LECTURE 7 The homomorphism and isomorphism theorems 7.1
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On a Theorem of Lovász that hom(.H) Determines the Isomorphism
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A homomorphism theorem for projective planes
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The Third Homomorphism Theorem on Trees
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On the Homomorphism Theorem for Semirings
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7 Homomorphisms and the First Isomorphism Theorem
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Lecture 4.3: The fundamental homomorphism theorem
Key observation. Q4/ Ker(?) ?= Im(?). M. Macauley (Clemson). Lecture 4.3: The fundamental homomorphism theorem. Math 4120 Modern Algebra.
GROUP THEORY (MATH 33300) 1. Basics 3 2. Homomorphisms 7 3
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We formalize and prove the theorem and use it to improve an O(n2) sorting algorithm to O(nlog n). 1 Introduction. List homomorphisms are those functions on
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[PDF] Lecture 43: The fundamental homomorphism theorem
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[PDF] Sec 43: The fundamental homomorphism theorem
The following is one of the central results in group theory Fundamental homomorphism theorem (FHT) If ?: G ? H is a homomorphism then Im(?) ?
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LECTURE 12: LIE'S FUNDAMENTAL THEOREMS
1.Lie Group Homomorphism v.s. Lie Algebra Homomorphism
Lemma 1.1.SupposeG,Hare connected Lie groups, and :G!His a Lie group homomorphism. Ifd :g!his bijective, thenis a covering map. Proof.SinceHis connected and is a Lie group homomorphism and is a local dieo- morphism near onto a neighborhood ofe2H, is surjective. By group invariance, it suces to check the covering property ate2H. Sinced :TeG!TeHis bijective, maps a neighborhoodUofeinGbijectively to a neighborhoodVofeinH.Let =
1(e)G. Then is a subgroup ofG. Moreover, for anya2,
La(g) = (ag) = (a)(g) = (g):
So1(V) =[a2LaU. The lemma is proved if we can showLa1U \La2U=;for
a16=a22. We show this by contradiction. Leta=a11a2. IfLa1U \La2U 6=;, then
L aU \U 6=;. Considerp2=ap12LaU \U, wherep1;p22 U. Then (p2) = (ap1) = (p1). However, is one-to-one onU. Sop1=p2. It followsa=e, anda1=a2.Contradiction.
Corollary 1.2.Let :G!Hbe a Lie group homomorphism withd :g!h bijective. SupposeG;Hare connected andHis simply connected. Thenis a Lie group isomorphism. Proof.Since is a covering map andHis simply connected, is homeomorphism. In particular, and1are both continuous Lie group homomorphisms. If follows that
both and1are smooth, and thus is a dieomorphism.
The main theorem in this section is the following \lifting" property: Theorem 1.3.LetG;Hbe Lie groups withGconnected and simply connected,g;h the Lie algebras ofG;H. If:g!his a Lie algebra homomorphism, then there is a unique Lie group homomorphism :G!Hsuch thatd =.Proof.Let
k= graph() =f(g;h)2gh:h=(g)g: Obviouslykis a vector space. It is actually a Lie subalgebra ofgh. In fact, if h i=(gi),i= 1;2, then the factis a Lie algebra homomorphism implies [h1;h2] = [(g1);(g2)] =([g1;g2]):It follows
[(g1;h1);(g2;h2)] = ([g1;g2];[h1;h2]) = ([g1;g2];([g1;g2])): 12 LECTURE 12: LIE'S FUNDAMENTAL THEOREMS
In other words,kis a Lie subalgebra ofgh.
By the theorem we proved in last time, there exists a unique connected Lie subgroup KofGHwithkas its Lie algebra. Consider the composition map ':KK,!GHpr1!G:
This is a Lie group homomorphism, sod'=dpr1dKis a Lie algebra homomorphism. Sincedpr1:gh!gis the projection map,d':k!gis a bijection. It follows that ':K!Gis a Lie group isomorphism.Now let :G!Hbe the composition
G '1!Kpr2!H: Then this is a Lie group homomorphism withd =:This completes the proof. Corollary 1.4.If connected and simply connected Lie groupsGandHhave isomorphicLie algebra, thenGandHare isomorphic.
2.Lie's Fundamental Theorems
We have seen that associated to each Lie groupGthere is a god-given Lie algebra g. A natural question is: To what extend will the Lie algebra determine this Lie group? On one hand, we have seen that the Lie algebras ofS1andR1are the same, so Lie groups are not determined by its Lie algebra. On the other hand, according to the B-C-H formula, the Lie group product near the identityeis totally determined by the Lie bracket. So at least the Lie algebragprovides the local information forG. The subtle relations between Lie groups and Lie algebras are described by the following theorems observed by S. Lie at the beginning of the whole subject. Recall that for any Lie group homomorphismf:G1!G2, there is an induced Lie algebra homomorphismdf:g1!g2. The assignmentf dfsatises the following functorial properties:Forf= Id :G!G,df= Id :g!g.
Iffi:Gi!Gi+1,i= 1;2, are Lie group homomorphisms, thend(f2f1) = df 2df1.To state the theorems, let's rst give a denition.
Denition 2.1.Alocal homomorphismbetween two Lie groupsG,His a smooth mapffrom a neighborhoodUofeGinGto a neighborhoodVofeHinHsuch that if g1;g22Uandg1g22U, thenf(g1g2) =f(g1)f(g2).fis called alocal isomorphismif it
is a dieomorphism fromUtoVsuch that bothfandf1are local homomorphisms. We know the Lie algebra, as the tangent space ate, is determined by the Lie group structure on any neighborhood ofeinG. So any local homomorphism determines a Lie algebra homomorphism as well. Moreover, the functorial properties above also holds for local homomorphisms.LECTURE 12: LIE'S FUNDAMENTAL THEOREMS 3
Theorem 2.2(Lie's rst fundamental theorem).IfGandHare locally isomorphicLie groups, thengandhare isomorphic Lie algebras.
Proof.Letfbe the local isomorphism betweenGandG. Sincedfe:g!his a Lie algebra homomorphism, We only need to showdfeis a bijection. However, this follows from the fact that exp is locally dieomorphism.Conversely,
Theorem 2.3(Lie's second fundamental theorem).Ifgandhare isomorphic Lie algebras, thenGandHare locally isomorphic Lie groups. Proof.Let:g!hbe the Lie algebra isomorphism map. As in the proof of theorem1.3, there exists a connected Lie subgroupKofGHwhose Lie algebra is
k=f(x;(x))jx2gg:We still have the facts that the composition map
':K ,!GHpr1!G is a Lie group homomorphism whose dierentiald':k!gis bijective. It follows that'is a local isomorphism and a dieomorphism, i.e. there is a neighborhoodU ofe2K, a neighborhoodVofe2Gand a dieomorphism :V!Usuch that '= 1Uand' = 1V.Similarly the map
:K ,!GHpr2!H is also a local dieomorphism, and a local isomorphism. Now the composition is the local isomorphism we want. Theorem 2.4(Lie's third fundamental theorem).For any nite dimensional Lie al- gebrag, there is a unique simply connected Lie groupGwhose Lie algebra isg. The proof is based on the following amazing theorem whose proof is beyond the scope of this course can can be found in books on Lie algebra representation theory. Theorem 2.5(Ado).Every nite dimensional Lie algebra is a Lie subalgebra of gl(n;R)fornlarge enough. Proof of Lie's third fundamental theorem.According to Ado's theoremgis a Lie sub- algebra of somegl(n;R). So there is a connected linear Lie groupG1whose Lie algebra isg. LetGbe the simply connected covering ofG1. ThenGis a simply connected Lie group. Its Lie algebra isgsince any covering map is a local isomorphism. The uniqueness follows from corollary 1.4.quotesdbs_dbs12.pdfusesText_18[PDF] homonyme liste deutsch
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