Lie group–Lie algebra correspondence Puns

Hey guys,

Please just a quick answer and since I need this for a report I am writing on, some source would be nice as well :)

If H denotes a seperable Hilbert space, End(H) = GL(H), the space of bounded invertible matrices. I guess its a Lie group as well, but what I need would be the corresponding Lie algebra sadly. I know that for finite dimensions this is just the space of n x n square matrices, no problemo here, but I am a little bit afraid of the "infinite-dimensional" stuff to be honest haha. Scares me, because things can get tight there. If you tell me its just the space of infinite dimensional matrices, this would be fine with me actually. If you then tell me that the Lie bracket induced on it is just the commutator AB-BA I would be even happier :D both is true for finite dimensions, but as said, infinte dimensions are strange sometimes.

Thank you very much for your help :)

Let L be the perfect Lie algebra. There's no further information about the Lie bracket given.

My approach is to consider a non-zero ideal I of L. Then since L is perfect, I is not the derived algebra of L, nor any other term in the derived series of L. But I'm having trouble thinking of what other possible ideals there could be (due to lack of information on the Lie bracket) and how to rule them out (though this step will probably be easier once I know what the ideals could be).

If I understand correctly the dimension of ideals of L must be less than or equal to three. Normally if there was info on the Lie bracket then I could consider a typical element of the ideal and manipulate it using the bracket to show that the ideal must equal the whole ring. Since that information isn't given, I'm assuming it must be something to do with the dimension of the algebra but I can't see what.

Hi. I'm a MSc Theoretical Physics student (we're in the maths department). I'm lost.

I'm into 3D computer vision and have been reading a book on slam algorithms. But I'm not quite good with lie algebra and the basic understanding of it. What concepts and books do you recommend? I've done linear algebra, probability, statistics and calculus. Do I need to start with abstract algebra? I'm looking for understanding concepts like SO(3) and se(3) relation, manifolds and topology? All the notes I searched start with heavy vocabulary.

I am speaking in the context of matrix representations, in the context of quantum mechanics.

1. I occasionally see people describe a representation as "the space that the transformation acts on". Act the transformation on a state can be done by acting the corresponding matrix on the corresponding column vector. Then the quote implies a representation is a choice of both the matrices and column vectors?
2. Highest weight representation - continuing from the above argument, using the ( j, m) notation, the weight is referring to the eigenvalue m of a state vector, but the matrix rep only depends on j and does not care about m. Then what that mean by highest weight (largest m) representation (matrices)?

I am a physics student so what I typed can be sloppy to mathematicians, sorry about that!

I'm currently reading slambook and realized that I need to get a good grip of lie algebra and groups. I tried khan academy and there aren't any. How about this youtube series? Can you please point out which videos I should actually go through cause the author makes videos related to quantum mechanics.

What are some books or other courses which I can refer to?

It is a theorem that given any real lie algebra, there is a unique simply connected group that has that lie algebra as its lie algebra.

I was wondering if there is any known analogue of thai for algebraic groups? Or if there can be no analogue?

I believe every lie group is algebraic (but I could be wrong, so let me know) so this algebraic statement would imply this original theorem. But the concept of being simply connected is messy over general fields.

Anyone who knows anything in this realm would be helpful.

1. What are the prerequisites?
2. How much should I learn?
3. What are some good resources answering above question?

Hi, I'm trying to understand how lie groups and their lie algebra work and how they're useful in understanding rigid body motion from mathematical perspective. I am reading from http://ethaneade.com/lie.pdf. I'd appreciate if someone could verify/correct my understanding of the following points.

1. SO(3) is an abstract space ∈ R^3 and only way to interact with its elements in 'real world' is to convert them to Lie algebra form so(3) which is represented as 3x3 matrix.

2. 'Adjoint' of Lie group means transforming an element in tangent space around one element to the tangent space of another (ie from one so(3) element to another so(3) without decomposing to SO(3) via exponentiation. i.e simple matrix multiplication). Is there a better explanation?

3. [this one is really confusing] When calculating jacobian, I didn't exactly understand how equation 28 came to be. Why did we first left multiply the SO(3) rotation by exp(ω) ? Isn't it it supposed to be other way around by taking log of R instead?

4. In the same section they derive dy/dR = -y. How exactly to interpret this?

5. In the next section(2.4.2) what do symbols ε and δ denote? is δ an se(3) element? what are their dimensions?

6. In jacobian derivation sections (2.4) and (3.4) for SO(3) and SE(3), what do equations (44) and (97) really mean? It is just a fancy way of saying "in order to reach transform at frame3 from origin you need to multiply transform at frame2 x frame1 ?"

Edit: spelling

A definition of a Lie Group states that a mapping exists such that G X G -> G. However, examples of Lie groups given include the unit circle (S1) and a real line (R1). However, a Cartesian product would not create a valid mapping, as S1 X S1 should return S2, and R1 X R1 should return R2, both of which force extra dimensions, thus making a mapping impossible. What am I doing wrong here?

I'm independently working through some quantum field theory texts right now, and I'm very unsatisfied with the haphazard way they cover lie groups. I am very interested in a more comprehensive introduction to these topics, with proofs and clear definitions, instead of the vague hand-waiving I get from physics books.

Since these groups are really at the heart of modern theoretical physics (and beautiful in their own right!), I wanted to embark on a quest to more thoroughly understand them in a rigorous way. I know basic differential geometry (enough for General Relativity) and basic group theory (a good chunk of Dummit and Foote). What are the next steps I should take towards the study of Lie groups?

I would imagine I may need a stronger foundation in differential geometry and group/representation theory, and I definitely need to learn a lot more about topology. Is this the case? What exactly are the dependencies here, and should I go about tackling these subjects in a particular order? Does anyone have any texts they could recommend?

I'm working on my own schedule, so I will take as much time as is necessary. Thanks for your help in advance!

I saw garrett lisi's video on his theory of everything. I want to understand but I don't know lie algebra. what the heck is lie algebra? I tried to watch a few of mit's gilbert's videos on lie but my attention span couldn't hold. could you make a simplified video on lie algebra?

Like the caption says, I have read like 5 books, 4 youtube videos. And the notion of weight decomposition, how to find weights of Lie Algebras is just not clicking in. Can anyone please help me out?

Hi, i'm actually studying Lie algebras and i've read that it has relations with quantum physics. Can anyone give a short introduction/explain how is this relation?

Today's topic is Lie Groups and Lie Algebras.

This recurring thread will be a place to ask questions and discuss famous/well-known/surprising results, clever and elegant proofs, or interesting open problems related to the topic of the week. Experts in the topic are especially encouraged to contribute and participate in these threads.

Next week's topic will be Probability Theory. Next-next week's topic will be on Monstrous Moonshine. These threads will be posted every Wednesday around 12pm EDT.

For previous week's "Everything about X" threads, check out the wiki link here.

Hi all,

Is the main motivation for using representation theory for Lie groups and Lie algebras because of the fact that this makes problems far easier to understand and solve, or is it so that we can continue using a linear algebraic mathematical framework for quantum physics? Or is there another reason that I'm missing?

Any help much appreciated! And reading materials would also be amazing.

Thanks!

We're having lectures about Lie Algebras right now, and those were the topics the last time. Our prof is not doing a good job at explaining, he basically reads down his notes without trying to explain the intuition, interpretation and motivation behind those concepts.

I think I get what the Lie Algebra is all about (basically, you have a system of equations/diffeomorphisms (whose IFG are the elements of the Lie Algebra), and using the Lie-bracket operation (so called "commutator"), you can create a group).

But we then went straight into the "Theory of prolongations, Criterion of invariance and splitting of defining equations". And I don't see how it is connected to the knowledge I already have about Lie Algebras.

The only thing I got was that we are now dealing with two systems of equations/diffeomorphisms

Where k=1,...,n and 𝛼=1,...,m and of course 𝑥∈ℝ^𝑛 and 𝑢∈ℝ^𝑚

Of course 𝜉 and 𝜂 are the coordinates of the corresponding IFG

This is where I got lost

We then define a manifold (𝑥,𝑢,∂𝑢,...,∂^𝑟 𝑢)

This will get us a system of equations:

𝑅^𝜎 (𝑥,𝑢,∂𝑢,...,∂^𝑟 𝑢)=0

For 𝜎=1,...,s

We are then creating a one-parameter group {𝐺𝛼} which is the set of those two systems of equations and is admitted to this 𝑅^𝜎 (𝑥,𝑢,∂𝑢,...,∂𝑟𝑢)=0 thing if it maps each solution of that into some other solution of this system

It continues that 𝑅^𝜎 (𝑥,𝑢,∂𝑢,...,∂𝑟𝑢)=0 implies for sufficiently small a that 𝑅^𝜎 (𝑥¯,𝑢¯,∂𝑢¯,...,∂^𝑟 𝑢¯)=0

Then we start talking about the Galilei Group, we go talking about the Theory of Prolongations and end on the Criterion of invariance & Splitting of defining equations

Because the script is in English I could send it to anyone interested here, it's a longer read but I really don't understand anything of it. I simply don't see the motivation, interpretation and intuition behind all of that, and what it still has got to do with "Solving ODEs with the Symmetry Methods"

Do you maybe know where I could inform myself better about it? Like which book/PDF/Youtube Videos/etc would you recommend?

Hey guys, first of all a disclaimer. I have some idea about the topic. I don't need a yes/no answer to this question, but since I am a postgraduate student of structural engineering I don't even know where to look for an answer. Having said that, I would like to get a more specific recommendation than "read that whole book". Also, I am not sure about the nomenclature so if I state the obvious, sorry.

The exponential operator maps an element of the Lie algebra to an element of the Lie group.

Rotation or orientation matrices in 3D are elements of SO(3) Lie group, which have some nice properties (SO means "special orthogonal"). If A is a 3x3 matrix such that A \in SO(3), then A^-1 =A^T and det A=+1. The exponential map between a Lie algebra so(3) which is a 3x3 skew symmetric matrix corresponding to a vector a=<a_1, a_2, a_3>^T has a CLOSED FORM, mostly known as the Rodrigues formula.

I am currently researching Lie group SR(6). Elements of this group are not orthogonal, but their determinant = +1. There are some (at least for me) remarkable analogies between SO(3) and SR(6), the most important one is that there exists a CLOSED FORM of the exponential map between SR(6) and their algebras sr(6). It was shown in an article by Bottasso and Borri (http://www.sciencedirect.com/science/article/pii/S0045782598000310).

So my question is - where could I find more information on the closed form of the exponential map, namely the so called SR(6)? Are there any other examples of closed forms of the exponential map?

Thank you so much!

__

If the question is to confusing please ask and I will give my best to clarify, for example the form of sr(6) and so on.

EDIT: The determinant of C \in SR(6) = +1

I often see statements like "The properties of an abstract Lie algebra are exactly those definitive of infinitesimal transformations, just as the axioms of group theory embody symmetry." I know the basics of Lie theory, and that the Lie algebra associated to a Lie group is the space of left-invariant vector fields, etc. But I don't intuitively "get" what infinitesimal transformations are and how they relate to Lie algebras and Lie groups.

If it matters, I come from a physics background and my motivation for learning it is to prepare myself for GR next year.

Hi, Just wanted to ask what textbooks / resources are best for teaching myself Lie Algebra. I’ve done a 1st course in Quantum Mechanics, where obviously commutation is thoroughly used. Any help where I can learn / teach myself Lie Algebra? All advice would be very much appreciated

i physics its pretty general to deal with Lie Algebra, ok almost everything works with it. i am look for a good book in that part of group theory. (i am taking most of the time particle physics courses and therefore it should be orientated maybe a bit in that direction, so that i can form a connection easier to physics)

thank you! ;)

http://www.its.caltech.edu/~yehgroup/NTU_2007%20Summer%20Lectures/NTU2007_Supplement_3.pdf

In the equation 3.13, how to come to that second equality? It must be something simple, but I just can't get it.

Also, more generally: How to get from adjoint representation of Lie algebra to adjoint representation of Lie group? I parameterized (e.g. three basis vectors of Heisenberg algebra) by exponentiating them to e^(tX), and so on. Adjoint representation of the group is defined as Ad_h(g)=hgh^(-1). That means that I will get to relations of type e^(tX)e^(rY)e^(-tX). Now, I apply Baker-Campbell-Hausdorff formula and get some new exponential function. Do I just acknowledge that this is some member of the group, and after that form the adjoint representation of the group in the same way adjoint representation of algebra is formed, or am I doing something wrong?

http://i.imgur.com/QDBvHHc.jpg

I think I can do (a)-(c) fine, (d) should be okay, but for (e) I was wondering, is the answer just that the connected Lie subgroup is just the intersection of the centre of G with the connected component of identity?

> From Agent Martin Skinner, Omega Sector, to the desk of Dr. James Albright, Epsilon Sector, Head of Research and Development.

“James, we have a problem. Those “oracles,” or whatever the hell the Bureau calls ‘em, have been flipping the fuck out. They’ve been sayin’ some really, really scary shit, like ‘End-of-Days’ type stuff, it’s creeping all of us out. It started about a week ago, we didn’t tell anyone because we didn’t wanna scare anybody, but I don’t know how much longer we can keep a lid on this. James, there’s something else on its way, something BIG. It ain’t like the other ones that showed up about fifteen years ago, it isn’t coming back because it needs to. It’s coming back because it wants to. The one-eyed paranoid creep in Götland and the rapey bird-boy in Rome wouldn’t have showed up if they didn’t have to, but this thing is ripping straight for us and the oracles know one thing: He. Is. Pissed. It ain’t just our guys hooked up to those machines in Virginia, our eyes on the other side have been sending back some stuff. Apparently Odin is really flippin’ shit, Zeus is in a tizzy and sending guys to the Holy Land, and I’ve heard that Ba’al and Moloch and the others are going dark. Some of our boys in Russia and Greece have said that the monks are talking about meeting angels and shit, seen some stuff that you’d swear came out of the Bible, like people being raised from the dead and shit. I ain’t always been much of a believer, but I’m starting to think that maybe I started skippin’ church too soon.

That’s not even the worst part. Apparently there’s two of these things headin’ our way. Some of those Worm freaks we caught tryin’ to break into Site-114 last week were singin’ glory glory hallelujah about somethin’ like “His maw will run red with the blood of the weak,” and “in triumph the Old Serpent will devour the fruits of Creation.” James, I’m scared. I don’t know what to do, but I have a real bad feelin’ that we ain’t gonna be able to keep a lid on this much longer. This didn’t start until we dug up that fuckin’ Ark, we should’ve burned the damn thing while we still could. I don’t know if you’ll even try to respond, but I’m sorry, just try to stay safe out there man, if you hear anything about it let me know. Thanks man, over and out.”

I'm a fourth year physics student currently writing a literature report on quantum weak values and I've found myself consistently running into mentions of Lie algebras and Lie groups. I was wondering if you guys have any recommendations for information on theses topics, be it books or videos etc.

As a side question how do you guys find quantum mechanics? Is it something you are interested in?

I'm interested in theoretical physics and it always seems to boil down to group theory. I have some college education and have taken math up to integral calculus. Differential equations and linear algebra were next on my plate before I dropped out due to various complications in life. Thanks!

I got a book by kirillov, but also almost no experience with topology. Learned about them in my tensors class during undergrad but never got the hang of it.

I've posted about Airlifts plenty of times before and I'll continue to do so as long as DICE remains quiet about it. Dan Mitre (/u/danmitre) has been teasing an Airlift blog for almost three weeks, and despite it appearing on this past Thurday's schedule slated in 'This Week in Battlefield', that blog never happened. It seems that DICE really doesn't give a fuck about how broken and uninspiring the Airlifts are. Bear in mind that you paid $80 for a Deluxe Edition under the pretense of being part of the Airlift system –– you can talk about how Deluxe edition included other benefits all you want, but an advertised feature of the Deluxe Edition that Dan has literally acknowledged as broken 26 days ago.

So as it stands:

• Since launch, they have not addressed player concerns. The purported Airlift blog hasn't happened for weeks and weeks, and when it was scheduled, it just decided to not happen.

• Some players haven't received any. 1 2 3

• The items released thusfar are fragments of whole skin sets, thusfar for the SG1-5 and KE7. Some people have gotten various pieces at varying times but there doesn't seem to be a lot of consistency between folks.

• One person reported getting a jacket, but recently Blockade Runner pants have come out... which are only a very slight recoloring of the Medic level 10 pants.

• People don’t really know what they are.

• People don’t know what to expect.

• They... sometimes?... show up in your Shipments without any indication. I received Blockade Runner pants on Wednesday and then also a duplicate of t

We're having lectures about Lie Algebras right now, and those were the topics the last time. Our prof is not doing a good job at explaining, he basically reads down his notes without trying to explain the intuition, interpretation and motivation behind those concepts.

I think I get what the Lie Algebra is all about (basically, you have a system of equations/diffeomorphisms (whose IFG are the elements of the Lie Algebra), and using the Lie-bracket operation (so called "commutator"), you can create a group).

But we then went straight into the "Theory of prolongations, Criterion of invariance and splitting of defining equations". And I don't see how it is connected to the knowledge I already have about Lie Algebras.

The only thing I got was that we are now dealing with two systems of equations/diffeomorphisms

Where k=1,...,n and 𝛼=1,...,m and of course 𝑥∈ℝ^𝑛 and 𝑢∈ℝ^𝑚

Of course 𝜉 and 𝜂 are the coordinates of the corresponding IFG

This is where I got lost

We then define a manifold (𝑥,𝑢,∂𝑢,...,∂^𝑟 𝑢)

This will get us a system of equations:

𝑅^𝜎 (𝑥,𝑢,∂𝑢,...,∂^𝑟 𝑢)=0

For 𝜎=1,...,s

We are then creating a one-parameter group {𝐺𝛼} which is the set of those two systems of equations and is admitted to this 𝑅^𝜎 (𝑥,𝑢,∂𝑢,...,∂𝑟𝑢)=0 thing if it maps each solution of that into some other solution of this system

It continues that 𝑅^𝜎 (𝑥,𝑢,∂𝑢,...,∂𝑟𝑢)=0 implies for sufficiently small a that 𝑅^𝜎 (𝑥¯,𝑢¯,∂𝑢¯,...,∂^𝑟 𝑢¯)=0

Then we start talking about the Galilei Group, we go talking about the Theory of Prolongations and end on the Criterion of invariance & Splitting of defining equations

Because the script is in English I could send it to anyone interested here, it's a longer read but I really don't understand anything of it. I simply don't see the motivation, interpretation and intuition behind all of that, and what it still has got to do with "Solving ODEs with the Symmetry Methods"

Do you maybe know where I could inform myself better about it? Like which book/PDF/Youtube Videos/etc would you recommend?