Adjoint endomorphism

In mathematics, the adjoint endomorphism or adjoint action is an endomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras and Lie groups. Given an element ''x'' of a Lie algebra $\backslash mathfrak\{g\}$, one defines the adjoint action of ''x'' on $\backslash mathfrak\{g\}$ as the endomorphism $\backslash textrm\{ad\}\_x\; :\backslash mathfrak\{g\}\backslash to\; \backslash mathfrak\{g\}$ with :$\backslash textrm\{ad\}\_x\; (y)\; =\; [x,y]$ for all ''y'' in $\backslash mathfrak\{g\}$. Note that ad_x is an group action and that it is linear. ==Adjoint representation== The mapping $\backslash textrm\{ad\}:\backslash mathfrak\{g\}\backslash rightarrow\; \backslash textrm\{End\}(\backslash mathfrak\{g\})$ given by $x\backslash mapsto\; \backslash textrm\{ad\}\_x$ is a representation of a Lie algebra and is called the adjoint representation of the algebra. Note that physics literature usually uses the notation ''gl''(''V'') instead of ''End''(''V'') to denote the set of linear maps of a vector space ''V'' (which is the Lie algebra of the general linear group over ''V''); we recall that, of course, $\backslash mathfrak\{g\}$ is a vector space. The Jacobi identity :$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$ takes the form :$\backslash textrm\{ad\}\_\{[x,y]\}\; =\; [\backslash textrm\{ad\}\_x,\backslash textrm\{ad\}\_y]$. Note that because $\backslash textrm\{End\}(\backslash mathfrak\{g\})$ is a set of linear transformations of a vector space, we can take the composition of two maps, and we can then write the Lie bracket as :$[\backslash textrm\{ad\}\_x,\backslash textrm\{ad\}\_y]=\backslash textrm\{ad\}\_x\; \backslash circ\; \backslash textrm\{ad\}\_y\; -\; \backslash textrm\{ad\}\_y\; \backslash circ\; \backslash textrm\{ad\}\_x$ where $\backslash circ$ denotes composition of linear maps. If a basis is chosen for $\backslash mathfrak\{g\}$, this corresponds to matrix multiplication. This last identity allows us to confirm that ''ad'' really is a Lie algebra homomorphism, in that the morphism ''ad'' commutes with the multiplication operator [,]. To see this, take an element ''z'' in ''g''. We then have :$\backslash left([\backslash textrm\{ad\}\_x,\backslash textrm\{ad\}\_y]\backslash right)(z)\; =\; [[x,y],z]\; =\; \backslash left(\backslash textrm\{ad\}\_\{[x,y]\}\backslash right)(z)$ ==Derivation== A derivation (abstract algebra) on a Lie algebra is a linear map $\backslash delta:\backslash mathfrak\{g\}\backslash rightarrow\; \backslash mathfrak\{g\}$ that obeys the Leibniz' law, that is, :$\backslash delta\; ([x,y])\; =\; [\backslash delta(x),y]\; +\; [x,\; \backslash delta(y)]$ for all ''x'' and ''y'' in the algebra. That ad_x is a derivation is a consequence of the Jacobi identity. This implies that the image of $\backslash mathfrak\{g\}$ under ''ad'' is a subalgebra of $\backslash operatorname\{Der\}(\backslash mathfrak\{g\})$, the space of all derivations of $\backslash mathfrak\{g\}$. ==Structure constants== The explicit matrix elements of the adjoint representation are given by the structure constants of the algebra. That is, let {eⁱ} be a set of basis vectors for the algebra, with :$[e^i,e^j]=\{c^\{ij\}\}\_k\; e^k$. Then the matrix elements for ad_eⁱ are given by :$\{\backslash left[\; \backslash textrm\{ad\}\_\{e^i\}\backslash right]\_k\}^j\; =\; \{c^\{ij\}\}\_k$. Thus, for example, the adjoint representation of su(2) is so(3). ==Relation to Ad== Note that Ad and ad are related through the exponential map; crudely, Ad = exp ad, where Ad is the adjoint representation for a Lie group. To be precise, let ''G'' be a Lie group, and let $\backslash Psi:G\backslash rightarrow\; \backslash textrm\{Aut\}\; (G)$ be the mapping $g\backslash mapsto\; \backslash Psi\_g$ with $\backslash Psi\_g:G\backslash to\; G$ given by the inner automorphism :$\backslash Psi\_g(h)=\; ghg^\{-1\}$. This is called the Lie group map. Define $\backslash textrm\{Ad\}\_g$ to be the tangent space of $\backslash Psi\_g$ at the origin: :$\backslash textrm\{Ad\}(g)\; =\; (d\backslash Psi\_g)\_e\; :\; T\_eG\; \backslash rightarrow\; T\_eG$ where ''d'' is the differential and ''T''_eG is the tangent space at the origin ''e'' (''e'' is the identity element of the group ''G''). Note that the Lie algebra ''g'' of ''G'' is ''g''=''T''_eG. Since $\backslash textrm\{Ad\}\_g\backslash in\backslash textrm\{Aut\}(\backslash mathfrak\{g\})$, $\backslash textrm\{Ad\}:g\backslash mapsto\; \backslash textrm\{Ad\}\_g$ is a map from ''G'' to Aut(''T''_e''G'') which will have a derivative from ''T''_e''G'' to End(''T''_e''G'') (the Lie algebra of Aut(''V'') is End(''V'')). Then we have :$\backslash textrm\{ad\}\; =\; d(\backslash textrm\{Ad\})\_e:T\_eG\backslash rightarrow\; \backslash textrm\{End\}\; (T\_eG)$. The use of upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector vector ''x'' in the algebra $\backslash mathfrak\{g\}$ generates a vector field ''X'' in the group ''G''. Similarly, the adjoint map ad_xy=[''x'',''y''] of vectors in $\backslash mathfrak\{g\}$ is homomorphic to the Lie derivative L_''X''''Y'' =[''X'',''Y''] of vector fields on the group ''G'' considered as a manifold. ==References== *William Fulton and Joe Harris, '' Representation Theory, A First Course'', (1991) Springer-Verlag, New York. ISBN 0-387-97495-4 Lie algebras Lie groups Representation theory

Adjoint endomorphism

The relationship between ad and Ad should really be its own article, and that article should develop the vector fields as well. Unfortunately, the articles on differential forms and manifolds need to be brought up to snuff as well. :I don't agree that the relationship between Ad and ad should have separate articles. I think the adjoint rep of a group and the adjoint rep of an algebra should be in the same article, along with the relationship between the two. User:Lethe | User talk:Lethe 03:05, Jan 31, 2005 (UTC) ::Err, maybe .. as long as the article stays short, they can stay together. But I think one can say a lot more about Ad, and that would end up making the article long .. too long. Also, what we really really need is a good article on exponential map and Exp and all of that tangled relationship. Ten years ago, I ditched my book on Lie algebras, now I am sorry I did... User:Linas 20:53, 7 Feb 2005 (UTC) Another generic problem I'm running into is how to figure out how to say this so that it holds for arbitrary fields, supersymmetric crap, and also for categories, and I don;'t see a way out of that kind of mess.User:Linas 02:02, 31 Jan 2005 (UTC) == notational problems == we have the problem in this article that we are using ''g'' for both an element of the Lie group ''G'', and also for the corresponding Lie algebra. To avoid this confusion, I suggest all instances of the Lie algebra be changed to $\backslash mathfrak\{g\}$ which has the unfortunate consequence of requiring us to use extensive math tags inline with the text. -User:Lethe | User talk:Lethe 02:48, Jan 31, 2005 (UTC) :Sounds good to me User:Linas 20:41, 7 Feb 2005 (UTC) :: Just some style remarks. You are inconsistent by using both ''g'' and mathfrac{g}. :::I have hopefully removed the last of the inconsistencies now. -User:Lethe | User talk:Lethe 07:56, Feb 9, 2005 (UTC) :: Also, I have just rudimentary knowledge of Lie algebras and conventions surrounding them; but, just for style, why not use a nice italic ''A'' for the algebra? User:Oleg Alexandrov 05:16, 9 Feb 2005 (UTC) ::::It's really nice to use a name for the algebra which indicates that it is related to the group. A lot of texts forego the mathfrak notation, and just use italics, but we don't have that luxury here. Also, ''A'' would lead to notational overlap with the Lie algebras ''A''_n. I'm sure there is a better notational solution to this problem, but I kinda like mathfrak, and I'm sick of waiting for browser support for mathml. Sick and tired! I'm going to go dunk my head in the sink now. -User:Lethe | User talk:Lethe 07:56, Feb 9, 2005 (UTC) ::: PS Just wait until User:Michael Hardy will see you abusing inline PNG. :) User:Oleg Alexandrov 05:18, 9 Feb 2005 (UTC) :::: I won't tell if you won't (-: User:Lethe | User talk:Lethe ::::: I had to consult an online dictionary to understand what "snarky" means. I know I am snarky sometimes, so I will try to always be "nonsnarky" as you suggest; if no other reason, at least to keep things consistent. User:Oleg Alexandrov 17:44, 9 Feb 2005 (UTC) ::::::Unfortunately, I can't make the same promise. (PS I liked some of the definitions at urbandictionary) User:Lethe | User talk:Lethe 19:51, Feb 9, 2005 (UTC) ::::::: Got it now. You meant snarky in a positive way. User:Oleg Alexandrov 20:35, 9 Feb 2005 (UTC) == regular representation == Is the adjoint represetation really also called the regular representation? wouldn't this cause some confusion the rep over at regular representation? Is this usage in Fulton & Harris? -User:Lethe | User talk:Lethe 03:01, Jan 31, 2005 (UTC) :My brain is off right now, but after skimming regular representation, it really sounds like the same thing as adjoint representation. Its the action of the group on itself. The article at regular representation is more general, whereas this article sticks to (and should stick to) Lie groups. User:Linas 20:41, 7 Feb 2005 (UTC) ::You think that the representation of a group acting by translations on the functional space on that group is the same thing as x\mapsto gxg^-1? I don't see anything in the article that could let you believe that. User:Lethe | User talk:Lethe 21:16, Feb 7, 2005 (UTC) ::I'm pretty sure that the regular representation (as defined here on wikipedia) is quite different from the adjoint representation. All I wanted to check with you is whether perhaps Fulton and Harris had given a different definition. Today I checked with them. They don't. I removed it. -User:Lethe | User talk:Lethe 17:42, Feb 8, 2005 (UTC) :::Lethe, you are confusing youself. And me too. Either that, or its late at night here, and I am confusing myself. Think about the adjoint rep of the algebra first ... its clearly just shoving vectors around. That is, a vector can act on another vector to generate a third vector. e.g. the adjoint rep of su(2) is so(3) which sounds like the regular representation over C to me. Now think "gosh, every vector in the algebra is a vector field on the manifold" ... you don't get just any representation, but a particular one, the one where the group acts on itself (which is why is SO(3)=S_3 and not SU(2)). For the Lie algebra, its the "topological group" case, where "translation on the functional space" means L_X the lie derivative. and the lie derivative L_g is just x mapsto gxg^-1. Of course, I could be hallucinating too... :::Ugh... I see, I think that actually that article is poorly written. I think the correct definition (for a finite group) is to take the ''set'' S to consist of the set of group elements. Then one builds a vector space from that set S over a field K. Then one asks "how does G act on that vector space". I think that is what the article tried to say, but that is not how that article is written. :::Worse, I'll say that article is poorly written ... you won't get a clue as to what a regular representation is unless you happen to already know what it is. :( User:Linas 08:35, 10 Feb 2005 (UTC) :::Why not just ask User:Charles Matthews directly? He wrote the article ...User:Linas 08:39, 10 Feb 2005 (UTC) ::::So you actually do think that the regular rep and the adjoint rep are the same thing? Could you provide some evidence? -User:Lethe | User talk:Lethe 11:22, Feb 10, 2005 (UTC) == merge == this article should really be merged with adjoint representation, I think -User:Lethe | User talk:Lethe 03:02, Jan 31, 2005 (UTC) :Yes, it should. I discovered that article after I got done writing this article, and was then hit with a bout of laziness. User:Linas 20:41, 7 Feb 2005 (UTC) :No, actually, I take that back. This article is about algebras, and that article is about groups. I am somwhat concerned that by trying to talk about algebras and groups at the same time, the article will get unweildy, and too long ... the algebra and the group really do behave differently. However, this article should be renamed adjoint representation of a Lie algebra or something like that. User:Linas 20:44, 7 Feb 2005 (UTC)

---

See other meanings of words starting from letter:

A

AB | AC | AD | AE | AF | AG | AH | AI | AJ | AK | AL | AM | AN | AO | AP | AR | AS | AT | AU | AW | AX | AY | AZ |

Words begining with Adjoint_endomorphism:

Adjoint_endomorphism
Adjoint_endomorphism

These materials are based on Wikipedia and licensed under the GNU FDL