except that the argument can now be an operator instead of just a number.
Furthermore, TODO confirm it is possible that a solution does not exist at all if and aren't sufficiently small.
This formula is likely the basis for the Lie group-Lie algebra correspondence. With it, we express the actual group operation in terms of the Lie algebra operations.
Notably, remember that a algebra over a field is just a vector space with one extra product operation defined.
Vector spaces are simple because all vector spaces of the same dimension on a given field are isomorphic, so besides the dimension, once we define a Lie bracket, we also define the corresponding Lie group.
Since a group is basically defined by what the group operation does to two arbitrary elements, once we have that defined via the Baker-Campbell-Hausdorff formula, we are basically done defining the group in terms of the algebra.
Cool data embedded in the Bitcoin blockchain Nelson-Mandela.jpg by Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16bitfossil.com/78f0e6de0ce007f4dd4a09085e649d7e354f70bc7da06d697b167f353f115b8e/ in block 273536 (2013-12-07).
This is one of the earliest AtomSea & EMBII uploads.
Nelson-Mandela.jpg"There is nothing like returning to a place that remains unchanged to find the ways in which you yourself have altered." - Nelson Mandela Nelson Rolihlahla Mandela was a South African anti-apartheid revolutionary, politician and philanthropist who served as President of South Africa from 1994 to 1999. - Wikipedia Born: July 18, 1918, Mvezo, South Africa Died: December 5, 2013.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
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- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. 
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All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
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