CHAPTER III LIE GROUPS - Math.ucdavis.edu
CHAPTER IIILIE GROUPS3.1. SCOPE OF THE CHAPTERThis chapter is devoted to a concise exposition of Lie groups that helpilluminate various structural peculiarities of mappings on manifolds. Thesegroups are so named because it was M. S. Lie who has first studied familyof continuous functions forming a group and recognised their effectivenessin revealing some very important and fundamental properties of differentialequations. We first define in Sec. 3.2 a Lie group as a smooth manifold endowed with a group operation in which multiplication and inversion operations are supposed to be smooth functions. Some of the salient features ofLie groups are then briefly examined. Next, in Sec. 3.3 we discuss left andright translations generated by an element of the group that are diffeomorphisms mapping the manifold onto itself. Left- and right-invariant vectorfields are introduced by means of differentials of these mappings and it isshown that they constitute Lie algebras. After that we briefly investigate inSec. 3.4 the group homomorphism between Lie groups that preserve groupoperations. We then consider in Sec. 3.5 one-parameter subgroups of a Liegroup that are homomorphisms between the commutative Lie group of realnumbers and an abstract Lie group. We then discuss the exponential mapping that may help characterise such one-parameter subgroups. Afterwardsin Sec. 3.6 the group of automorphisms mapping the Lie group onto itselfand generated by elements of the Lie group itself is defined and it is shownthat this group, which is called adjoint representation, is isomorphic to theLie group. In Sec. 3.5 we examine some notable properties of Lie transformation groups that map a smooth manifold onto itself and form also a Liegroup. Finally, Killing vector fields were introduced.3.2. LIE GROUPSWe assume that a binary operation ‡ À K ‚ K Ä K on a set K, whichExterior Analysis, DOI: 10.1016/B978-0-12-415902-0.50003-7 2013 Elsevier Inc. All rights reserved.175
176III Lie Groupswill be called briefly as a product, satisfy the following conditions:Ð3ÑÞ Operation is closed: 1" ‡1# K for all 1" ß 1# K.Ð33ÑÞ Operation is associative: 1" ‡Ð1# ‡1 Ñ œ Ð1" ‡1# ч1 for all 1" ß 1# ß 1 K.Ð333ÑÞ There is an identity element / K: /‡1 œ 1‡/ œ 1 for all 1 K.Ð3@ÑÞ For each 1 K there is an inverse 1 " K: 1 ‡1 " œ 1 " ‡1 œ /.Then ÐKß ‡Ñ is called a group. It is easily observed that the identity element/ and the inverse element 1 " of an element 1 K are uniquely specified. ALie group K is also a smooth manifold and the mappings5 ÀK‚K ÄKand ÀKÄKdefined by 5Ð1" ß 1# Ñ œ 1" ‡1# and Ð1Ñ œ 1 " are smooth mappingsÞThese two last conditions can be combined into a single one imposingthat the mapping 5 À K ‚ K Ä K defined by the rule 5Ð1" ß 1# Ñ œ 1" ‡1# " issmooth. To prove this proposition, let us first introduce the smooth mapping\ À K Ä K ‚ K by the simple rule \ Ð1Ñ œ Ð/ß 1Ñ. We see that 5 ‰ \ œ . " "Indeed, we find at once that Ð5 ‰ \ ÑÐ1Ñ œ 5 Ð/ß 1Ñ œ /‡1 œ 1 œ Ð1Ñ forall 1 K. Since is now written as the composition of two smooth mappings, it turns out to be a smooth mapping as well. Similarly, let us introduce the smooth mapping ¼ À K ‚ K Ä K ‚ K through the relation¼Ð1" ß 1# Ñ œ Ð1" ß 1# " Ñ œ ˆ1" ß Ð1# щ "from which it follows that 5 ˆ¼Ð1" ß 1# щ œ 5 Ð1" ß 1# Ñ œ 1" ‡1# œ 5 Ð1" ß 1# Ñfor all 1" ß 1# K. Thus the mapping 5 œ 5 ‰ ¼ is also smooth. If K is afinite 7-dimensional manifold, then it is called an 7-parameter Lie group.Let ÐKß ‡Ñ and ÐLß ˆ Ñ be two Lie groups. The Cartesian productK ‚ L of the manifolds K and L can easily be equipped with a groupstructure by defining the product of elements Ð1" ß 2" Ñ and Ð1# ß 2# Ñ of theproduct manifold K ‚ L where 1" ß 1# K and 2" ß 2# L in the followingfashionÐ1" ß 2" Ñ ì Ð1# ß 2# Ñ œ Ð1" ‡1# ß 2" ˆ 2# Ñ K ‚ L .One checks readily that the binary operation ì is a group operation since itis solely determined by group operations on the Lie groups K and L andsmoothness requirements are clearly met. If K and L are 7- and 8-parameter Lie groups, respectively, then the product manifold K ‚ L turnsout to be an Ð7 8Ñ-parameter Lie group. Such a group is called a directproduct of groups K and L .Let us now consider some examples to Lie groups.Example 3.2.1. The smooth manifold ‘8 (see Example 2.2.1) is a
3.2 Lie Groups177commutative Lie group with respect to the operation of addition in ‘8 . Ifxß y ‘8 , then we have y " œ y so that we obtain x‡y " œ x y œÐB" C" ß á ß B8 C8 Ñ. This is obviously a smooth function.Example 3.2.2. Let us consider the manifold KPÐ8ß ‘Ñ which we hadintroduced in Example 2.2.2 and we had already called the general lineargroup of degree n. It is immediately seen that this manifold becomes also anon-commutative group with respect to the usual matrix multiplication. LetAß B KPÐ8ß ‘Ñ. With the coordinates 34 ß ,43 ‘ß 3ß 4 œ "ß á ß 8 thesematrices are represented by A œ Ò 34 Óß B œ Ò,43 Ó and we know that the matrixAB " is expressed as follows5AB " œ Ò 35 , " 4 Ó œ Ò 35 Ðcofactor ,45 ÑT Îdet BÓ.Nevertheless, this is a smooth function because it is obviously the ratio oftwo polynomials. Hence KPÐ8ß ‘Ñ is a Lie group of dimension 8# .Let us now define a subset of the general linear group given byWPÐ8ß ‘Ñ œ ÖA KPÐ8ß ‘Ñ À det A œ " It is clear that this subset is also a group with respect to matrix multiplication. In view of Theorem 2.4.1, WPÐ8ß ‘Ñ is a submanifold of dimension8# " of the general linear group. Hence, it is a Lie group. This group iscalled the special linear group or the unimodular group.We now consider the following subsetSÐ8Ñ œ ÖA KPÐ8ß ‘Ñ À AAT œ I of the group KPÐ8ß ‘Ñ which is formed by orthogonal matrices. Since theproduct of two orthogonal matrices is again an orthogonal matrix, SÐ8Ñ is agroup and Theorem 2.4.1 implies that it is a submanifold of KPÐ8ß ‘Ñ withthe dimension 8# 8Ð8 "ÑÎ# œ 8Ð8 "ÑÎ#. Thus, it is a Lie group.SÐ8Ñ is called the orthogonal group. If A SÐ8Ñ, then Ðdet AÑ# œ " sothat det A œ „ ". The Lie groupWSÐ8Ñ œ ÖA SÐ8Ñ À det A œ " whose dimension is also 8Ð8 "ÑÎ# is known as the special orthogonalgroup because it preserves the length of a vector x and volumes in ‘8 . Infact, we obtain for A SÐ8ÑÐAxÑT Ax œ xT AT Ax œ xT x.The orthogonal group is in fact a disconnected Lie group that is expressibleas the union of two disjoint connected groups as
III Lie Groups178SÐ8Ñ œ WSÐ8Ñ HWSÐ8Ñwhere H is the 8 ‚ 8 matrixÔ "Ö !HœÖãÕ !!"ã!â ! â !ÙÙã ãâ "Øso that det H œ ".Example 3.2.3. The complex plane ‚ Ö! is the #-dimensionalsmooth manifold ‘# ÖÐ!ß !Ñ . This manifold is also a group with respectto the complex multiplication. On the other hand, if D" ß D# ‚ Ö! , thenD" D# " is a smooth function of real coordinates. Hence, this manifold is a Liegroup.Example 3.2.4. Let us consider the smooth manifold ’" , the unitcircle. The points of this manifold can be determined by complex numberswith unit moduli such as lDl œ ". If D" ß D# ’" , then lD" D# l œ lD" llD# l œ "and this means that D" D# ’" . This is tantamount to say that the manifold’" is a Lie group.Example 3.2.5. The 7-torus defined as “7 œ Ð’" Ñ7 is a Lie groupbecause it is the 7-fold Cartesian product of a Lie group.Subgroup. A submanifold L of a Lie group K is called a subgroup iffor all elements 2" ß 2# L one finds 2" ‡2# L and 2" " L . Therefore,a subgroup is a submanifold of a Lie group that is closed with respect tooperations of group multiplication and inversion.If a Lie group is connected, then the following theorem states that itcan be generated by an open neighbourhood of its identity element.Theorem 3.2.1. Let K be a connected Lie group and Y be an openneighbourhood of the identity element /. We denote the set of all 8-foldproducts of elements of Y by Y 8 œ Ö?" ‡?# ‡â‡?8 À ?3 Y . Then onecan writeK œ Y 8.8œ"In other words, each group element 1 K is expressible as a finite productof some elements in the open set Y . Hence, we can say that Y generates thegroup KÞLet us choose a fixed 1 K, and define a function 51 À K Ä K by therule 51 Ð2Ñ œ 5 Ð1ß 2Ñ œ 1‡2. 51 is a diffeomorphism [see Sec. 3.3]. Hence,if Y is an open set, then the set 51 ÐY Ñ œ Ö1‡? À ? Y § K will also be
3.2 Lie Groups179open. Consequently, the set Y 8 is open for all 8. Since is a diffeomorphism, the set Y " œ Ö? " œ Ð?Ñ À ? Y is also open. We then concludethat the sets Z œ Y Y " § Y and Z 8 are all open. Furthermore, theobvious relationship Z œ Z " would be valid. Because / Y and / œ / " ,we see at once that / Z , i.e., Z is not empty. Let us now define the set8œ"8œ"L œ Z 8 Y 8 G.L is an open set since it is the union of countably many open sets, and it is,consequently, an open submanifold [see :Þ 77]. Due to the property of theset Z , L will be a subgroup. We now consider the family of open sets51 ÐLÑ œ Ö1‡2 À 2 L defined for all 1 K. One has evidently the relation L œ 51 L ÐLÑ. Thus we can obviously writeKœL 1 Kß1ÂL51 ÐLÑ.But the open set L is the complement of the open set 1 Kß1ÂL51 ÐLÑ withrespect to K so it must also be a closed set. In a connected topological spaceonly the empty set or the space itself can be both open and closed. L cannotbe empty since / L so it must be equal to K. We therefore reach to theconclusion that K œ Y 8 .8œ" The above theorem indicates that if a Lie group is a connected topological space, then an open neighbourhood of the identity element determinesthe entire group.A subgroup L of the group K is called a normal or invariant subgroup if for all 2 L we get 21 œ 1 " ‡2‡1 L for all 1 K so that L isinvariant under conjugation. In other words, if L is a normal subgroup, aconjugate element 21 L corresponds to each element 2 L so that1‡21 œ 2‡1 for each 1 K.This property is symbolically reflected by the notation 1‡L œ L‡1 for all1 K. Let L be a normal subgroup, the quotient group is defined as the setKÎL œ Ö1‡L À 1 K . The coset 1‡L is the subset of K defined byÖ1‡2 À a2 L . It is easy to verify that KÎL is actually a group. Let usconsider the direct product which can be written as followsÐ1" ‡LчÐ1# ‡LÑ œ Ð1" ‡1# чÐL‡LÑ œ Ð1" ‡1# чL KÎLsince one obviously observe the symbolic relation L‡L œ L because L isa subgroup.
III Lie Groups1803.3. LIE ALGEBRASLet K be a Lie group. We choose a fixed element 1 K to define amapping P1 À K Ä K in such a way thatP1 Ð2Ñ œ 5Ð1ß 2Ñ œ 1‡2(3.3.1)for all 2 K. P1 is evidently a smooth mapping on the manifold K. Themapping P1 is called the left translation of the Lie group K by the element1 K. We can obviously define a left translation for each element 1 of thegroup K. It can easily be seen that the relation ÐP1 Ñ " œ P1 " is valid.Indeed, for each 2 K we can writeP1 ˆP1 " Ð2щ œ 1‡1 " ‡2 œ /‡2 œ 2so that we obtain P1 ‰ P1 " œ 3K . Similarly, it is found that P1 " ‰ P1 œ 3K .Hence, the inverse mapping ÐP1 Ñ " œ P1 " is also smooth. Consequently,the left translation P1 is a diffeomorphism. The set of mappingsK" œ ÖP1 À 1 K constitutes a group with respect to the operation of composition of mappings. In fact, if P1" ß P1# K" , then owing to the relationP1" ˆP1# Ð2щ œ 1" ‡1# ‡2 œ P1" ‡1# Ð2Ñfor all 2 K, we obtain P1" ‰ P1# œ P1" ‡1# K" since 1" ‡1# K. BecauseP/ œ 3K , it then follows thatP / ‰ P 1 œ P1 ‰ P / œ P 1 .Thus, the identity element of K" is P/ and the inverse of P1 in K" is clearlyP1 " . Since the composition is an associative binary operation, we finalisethe realisation of the group structure of K" . Therefore, there exists a mapping À K Ä K" such that Ð1Ñ œ P1 . This mapping is evidently surjective. Let us further suppose that Ð1" Ñ œ Ð1# Ñ. If P1" Ð2Ñ œ P1# Ð2Ñ forall 2 K, the relation 1" ‡2 œ 1# ‡2 then leads to 1" œ 1# if we multiplyboth sides by 2 " from left which means that is injective, and consequently is bijective. On the other hand, due to the relation Ð1" ‡1# Ñ œP1" ‰ P1# œ Ð1" Ñ ‰ Ð1# Ñ, we infer that that the mapping preserves groupoperations. In other words, it is a group isomorphism. Hence, the groups Kand K" are isomorphic.In exactly same fashion, we can define the right translation of the Liegroup K by the element 1 K as the mapping V1 À K Ä K such that
3.3 Lie AlgebrasV1 Ð2Ñ œ 2‡1181(3.3.2)for all 2 K. We can readily verify that a right translation is also a diffeomorphism and due to the relation V1" ˆV1# Ð2щ œ 2‡1# ‡1" œ V1# ‡1" Ð2Ñ forall 2 K, one obtains V1" ‰ V1# œ V1# ‡1" . It is then straightforward toobserve that the set of mappings K# œ ÖV1 À 1 K constitutes a groupwith respect to the operation of composition. The identity element of thisgroup is V/ œ 3K and the inverse of an element is given by ÐV1 Ñ " œ V1 " .It is clear that this group is also isomorphic to K. Therefore, the groups K"and K# are isomorphic to one another as well. It is now evident that left andright translations are connected through the following relationV1 Ð2Ñ œ 1 " ‡1‡2‡1 œ 1 " ‡P1 Ð2ч1.Therefore, a right translation of an element of the group K is conjugate to itsleft translation, and vice versa. Moreover, it follows from ÐP1" ‰ V1# ÑÐ2Ñ œ1" ‡Ð2‡1# Ñ œ Ð1" ‡2ч1# œ ÐV1# ‰ P1" ÑÐ2Ñ for all 2 K that these mappingscommute, that is,P1" ‰ V1# œ V1# ‰ P1" .(3.3.3)In case K is a commutative group, we find that P1 Ð2Ñ œ 1‡2 œ 2‡1 œV1 Ð2Ñ for all 2 K. Hence, we deduce that P1 œ V1 for all 1 K in suchan Abelian group.Inasmuch as the mapping P1 is a diffeomorphism on K, its differential.P1 k2 À X2 ÐKÑ Ä X1‡2 ÐKÑ is an isomorphism [see :. 124] transforming vector fields onto vector fields. A vector field Z on the Lie group K is called aleft-invariant vector field if it satisfies the equality.P1 ˆZ Ð2щ œ Z ˆP1 Ð2щ œ Z Ð1‡2Ñ(3.3.4)for all 1ß 2 K. This means that the image of a vector of such a field at thepoint 2 under the linear operator .P1 will be a vector of the same field atthe point 1‡2. Thus the operator .P1 transforms a left-invariant vector fieldonto itself. So it is permissible to write symbolically.P1 ÐZ Ñ œ Zfor all 1 K. If we take 2 œ / in (3.3.4), we obtain.P1 ˆZ Ð/щ œ Z Ð1Ñ(3.3.5)for all 1 K. This relation implies that a left-invariant vector field on K iscompletely determined by a vector in the tangent space X/ ÐKÑ of the
III Lie Groups182identity element / of the Lie group K. So it becomes quite reasonable tointerpret left-invariant vector fields as 'constant vector fields' on the manifold K [Fig. 3.3.1].Conversely, let us suppose that the relation .P1 ˆZ Ð/щ œ Z Ð1Ñ issatisfied for all 1 K. We then easily deduce thatZÐ2Ñì2.P1ZÐ1‡2Ññ1‡2X1‡2ÐKÑKFig. 3.3.1. A left-invariant vector field.Z Ð1‡2Ñ œ .P1‡2 ˆZ Ð/щ œ .ÐP1 ‰ P2 шZ Ð/щœ .P1 .P2 ˆZ Ð/щ‘ œ .P1 ˆZ Ð2щ.(3.3.6)According to (3.3.4), such a vector field Z is a left-invariant vector field.We now denote the set of all left-invariant vector fields by . It is seen atonce that is a linear vector space on real numbers. Indeed, if Z" ß Z# and !" ß !# ‘, the linearity of the operator .P1 on real numbers leads tothe result.P1 Ð!" Z" !# Z# Ñ œ !" .P1 ÐZ" Ñ !# .P1 ÐZ# Ñ œ !" Z" !# Z#from which !" Z" !# Z# follows. If we assume, instead, !" are !# aresmooth functions on K, we realise that the invariance requirement can onlybe fulfilled if admissible functions are merely constant. The foregoing observations bring to mind the possibility of the existence of a bijective mapping between and X/ ÐKÑ. To this end, we presently introduce a mappingZ À Ä X/ ÐKÑ by the rule Z ÐZ Ñ œ Z Ð/Ñ. Owing to (3.3.5), the operator Z
3.3 Lie Algebras183must be linear. Indeed, one can writeÐ!" Z" !# Z# ÑÐ1Ñ œ .P1 ˆ!" Z" Ð/Ñ !# Z# Ð/щœ !" .P1 ˆZ" Ð/щ !# .P1 ˆZ# Ð/щ œ !" Z" Ð1Ñ !# Z# Ð1Ñfor all 1 K. Thus we find that Z Ð!" Z" !# Z# Ñ œ !" Z ÐZ" Ñ !# Z ÐZ# Ñ.The mapping Z is injective. Suppose that Z ÐZ" Ñ œ Z ÐZ# Ñ. We then haveZ" Ð1Ñ œ .P1 ˆZ" Ð/щ œ .P1 ˆZ# Ð/щ œ Z# Ð1Ñfor all 1 K and we conclude that Z" œ Z# . Z is surjective. Let us considera vector Z Ð/Ñ X/ ÐKÑ. The vector field defined by .P1 ˆZ Ð/щ œ Z Ð1Ñ forall 1 K is a left-invariant vector field in view of (3.3.6), hence it is an element of . In conclusion, Z is an isomorphism and the vector spaces andX/ ÐKÑ are isomorphic. This result dictates that the dimension of will bethe same as that of X/ ÐKÑ. It is, of course, the same as the dimension of themanifold K.At the identity element /, one writes .P1 À X/ ÐKÑ Ä X1 ÐKÑ so that wehave Ð.P1 Ñ " À X1 ÐKÑ Ä X/ ÐKÑ. Because of the relation P1 " Ð1Ñ œ /, weobtain .P1 " À X1 ÐKÑ Ä X/ ÐKÑ. On the other hand, the identities P1 ‰ P1 "œ P1 " ‰ P1 œ 3K will result in the relations .P1 ‰ .P1 " œ MX1 ÐKÑ and.P1 " ‰ .P1 œ MX/ ÐKÑ . It then follow that Ð.P1 k/ Ñ " œ .P1 " k1 .Since K is a smooth manifold of dimension 7, each point of K is contained in an open neighbourhood in K and there is a homeomorphism :mapping this open set onto an open set of ‘7 . If local coordinates of a point2 K are prescribed by x œ ÐB" ß á ß B7 Ñ and local coordinates of a pointP1 Ð2Ñ œ 1‡2 K are given by y œ ÐC" ß á ß C7 Ñ, then we know that thereexists a functional relationship in the form y œ Ð: ‰ P1 ‰ : " ÑÐxÑ œ L1 ÐxÑ,or C3 œ P31 ÐxÑ. Hence the definition (3.3.4) implies that the local components of a left-invariant vector field must satisfy the following expressions@3 ÐyÑ œ P31 ÐxÑ 4@ ÐxÑß y œ L1 ÐxÑ B4(3.3.7)for all x ‘7 in respective charts.We now demonstrate that the Lie bracket of vector fields Z" ß Z# also a left-invariant vector field. If we recall (2.10.21) we find thatis.P1 ÐÒZ" ß Z# ÓÑ œ Ò.P1 ÐZ" Ñß .P1 ÐZ# ÑÓ œ ÒZ" ß Z# Ó,hence, ÒZ" ß Z# Ó . As a result of this, we see that left-invariant vectorfields constitute a Lie algebra. or X/ ÐKÑ that is isomorphic to is calledthe Lie algebra of the Lie group K. Indeed, since we have ÒZ" ß Z# Ó if
184III Lie GroupsZ" ß Z# , we understand that the relationZ ÐÒZ" ß Z# ÓÑ œ ÒZ" Ð/Ñß Z# Ð/ÑÓ œ ÒZ ÐZ" Ñß Z ÐZ# ÑÓwould also be valid. If the dimension of the manifold K is 7, a basis of thevector space are determined by 7 linearly independent left-invariant vector fields ÖZ3 À 3 œ "ß á ß 7 . Properties of a Lie algebra will impose thefollowing restriction on these vectors for all 3ß 4ß 5 œ "ß á ß 7ÒZ3 ß Z4 Ó ÒZ4 ß Z3 Ó œ !ß . Z3 ß ÒZ4 ß Z5 Ó‘ Z4 ß ÒZ5 ß Z3 Ó‘ Z5 ß ÒZ3 ß Z4 Ó‘ œ !Since(3.3.8)is a Lie algebra, there must exist constants -345 so that one hasÒZ3 ß Z4 Ó œ -345 Z5 .(3.3.9)These constants are called structure constants of the Lie algebra with respect to the basis ÖZ3 . Because of the relations (3.3.9) and (3.3.8), thestructure constants should meet the conditions-345 -435 œ !ß(3.3.10)8 68 66-45-38 -53-48 -348 -58œ!for all 3ß 4ß 5ß 6 œ "ß á ß 7 [see (2.11.4)]. Structure constants holding theconditions (3.3.10) completely determines the Lie algebra. It is clear that thestructure constants depend on the selected basis. Let us choose another basisby the transformation Z4w œ 43 Z3 where A œ Ò 43 Ó is a regular matrix. If wewrite ÒZ3w ß Z4w Ó œ -34w5 Z5w , we easily find that the following expressions mustbe satisfied -34w5 5 Z œ Ò :3 Z: ß 4; Z; Ó œ :3 ;4 ÒZ: ß Z; Ó œ :3 4; -:;Z .Since the vectors Z are linearly independent, we conclude that -34w5 œ :3 4; , 5 -:;(3.3.11)where B œ A " œ Ò,43 Ó. (3.3.11) clearly indicates that structure constants arecomponents of a third order mixed tensor. This tensor is called the structuretensor of the Lie algebra. We have seen that the Lie algebra of left-invariantvector fields is isomorphic to the tangent space X/ ÐKÑ at the identity element / and the integral manifold of that tangent space locally determines themanifold K. This is tantamount to say that the Lie algebra fully determinesthe Lie group locally in a neighbourhood of /. However, the correspondencebetween the Lie groups and the Lie algebras is not unique. Although a given
3.3 Lie Algebras185Lie group determines uniquely its Lie algebra, several Lie groups maygenerate the same Lie algebra. But, it can be shown that among all the Liegroups with the same Lie algebra, there is only one Lie group that is simplyconnected. Therefore, a given Lie algebra gives rise to a unique simply connected Lie group locally in a neighbourhood of /Þ Then in view ofTheorem 3.2.1 it determines the Lie group globally if the manifold K is connected. Because features of a Lie algebra are entirely elucidated by its structure constants, to investigate the properties of constants satisfying the algebraic relations (3.3.10) provides quite a significant information about theassociated Lie group itself.If structure constants are all zero, we then have ÒZ3 ß Z4 Ó œ ! so thatbecomes a commutative Lie algebra. Such algebras are named as AbelianLie algebras.In exactly the same fashion as we have introduced the left-invariantvectors, we can define the right-invariant vector fields through the relation.V1 ÐZ Ñ œ Z . We immediately observe that these vector fields constitute aLie algebra that is isomorphic to the vector space X/ ÐKÑ. Let us denote Liealgebras of left- and right-invariant vectors by P and V , respectively.Since both algebras are isomorphic to the tangent space X/ ÐKÑ, they are ofcourse isomorphic to one another through the isomorphism ZV " ‰ ZP .In view of (2.7.7), the relation (3.3.3) yields.P1 ‰ .V1 œ .V1 ‰ .P1 .If Z is a left-invariant vector field, we find.P1 ˆ.V1 ÐZ щ œ .V1 ˆ.P1 ÐZ щ œ .V1 ÐZ Ñfor all 1 K. This result means that the vector field .V1 ÐZ Ñ turns out alsoto be a left- invariant vector field. Conversely, if Z is a right-invariant vector field, then the same expression implies that the vector field .P1 ÐZ Ñ is aright-invariant vector field.Example 3.3.1. Consider the affine space ‘8 [see Example 2.2.1].This smooth manifold is obviously a commutative Lie group with respect tothe following addition operationB C œ ÐB" C" ß á ß B8 C8 Ñfor all Bß C ‘. In this case left and right translations are not different andthey are given byPB ÐCÑ œ VB ÐCÑ œ B C.Let us denote a left-invariant vector field by Z ÐBÑ œ @3 ÐBÑ 3 . Then (3.3.7)
III Lie Groups186leads to the relation@3 ÐB CÑ œ ÐC3 B3 Ñ 43@ ÐBÑ œ 4 @4 ÐBÑ œ @3 ÐBÑ. B4Hence the left-invariant vector fields are constant vector fields whose components merely @3 ‘. Of course, they generate a commutative Lie algebraèwith vanishing structure constant.Example 3.3.2. We wish to compute the Lie algebra of the Lie groupKPÐ8ß ‘Ñ. Inasmuch as KPÐ8ß ‘Ñ is an open submanifold of the manifold16Ð8ß ‘Ñ, its dimension is 8# . Hence, the tangent space at the identity element / œ I is an 8# -dimensional vector space. We can thus identify the associated Lie algebra with the space 16Ð8ß ‘Ñ that consists of all 8 ‚ 8matrices. We can choose as basis vectors the set of following linearly independent 8 ‚ 8 matrices whose only one entry is " and all the other entriesare !:3 6Z43 Ð/Ñ œ 5 4 ß 3ß 4ß 5ß 6 œ "ß á ß 8 B65where 8# matrix entries B56 represent the local coordinates of KPÐ8ß ‘Ñ.Left translation is naturally defined as the matrix product P1 Ð2Ñ œ GH or5 7ˆP1 Ð2щ5 œ 1726 in terms of components of G œ Ò143 Ó and H œ Ò243 Ó.6Hence, according to (3.3.7), the components of a left-invariant vector fieldmust obey the equalityˆZ43 Ð1щ5 œ65 7 Ð17B6 Ñ 335 7 ; 3 :ˆZ4 Ð/щ: œ 17 : 6 ; 4 œ 145 6 .; B:;Consequently we can construct left-invariant vector fields by making use ofthe basis vectors3Z43 Ð1Ñ œ 145 6 œ 145 55 16 13for all 1 KPÐ8ß ‘Ñ. An element of the Lie algebra Ð8Ñ will now beexpressible as3 6ZA œ 43 Z43 Ð/Ñ œ 34 5 4 œ 43 46 B5 B3where the numbers 43 are entries of a matrix A. Next, we determine thestructure constants of the Lie algebra by evaluating
3.4 Lie Group Homomorphisms187ÒZ43 ß Z65 Ó œ ’14: ; : ; ; : ‹: ß 16; “ œ 14: Š16; ‹ 16; Š14 13 15 13 15 15 13: # #; 3 5 œ 14: : 6 ; 14: 16; : ; 16; ;: 4 : 16; 14: ; : 15 13 15 13 15 13 3535œ 6 14: : 4 16: : œ 6 Z45 4 Z63 . 15 13It then follows that35; :ÒZ43 ß Z65 Ó œ ˆ 6 : 4 4 : 6 ‰Z;: œ -46:Z; .3 5 ;5 3 ;Since ˆZA Ð/щ4 œ 34 , the left-invariant vector field generated by avector ZA becomes33ZA Ð1Ñ œ ˆZA Ð1щ43 7 Ð17B4 Ñ 5 œ 6 3 œ 153 45 3 .5 143 1 1 B644Therefore the Lie product (bracket) of left-invariant matrices correspondingto matrices A and B is found to be3 7ÒZA ß ZB ÓÐ1Ñ œ ’17 4 3 7 4 ß 1:5 ,6: 5 “ œ 17 4 ,6 3 1:3 ,6: 46 33 14 16 14 16 47 4 3 3œ Ð 7œ 17œ ZÒAßBÓ Ð1Ñ.ÒAß BÓ74 ,6 ,4 6 Ñ1763 16 163where ÒAß BÓ œ AB BA is the matrix commutator. These results clearlyindicate that the Lie algebra Ð8Ñ is actually generated by the elements ofthe vector space 16Ð8ß ‘Ñ on which the Lie product of matrices Aß B isdefined as the matrix commutator ÒAß BÓ.è3.4. LIE GROUP HOMOMORPHISMSLet ÐKß ‡Ñ and ÐLß ˆ Ñ be Lie groups, and 9 À K Ä L be a smoothfunction. If, for all 1" ß 1# K, the relation 9Ð1" ‡1# Ñ œ 9Ð1" Ñ ˆ 9 Ð1# Ñ isvalid, then the function 9 is called a Lie group homomorphism. Moreover,if the homomorphism 9 is also a diffeomorphism, 9 is then a Lie groupisomorphism. For the identity element / K, we simply obtain9Ð1Ñ œ 9Ð/‡1Ñ œ 9Ð/Ñ ˆ 9Ð1Ñ.Hence, the unique identity element /w of the Lie group L will necessarily be
188III Lie Groups/w œ 9Ð/Ñ. Moreover, we can write /w œ 9Ð1‡1 " Ñ œ 9Ð1Ñ ˆ 9Ð1 " Ñ so that "we deduce the relation ˆ9Ð1щ œ 9Ð1 " Ñ. Thus, 9ÐKÑ L is a subgroup.If a left translation on K is P1 , then we obtain9ˆP1 Ð1" щ œ 9Ð1‡1" Ñ œ 9Ð1Ñ ˆ 9Ð1" Ñ œ P9Ð1Ñ ˆ9 Ð1" щfor all 1ß 1" K from which it follows that 9 ‰ P1 œ P9Ð1Ñ ‰ 9 À K Ä L forall 1 K. The expression (2.7.7) now leads to the rule. 9 ‰ .P1 œ .P9Ð1Ñ ‰ . 9.(3.4.1)Let us consider a left-invariant vector field Z on K. Since Z satisfies therelation .P1 ÐZ Ñ œ Z , (3.4.1) now yields the result. 9ˆ.P1 ÐZ щ œ . 9ÐZ Ñ œ .P9Ð1Ñ ˆ.9ÐZ щvalid for all 1 K. This means that the vector field . 9ÐZ Ñ is a left-invariant vector field of L on the subgroup 9ÐKÑ L . Let Z" ß Z# be left-invariant vector fields on K. On taking into account the relation .P1 ÐÒZ" ß Z# ÓÑœ ÒZ" ß Z# Ó, (3.4.1) leads to the conclusion. 9ˆ.P1 ÐÒZ" ß Z# Óщ œ . 9ÐÒZ" ß Z# ÓÑ œ .P9Ð1Ñ ˆ. 9ÐÒZ" ß Z# Óщwhich expresses the fact that . 9ÐÒZ" ß Z# ÓÑ œ Ò. 9ÐZ" Ñß .9ÐZ# ÑÓ is a leftinvariant vector field on L . In other words, images of left-invariant vectorfields under the differential mapping .9 where 9 is a homomorphism areelements of a Lie algebra on L . Since the homomorphism 9 transports theidentity element / in K to the identity element 9Ð/Ñ in L , we find that. 9 À X/ ÐKÑ Ä X9Ð/Ñ ÐLÑ. We denote the Lie algebras on K and L by and¡, respectively. Via isomorphisms Z À Ä X/ ÐKÑ and [ À ¡ Ä X9Ð/Ñ ÐLÑ,which we have discussed on :Þ 182, we can introduce a linear operator œ [ " ‰ . 9 ‰ Z À Ä ¡. It is straightforward to see that this operatorfulfil the relation ÐÒZ" ß Z# ÓÑ œ Ò ÐZ" Ñß ÐZ# ÑÓ(3.4.2)for all Z" ß Z# , that is, preserves the Lie product. We thus concludethat so defined is a Lie algebra homomorphism. The image Ð Ñ of isclearly a subalgebra of ¡.When 9 is an isomorphism, turns out to be likewise an isomorphismand we find that ¡ œ Ð Ñ. In that situation, if the set of vector fields ÖZ3 is a basis for the Lie algebra , then the set of vector fields Ö ÐZ3 Ñ becomes a basis for the Lie algebra ¡. Because of relations ÒZ3 ß Z4 Ó œ -345 Z5 itfollows from (3.4.2) that
3.5 One-Parameter SubgroupsÒ ÐZ3 Ñß ÐZ4 ÑÓ œ ÐÒZ3 ß Z4 ÓÑ œ Ð-345 Z5 Ñ œ -345 ÐZ5 Ñ.189(3.4.3)Hence, such an isomorphism preserves structure constants.3.5. ONE-PARAMETER SUBGROUPSWe consider a Lie group K. As is well known, the set ‘ is an AbelianLie group with respect to the operation of addition. Let 9 À ‘ Ä K be a Liegroup homomorphism. The subset Ö9Ð Ñ À ‘ œ 9Ð‘Ñ K is called aone-parameter subgroup of K. By definition, the function 9 must satisfy thecondition9Ð Ñ œ 9Рч9Ð Ñ œ 9Рч9Ð Ñ(3.5.1)for all ß ‘ because œ . Therefore, one-parameter subgroupswould necessarily be commutative. Inasmuch as 9 is a homomorphism, we "observe that / œ 9Ð!Ñ and ˆ9Рщ œ 9Ð Ñ. The smooth function 9 willevidently describe a smooth curve on the manifold K through the point /.Theorem 3.5.1. A curve on a Lie group K is a one-parameter subgroup if and only if it is an integral curve of a left-invariant or a rightinvariant vector field through the identity element /.Let 9 À ‘ Ä K give rise to a one-parameter subgroup. As in (2.9.1),we represent symbolically a tangent vector at an element 1 œ 9Ð Ñ in thefollowing mannerZ ˆ9Рщ œ. 9Ð Ñ. (3.5.2)Owing to the formula P9Ð Ñ ˆ9Рщ œ 9Рч9Ð Ñ œ 9Ð Ñ, the vector fieldZ ˆ9Рщ under the differential operator .P9Ð Ñ must satisfy the relation.P9Ð Ñ Š. 9Ð Ñ. 9Ð Ñ.9Ð Ñœ‹œ. . . If we insert œ ! into this expression, we obtain.P9Ð Ñ ˆZ Ð/щ œ Z ˆ9Рщ.which indicates that (3.5.2) is a left-invariant vector field. It is evident that9Ð Ñ is an integral curve of this vector field through the point / K. If weassociate each point 1 K with a curve defined by91 Ð Ñ œ 1‡9Ð Ñ
190III Lie Groupswe produce a congruence on K that is tangent to the left-invariant vectorfield Z . However, it is evident that only the curve of this congruencethrough the point / corresponds to a one-parameter subgroup.Conversely, let us now consider a left-invariant vector field Z .This vector field associated with a vector in X/ ÐKÑ generates a flow on Kwhose member through the identity element / K will be given just like in(2.9.11) by1 Ð/Ñ œ / Z Ð/Ñ K.(3.5.3)If we make use of the relation (2.9.17) it follows from (3.5.3) that1 ‡1 œ P1 Ð1 Ñ œ P1 ˆ/ Z Ð/щ œ / .P1 ÐZ Ñ P1 Ð/Ñ œ / Z Ð1 Ñœ / Z / Z Ð/Ñ œ / Z Ð/ч/ Z Ð/Ñ œ /Ð ÑZ Ð/Ñ œ 1 .This clearly shows that the subset (3.5.3) is a one-parameter subgroup, andwe have / œ 1! and Ð1 Ñ " œ 1 .The case of right-invariant vector fields can be treated in exactly thesame manner. Let 9 À ‘ Ä K be a one-parameter subgroup. If we write 1Ð Ñ œ 9Ð Ñ,this subgroup gives rise to a one-parameter group of t
CHAPTER III LIE GROUPS 3.1. SCOPE OF THE CHAPTER This chapter is devoted to a concise exposition of Lie groups that help illuminate various structural peculiarities of mappings on manifolds. These groups are so named because it was M. S. Lie who has first studied family of continuous functions forming a group and recognised their effectiveness
Chapter II. Lie groups and their Lie algebras33 1. Matrix Lie groups34 1.1. Continuous symmetries34 1.2. Matrix Lie groups: de nition and examples34 1.3. Topological considerations38 2. Lie algebras of matrix Lie groups43 2.1. Commutators43 2.2. Matrix exponentiald and Lie's formulas43 2.3. The Lie algebra of a matrix Lie group45 2.4.
Chapter 1. Introduction 7 Chapter 2. Lie Groups: Basic Definitions 9 §2.1. Lie groups, subgroups, and cosets 9 §2.2. Action of Lie groups on manifolds and representations 12 §2.3. Orbits and homogeneous spaces 13 §2.4. Left, right, and adjoint action 14 §2.5. Classical groups 15 Exercises 18 Chapter 3. Lie Groups and Lie algebras 21 §3.1 .
Chapter 1. Introduction 7 Chapter 2. Lie Groups: Basic Definitions 9 §2.1. Lie groups, subgroups, and cosets 9 §2.2. Action of Lie groups on manifolds and representations 12 §2.3. Orbits and homogeneous spaces 13 §2.4. Left, right, and adjoint action 14 §2.5. Classical groups 15 Exercises 18 Chapter 3. Lie Groups and Lie algebras 21 §3.1 .
call them matrix Lie groups. The Lie correspondences between Lie group and its Lie algebra allow us to study Lie group which is an algebraic object in term of Lie algebra which is a linear object. In this work, we concern about the two correspondences in the case of matrix Lie groups; namely, 1.
Chapter 1. Lie Groups 1 1. An example of a Lie group 1 2. Smooth manifolds: A review 2 3. Lie groups 8 4. The tangent space of a Lie group - Lie algebras 12 5. One-parameter subgroups 15 6. The Campbell-Baker-HausdorfT formula 20 7. Lie's theorems 21 Chapter 2. Maximal Tori and the Classification Theorem 23 1. Representation theory: elementary .
(1) R and C are evidently Lie groups under addition. More generally, any nite dimensional real or complex vector space is a Lie group under addition. (2) Rnf0g, R 0, and Cnf0gare all Lie groups under multiplication. Also U(1) : fz2C : jzj 1gis a Lie group under multiplication. (3) If Gand H are Lie groups then the product G H is a Lie group .
Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .
Leo Aronson. My hope is that your wonderful capacity for empathy and compassion will help make the world a better place. —E.A. To my family, Deirdre Smith, Christopher Wilson, and Leigh Wilson —T.D.W. To my children, Genevieve and Everett —B.F. To my mentor, colleague, and friend, Dane Archer —R.M.A.
