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The Covariant Picard Groupoid in Di?erential Geometry ?Stefan Waldmann ??Fakult¨a t f¨u r Mathematik und Physik Albert-Ludwigs-Universit¨a t Freiburg Physikalisches Institut Hermann Herder Stra?e 3D 79104Freiburg Germany September 2005FR-THEP 2005/10Abstract In this article we discuss some general results on the covariant Picard groupoid in the context of di?erential geometry and interpret the problem of lifting Lie algebra actions to line bundles in the Picard groupoid approach.Keywords:Morita equivalence,?-Algebras,Picard groupoid,Hopf algebra actions.MSC (2000):16D90,16W30,16W10,53D55Contents 1Introduction 12From Geometry to Algebra 2
3Morita Equivalence in Di?erent Flavours 44The Covariant Situation 75The case of a Lie algebra 81Introduction
In this work we would like to illustrate and exemplify some general results from [16]where the general framework of a Morita theory which is covariant under a given Hopf algebra was studied.One main motivation to do so is coming from (deformation)quantization theory [3],see e.g [12,14]for recent reviews.Here Morita equivalence provides an important notion of equivalence of
observable algebras[6,31].In particular,on cotangent bundles the condition for star products to be Morita equivalent is shown to coincide with Dirac’s integrality condition for magnetic charges of a background magnetic?eld[6]leading to a natural interpretation of Morita equivalence also in more general situations.
From the di?erential geometric point of view,it is a natural question whether all these techniques as developed in[5,7–9,31]can be made compatible with a certain given symmetry of the underlying manifold.On the purely algebraic level,a fairly general notion of‘symmetry’is that of a Hopf algebra action of a given Hopf algebra.In[16]we studied this type of symmetry in the general situation.
From this general framework we shall specialize now into two directions:on one hand,the algebras on which the symmetry acts and whose Morita theory shall be studied will now be com-mutative:we are interested in the algebra of functions C∞(M)on a manifold.On the other hand, the symmetry in question will either be coming from a Lie group action on M or from a Lie algebra action as its in?nitesimal counterpart.
It is well-known that two commutative algebras are Morita equivalent if and only if they are isomorphic,see e.g.the textbook[19],whence for commutative algebras Morita equivalence seems to be a useless notion.However,this is not true as things become interesting if one asks in addition in how many ways two algebras can be Morita equivalent compared to the ways in which they can be isomorphic.It turns out that in general there are new possibilities which makes Morita theory interesting even in the commutative framework.
This phenomenon is precisely encoded in the so-called Picard groupoid which we shall compute for the case of function algebras.This way,we?nd an interesting and non-trivial class of examples illustrating the general ideas of[16].Moreover,it will also be of independent interest as we are now able to re-interpret several well-known problems and results in di?erential geometry from a Morita theoretic point of view.Finally,the commutative situation with the algebras being function algebras C∞(M)is expected to be the starting point for a discussion of Morita equivalence of star products as in[6]but now being compatible with a symmetry of the classical phase space[15].
The article is organized as follows:In Section2we recall some well-known arguments why one should and how one can pass from a geometric to a more algebraic description of di?erential geometry.The next section is devoted to a general discussion on Morita theory in di?erent?avours, taking into account speci?c structures of the algebras in question.Here we are mainly interested in?-involutions and notions of positivity.In Section4we add one more structure to be preserved by Morita theory,namely a symmetry which we model by a Hopf algebra action.This can be specialized to group actions and Lie algebra actions.The last section contains some new material, namely the explicit computation of a certain part of the Lie algebra covariant Picard group. Acknowledgments:It is a pleasure for me to thank the organizers and in particular Michel Cahen and Willy Sarlet for their kind invitation to the20th International Workshop on Di?erential Geo-metric Methods in Theoretical Mechanics in Ghent where the content of this work was presented. Moreover,I would like to thank Stefan Jansen and Nikolai Neumaier for valuable discussions and comments on the manuscript.
2From Geometry to Algebra
In some sense,di?erential geometric methods in mathematical physics correspond mainly to clas-sical theories:Hamiltonian mechanics on a symplectic or Poisson manifold M is one prominent example.On the other hand,quantum theories require a more algebraic approach:here the un-certainty relations in physics are modelled mathematically by non-trivial commutation relations
between observables in some noncommutative algebra,the observable algebra.Thus quantization in a very broad sense can be understood as the passage from geometric to noncommutative algebraic structures.An intermediate step is of course to encode the geometric structures on M in algebraic terms based on the commutative algebra of functions C∞(M)where,in view of applications to quantization,it is convenient to consider complex-valued functions.
Then it is a folklore statement(Milnor’s exercise)that one can recover the smooth manifold M from the?-algebra C∞(M).More speci?cally:every?-homomorphismΦ:C∞(M)?→C∞(N) between function algebras is actually of the formΦ=φ?with some smooth mapφ:N?→M between the underlying manifolds,see e.g.[13,24]for a recent discussion.Thus the category of ?-algebras with?-homomorphisms as morphisms becomes relevant to di?erential geometry.
The‘dictionary’to translate geometric to algebraic terms,which is also one of the cornerstones of Connes’noncommutative geometry[11],can be extended in various directions.We mention just one further example also relevant to Morita theory:by the well-known Serre-Swan theorem, see e.g.[29],the(complex)vector bundles E?→M correspond to?nitely generated projective modules over the algebra C∞(M)via E?Γ∞(E).In more geometric terms this means that for any vector bundle there exists another vector bundle F?→M such that E⊕F is a trivial vector bundle.This is of course the key to relate the algebraic K0-theory of C∞(M)to the topological K0-theory of M.
Let us now turn to Morita theory.Its?rst motivation came from the question what one can say about two algebras A and B provided one knows that their categories of(left)modules are equivalent,see[19,23].Clearly,in view of applications to quantum mechanics a good understanding of the more speci?c modules given by?-representations of the observable algebras on(pre)Hilbert spaces is crucial for any physical interpretation.As we shall not start to de?ne what a reasonable category of modules over the?-algebra C∞(M)should be—though this can perfectly be done,see e.g.[7,27,31]and references therein—we take a di?erent motivation which will lead essentially to the same structures.The idea is to take the category of?-algebras,keep the objects and enhance the notion of morphisms.This can be expected to be interesting as for function algebras we already know what the‘ordinary’morphisms are:pull-backs by smooth maps.Thus a generalization would lead to a generalization of smooth maps between manifolds,when we translate things back using our‘dictionary’.In particular,it might happen that algebras become isomorphic in this new, enhanced category(which will turn out to be not the case for function algebras)and one might have more‘automorphisms’of a given algebra(which will indeed be the case for function algebras). In general,the invertible morphisms in a category form a(large)groupoid in the obvious sense which is called the Picard groupoid of the category.Thus a major step in understanding the whole category is to consider its Picard groupoid of invertible arrows?rst.
In principle,the whole idea should be familiar from geometric mechanics as one example of en-hancing a category by allowing more general morphisms is given by the symplectic‘category’:?rst one considers symplectic manifolds as objects and symplectomorphisms as morphisms.Though this is a reasonable choice to look at,it turns out to be rather boring as the choice for the morphisms is too restrictive.More interesting is the‘category’where one considers morphisms to be canonical relations,see e.g.[2].However,this is no longer an honest category since the composition of mor-phisms is only de?ned when certain technical requirements like clean intesections of the canonical relations are ful?lled.Nevertheless this‘symplectic category’is by far more interesting now.
Other examples are the Morita theory for(integrable)Poisson manifolds by Xu[32]as well as the Morita theory of Lie groupoids,see e.g.[22].
3Morita Equivalence in Di?erent Flavours
After having outlined the general ideas in the previous section we should start being more concrete now.As warming up we discuss the‘enhancing of the category’for the category of unital algebras with usual algebra morphisms?rst,see e.g.[4,19]for this classical approach.
Here the generalized morphisms are the bimodules:For two algebras A and B a(B,A)-bimodule
E,which we shall frequently denote by
B E
A
to indicate that B acts from the left while A acts from
the right,is considered as an arrow A?→B.
Why does this give a reasonable notion of morphisms?In particular,we have to de?ne the com-
position of morphisms.Thus let
B E
A
and
C
F
B
be bimodules then their tensor product
C
F
B
?B
B
E
A
over B is a(C,A)-bimodule and hence an arrow A?→C.However,this is not yet an associative
composition law as for three bimodules
D G
C
,
C
F
B
,
B
E
A
we have a canonical isomorphism
D G
C
?C(
C
F
B
?B
B
E
A
)～=(
D
G
C
?C
C
F
B
)?B
B
E
A
(3.1)
as(D,A)-bimodules but not equality.The way out is to use isomorphism classes of bimodules as arrows instead of bimodules themselves.Then the tensor product becomes indeed associative and
the isomorphism class of the canonical bimodule
A A
A
serves as the identity morphism of the object
A since we use unital algebras for simplicity.
The?nal restriction we have to impose is that in a category the morphism space between two objects has to be a set,which is a priori not clear in our enhanced category.Therefor one should pose additional constraints on the bimodules like?nitely generatedness.However,we shall ignore these subtleties in the following as the new notion of isomorphisms in this category will be una?ected anyway.
However,we still have to show that we really get an extension of our previous notion of mor-phisms.Thus letΦ:A?→B be an algebra homomorphism.Then on B we de?ne a right A-module
structure by b·Φa=bΦ(a)and obtain a bimodule
B BΦ
A
.Its isomorphism class is denoted by?(Φ).
It is easy to see that?(Φ?Ψ)=?(Φ)??(Ψ)and?(id A)is the class of
A A
A
whence our previous
notion of morphisms is indeed contained in the new one.
If we denote this new category by ALG then two unital algebras A and B are called Morita equivalent i?they are isomorphic in ALG.Without going into the details this is equivalent to the existence of a certain bimodule which is‘invertible’with respect to the composition?.In fact, such bimodules can be characterized rather explicitly,see e.g.[19].
The isomorphism classes of these invertible bimodules constitute now the Picard groupoid of this category ALG which we shall denote by Pic.The invertible arrows from A to B are denoted by Pic(B,A)while the isotropy group of this groupoid at the local unit A is denoted by Pic(A),the Picard group of A.
The map?induces now a group homomorphism such that
1?→InnAut(A)?→Aut(A)??→Pic(A)(3.2) is exact,whence in the commutative case,the automorphism group of A is a subgroup of the Picard group Pic(A).Finally,it can be shown that for commutative A,the exact sequence(3.2)is split whence
Pic(A)=Aut(A)?Pic A(A),(3.3) where the subgroup Pic A(A)consists of the symmetric invertible bimodules,i.e.those where a·x=x·a for all x∈E and a∈A.Then Pic A(A)is called the commutative or static Picard group, see e.g.[8,10]for a discussion and further references.
It is a well-known theorem in Morita theory that for unital algebras A,B the equivalence
bimodules
B E
A
are certain?nitely generated projective right A-modules such that Hom A(E
A
)～=B.
Coming back to our example A=C∞(M)we see,using the Serre-Swan theorem,that the only candidates for the symmetric self-equivalence bimodules are the sectionsΓ∞(L)of a complex line bundle.In fact,it turns out thatΓ∞(L)is indeed invertible with inverse given by the class of Γ∞(L?)sinceΓ∞(L)?C∞(M)Γ∞(L)～=Γ∞(L??L)～=C∞(M)as C∞(M)-bimodules.This shows that the static Picard group of C∞(M)is just the‘geometric’Picard group,i.e.the group of isomorphism classes of complex line bundles with the tensor product as https://www.doczj.com/doc/0917288652.html,ing the Chern class to classify complex line bundles then gives according to(3.3)
Pic(C∞(M))=Di?eo(M)?ˇH2(M,),(3.4) where the semidirect product structure comes from the usual action of di?eomorphisms onˇH2(M,).
In general,all Morita equivalence bimodules
B E
A
for A=C∞(M)are isomorphic to someΓ∞(E)
with a vector bundle E?→M of non-zero?bre dimension.Moreover,B has to be isomorphic toΓ∞(End(E)).Thus for function algebras C∞(M)we have a complete description of the Picard groupoid.
We shall now specialize our notion of Morita equivalence:we have already argued that the ?-involution of C∞(M)should be taken into account when having applications to quantization in mind.Moreover,one can include notions of positivity into Morita theory.One de?nes a linear functionalω:A?→to be positive ifω(a?a)≥0for all a∈A.Then an element a∈A is called positive ifω(a)≥0for all positive linear functionalsωof A,see[7,27,30,31]for a detailed discussion.It is clear that for applications to quantum theories such notions of positive functionals are crucial as they encode expectation value functionals and hence the physical states for the observable algebra.
In particular,for A=C∞(M)one?nds that positive linear functionals are precisely the inte-grations with respect to compactly supported positive Borel measures.This follows essentially from Riesz’representation theorem,see[5,App.B].From this it immediately follows that f∈C∞(M) is positive i?f(x)≥0for all x∈M,whence the above,purely algebraic de?nition reproduces the usual notion.
We can now state the de?nition of?-Morita equivalence[1]and strong Morita equivalence bimodules,see[26]as well as[20]for Rie?el’s original formulation in the context of C?-algebras and[5,7]for the general case of?-algebras.Instead of describing the‘enhanced category’way,we directly give the de?nition in terms of bimodules which is entirely equivalent,see[7].
De?nition3.1A?-Morita equivalence bimodule
B E
A
is a(B,A)-bimodule together with inner
products
·,·
A
:E×E?→A(3.5) and
B ·,· :E×E?→B(3.6) such that for all x,y,z∈E,a∈A and b∈B we have:
1. ·,·
A
(resp.B ·,· )is linear in the right(resp.left)argument.
2. x,y·a
A = x,y
A
a and B b·x,y =
b B x,y .
3. x,y
A = y,x
A
?and
B x,y =B y,x
?.
4. ·,·
A
and B ·,· are non-degenerate.
5. ·,· A and B ·,· are full.
6. x,b ·y A = b ?·x,y A and
B x,y ·a =B x ·a ?,y .7.B x,y ·z =x · y,z A .
If in addition the inner products are completely positive then
B E A is called a strong Morita equiv-
alence bimodule.Here ·,· A is called full if the -span of all elements x,y A is the whole algebra A ;in general
these elements constitute a ?-ideal.Moreover, ·,· A is called completely positive if for all n ∈
and all x 1,...,x n ∈E the matrix ( x i ,x j A )∈M n (A )is positive in the ?-algebra M n (A ).
The composition of bimodules is again the tensor product where on C F B ?B B E A the A -valued
inner product is now de?ned by Rie?el’s formula
x ?φ,y ?ψ F ?E A = φ, x,y F B ·ψ E A
,(3.7)and analogously for the C -valued inner product.It is then a non-trivial theorem that this is indeed completely positive again,if the inner products on E and F have been completely positive [7].Passing to isometric isomorphism classes one can show that this gives a groupoid:the ?-Picard groupoid Pic ?and the strong Picard groupoid Pic str ,respectively.In particular,the local unit at A is given by the isometric isomorphism class of A A A equipped with the inner products
a,b A =a ?b and A a,b =ab ?.(3.8)
More generally,A n ,viewed as (M n (A ),A )-bimodule equipped with the canonical inner products
M n (A ) x,y =n
i =1x · y,· A and x,y A =n
i =1x ?i y i (3.9)
implements the strong Morita equivalence between A and M n (A ).
Since we simply can forget the additional structures we obtain canonical groupoid morphisms
Pic str
Pic ?Pic ,(3.10)
which have been studied in [7]:in general,none of them is surjective nor injective,even on the level of the Picard groups.
The geometric interpretation of the inner products is that they correspond to Hermitian ?ber metrics on the corresponding line bundles or vector bundles,respectively:Indeed,this can be seen easily from the very de?nitions.Since up to isometry there is only one positive Hermitian ?ber metric on a given line bundle we have in the case of A =C ∞(M )
Pic str (C ∞(M ))=Di?eo(M )?ˇH 2(M,)=Pic (C ∞(M )).(3.11)
Remark 3.2In the approach of [5,7–9]one main point was to replace the real numbers by an arbitrary ordered ring R and by the ring extension C =R (i)with i 2=?1.This allows to include also the formal star product algebras from deformation quantization into the game.They are de?ned as algebras over the formal power series [[λ]].Surprisingly,essentially all of the constructions involving positivity go through without problems.
4The Covariant Situation
Let us now pass to the covariant situation:we want to incorporate some given symmetry of the?-algebras in question.Here we have two main motivations and examples from di?erential geometry: First,a smooth actionΦ:M×G?→M of a Lie group G on M,where by convention we choose a right action in order to have a left action g→Φ?g on C∞(M)by?-automorphisms.Second,as in?nitesimal version ofΦ,a Lie algebra action,i.e.a Lie algebra homomorphism?:g?→X(M) from a real?nite dimensional Lie algebra g into the Lie algebra of real vector?elds,which correspond to the?-derivations of C∞(M).
In order to formalize and unify both situations it is advantageous to consider Hopf?-algebras and their actions on algebras.Thus let H be a Hopf?-algebra,i.e.a unital?-algebra with a coassociative coproduct?,a counit?and an antipode S such that?:H?→H?H as well as ?:H?→are?-homomorphisms and S(S(g?)?)=g for all g∈H,see e.g.[17,Sect.IV.8].For the coproduct we shall use Sweedler’s notation?(g)=g(1)?g(2).
The two geometric examples we want to discuss are now encoded in the following Hopf?-algebras:
First,recall that any group G de?nes its group algebra[G]which becomes a Hopf?-algebra by setting?(g)=g?g,?(g)=1and S(g)=g?1=g?for g∈G?[G].In this case H=[G]is even cocommutative,i.e.?=?opp where the opposite coproduct is de?ned by?opp(g)=g(2)?g(1).
Second,for any real Lie algebra the complexi?ed universal enveloping algebra U(g)?= U(g)becomes a Hopf?-algebra by setting?(ξ)=ξ?+?ξ,?(ξ)=0and S(ξ)=?ξ=ξ?together with the resulting extensions to all of U(g).Again,U(g)is cocommutative.
The situation that a group G acts by?-automorphisms on a?-algebra as well as a Lie algebra representation by?-derivations can be uni?ed in terms of Hopf?-algebras as follows:A?-action?of H on A is a bilinear map?:H×A?→A such that g?(h?a)=(gh)?a and H?a=a,i.e.A is a left H-module,and g?(ab)=(g(1)?a)(g(2)?b),g?A=?(g)A,and(g?a)?=S(g)??a?for all g,h∈H and a,b∈A.Then it is well-known and easy to see that for our two examples[G] and U(g)this indeed generalizes and uni?es the action by?-automorphisms and?-derivations, respectively.The interesting relations are all encoded in the di?erent coproducts.
In principle and probably even more naturally,one should consider coactions instead of actions of H,see e.g.[17,Sect.III.6].Nevertheless,we stick to the more intuitive point of view where H ‘acts’.
Now suppose we have?-algebras A,B with a?-action of H.Let furthermore
B E
A
be a strong
or?-Morita equivalence bimodule.Then we call the bimodule H-covariant if there is an H-module structure on E denoted by?,too,such that we have the following compatibilities
g?(b·x)=(g
(1)?b)·(g
(2)
?x)(4.1)
g?(x·a)=(g
(1)?x)·(g
(2)
?a)(4.2)
g?B x,y =B g(1)?x,S(g(2))??y (4.3)
g? x,y
A = S(g(1))??x,g(2)?y
A
(4.4)
for all x,y∈E,a∈A,b∈B and g,h∈H.Of course,in the case of ring-theoretic Morita theory one only requires(4.1)and(4.2).Taking isometric isomorphism classes also respecting the action of H gives the H-covariant?avours of the Picard groupoids,denoted by Pic H,Pic?H,and
Pic str H ,respectively.Since we can successively forget the additional structures we get the following commuting diagram of canonical groupoid morphisms:
Pic str H
Pic ?H
Pic H
Pic Pic str
Pic ?,(4.5)Now let us interpret the diagram on the level of Picard groups and in our geometric situation:Let e.g.H =U (g )and A =C ∞(M )be as before,equipped with an action of g by ?-derivations.Then the kernel and the image of the group morphism
Pic H (A )?→Pic (A )(4.6)
encodes on which line bundles we can lift the g -action and if so,in how many di?erent ways up to isomorphism.Analogously,in the strong situation one requires in addition compatibility with the Hermitian ?ber metric.The case of H =[G ]leads to the question of existence and uniqueness of liftings of the group action on M to a group action on L by vector bundle automorphisms.In the strong case one requires the lift in addition to be unitary with respect to the ?ber metric.All this can easily be seen from the compatibility requirements (4.1),(4.2),(4.3),and (4.4)applied to our situation.
Clearly,all these lifting problems are very natural questions in di?erential geometry and have been discussed by various authors,see in particular [18,21,25,28],whence one can rely on the tech-niques developed there.Even though our approach does not give essential new techniques to attack the (in general quite di?cult)lifting problem,it shines some new light on it and unreveals some additional structure of the problem,namely the groupoid structures together with the canonical groupoid morphisms (4.5).Moreover,this point of view embeds the lifting problem in some larger and completely algebraic context since neither the ?-algebras have to be commutative nor has the Hopf ?-algebra to be cocommutative.As remarked already,we can even replace and by R and C ,respectively,and incorporate in particular the formal star product algebras from deformation quantization as well.
5The case of a Lie algebra
In this last section we consider the case of H =U (g )more closely and develop some general results from [16]slightly further.
Assume that B E A is a ?-Morita equivalence bimodule which allows for a lift of the actions of
H on A and B .Then it was shown in [16,Thm 4.14]in full generality that the possible lifts are parametrized by the following group U (H,A ):
We consider linear maps Hom (H,A )with the usual convolution product given by (a ?b )(g )=a (g (1))b (g (2)).This makes Hom (H,A )an associative algebra with unit e (g )=?(g )A ,see e.g.[17,Sect.III.3].Then we consider the following conditions for a ∈Hom (H,A )
a (H )=A ,(5.1)
a (gh )=a (g (1))(g (2)?a (h ))
for all g,h ∈H,
(5.2)
(g(1)?b)a(g(2))=a(g(1))(g(2)?b)for all b∈A,g∈H,(5.3)
)a S(g?(2)) ?=?(g)A for all g∈H.(5.4)
a(g
(1)
Then U(H,A)is de?ned to be the subset of those a∈Hom(H,A)which satisfy(5.1)–(5.4)and it turns out that U(H,A)is a group with respect to the convolution product[16,App.A].Moreover, for unitary central elements c∈U(Z(A))one de?nes?c∈U(H,A)by?c(g)=c(g?c?1).Then one obtains the exact sequence
1?→U(Z(A))H?→U(Z(A)) ?→U(H,A)(5.5) U(Z(A))?U(H,A)is a central and hence normal subgroup.Thus we can de?ne and the image
U(Z(A)).
the group U0(H,A)=U(H,A)
The parametrization of all possible lifts is obtained by a free and transitive group action?→?b of U(H,B)on the set of lifts given by
)·(g(2)?x),(5.6)
g?b x=b(g
(1)
where b∈U(H,B),g∈H and x∈E.In fact,for H-covariantly?-Morita equivalent algebras A and B we have U(H,A)～=U(H,B).Moreover,?b and?give isomorphic actions i?b=?c for some c∈U(Z(B))whence the isomorphism classes of lifts are parametrized by the group U0(H,B)～=U0(H,A)which acts freely and transitively via(5.6)on the isomorphism classes of lifts.
While the above characterization works in full generality we want to specialize now to Lie algebra actions where H=U(g).First,it follows from[16,Prop.A.7]that U(U(g),A)= U(U(g),Z(A))and hence U0(U(g),A)=U0(U(g),Z(A))since U(g)is cocommutative.Thus the values of a∈U(U(g),A)are automatically central,essentially by(5.3).Moreover,the groups U(U(g),A)and U0(U(g),A)are abelian.Since the center Z(A)is invariant under the g-action (by derivations!)we can restrict a∈U(U(g),A)to g?U(g)and obtain a Chevalley-Eilenberg cochainα=a g∈C1CE(g,Z(A)).Evaluating the conditions(5.2),(5.3)and(5.4)on elements ξ,η∈g we?nd by a simple computation the following lemma:
Lemma5.1The restriction gives an injective group homomorphism
U(U(g),A)?a→α=a g∈Z1CE(g,Z(A)antiHermitian)(5.7) into the Chevalley-Eilenberg one-cocycles with values in the anti Hermitian central elements of A. The injectivity easily follows from successively applying(5.2).
Conversely,givenα∈Z1CE(g,Z(A)antiHermitian)we can construct an element a∈U(U(g),A) with a g=α:Let T k(g)denote the k-th complexi?ed tensor power of g and de?ne a(k):T k(g)?→A inductively by
a(0)=A and a(k)(ξ?Y)=α(ξ)a(k?1)(Y)+ξ?a(k?1)(Y)for k≥1,(5.8) where Y∈T k?1(g).Then a lenghty but straightforward computation usingδCEα=0shows that a= ∞k=0a(k)passes to the universal enveloping algebra U(g),viewed as a quotient of T?(g)in the usual way,and ful?lls(5.1)to(5.4).Thus we have:
Theorem5.2The map(5.7)is an isomorphism of abelian groups.
Note that this is in some sense surprising as the condition for αto be a cocycle is linear while the condition (5.2)for a is highly non-linear .It only becomes linear when evaluated on ξ,η∈g ?U (g )thanks to the fact that these elements are primitive,i.e.satisfy ?(ξ)=ξ?+?ξ.Thus a simpli?cation like in Theorem 5.2cannot be expected for more non-trivial Hopf ?-algebras.
Moreover,under the identi?cation (5.7)the elements ?c give just the cocycles ?c (ξ)=c (ξ?c ?1)as
usual.Note that in general U (Z (A ))?Z 1CE (g ,Z (A )antiHermitian )are not CE-coboundaries.Thus,
if we want to relate U 0(U (g ),A )to Lie algebra cohomology we have to assume an additional structure for A :
De?nition 5.3Let A be a unital ?-algebra.Then an exponential function exp is a map exp :Z (A )?→Z (A )such that
1.exp(a +b )=exp(a )exp(b ),
2.exp(0)=A ,
3.D exp(a )=exp(a )Da ,
4.exp(a ?)=exp(a )?,
for all a,b ∈Z (A )and D ∈Der (A ).Note that D ∈Der (A )induces an outer derivation D Z (A )of the center.
We shall now assume that A has an exponential function where our motivating example is of course A =C ∞(M )with the usual exponential.
The ?rst trivial observation is that for a ∈Z (A )we have
exp(a )(ξ)=?(δCE a )(ξ),(5.9)
and for a =?a ?∈Z (A )antiHermitian we clearly have exp(a )∈U (Z (A )).Thus in this case U (Z (A ))
contains all anti Hermitian CE-coboundaries in Z 1CE (g ,Z (A )antiHermitian ).Note however,that in
general, U (Z (A ))is strictly larger.To measure this we consider those elements in U (Z (A ))which are not in the image of exp:we de?ne the abelian group
H 1dR (Z (A ),2πi )=U (Z (A ))
H 1dR (Z (A ),2πi ).(5.12)
In particular,this applies to A =C ∞(M )whence we obtain the full classi?cation of the in-equivalent lifts of the Lie algebra action to line bundles.Note that in general,U 0(U (g ),A )does not depend on the line bundle itself but is universal for all line bundles.
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The way 的用法 Ⅰ常见用法： 1)the way+ that 2)the way + in which（最为正式的用法） 3)the way + 省略（最为自然的用法） 举例：I like the way in which he talks. I like the way that he talks. I like the way he talks. Ⅱ习惯用法： 在当代美国英语中，the way用作为副词的对格，“the way+ 从句”实际上相当于一个状语从句来修饰整个句子。 1)The way =as I am talking to you just the way I’d talk to my own child. He did not do it the way his friends did. Most fruits are naturally sweet and we can eat them just the way they are—all we have to do is to clean and peel them. 2）The way= according to the way/ judging from the way The way you answer the question, you are an excellent student. The way most people look at you, you’d think trash man is a monster. 3）The way =how/ how much No one can imagine the way he missed her. 4）The way =because
The way的用法及其含义（二） 二、the way在句中的语法作用 the way在句中可以作主语、宾语或表语： 1.作主语 The way you are doing it is completely crazy.你这个干法简直发疯。 The way she puts on that accent really irritates me. 她故意操那种口音的样子实在令我恼火。The way she behaved towards him was utterly ruthless. 她对待他真是无情至极。 Words are important, but the way a person stands, folds his or her arms or moves his or her hands can also give us information about his or her feelings. 言语固然重要,但人的站姿,抱臂的方式和手势也回告诉我们他(她)的情感。 2.作宾语 I hate the way she stared at me.我讨厌她盯我看的样子。 We like the way that her hair hangs down.我们喜欢她的头发笔直地垂下来。 You could tell she was foreign by the way she was dressed. 从她的穿著就可以看出她是外国人。 She could not hide her amusement at the way he was dancing. 她见他跳舞的姿势，忍俊不禁。 3.作表语 This is the way the accident happened.这就是事故如何发生的。 Believe it or not, that's the way it is. 信不信由你, 反正事情就是这样。 That's the way I look at it, too. 我也是这么想。 That was the way minority nationalities were treated in old China. 那就是少数民族在旧中
定冠词the的用法： 定冠词the与指示代词this ,that同源,有“那(这)个”的意思,但较弱,可以和一个名词连用,来表示某个或某些特定的人或东西. (1)特指双方都明白的人或物 Take the medicine.把药吃了. (2)上文提到过的人或事 He bought a house.他买了幢房子. I've been to the house.我去过那幢房子. (3)指世界上独一无二的事物 the sun ,the sky ,the moon, the earth (4)单数名词连用表示一类事物 the dollar 美元 the fox 狐狸 或与形容词或分词连用,表示一类人 the rich 富人 the living 生者 (5)用在序数词和形容词最高级,及形容词等前面 Where do you live?你住在哪? I live on the second floor.我住在二楼. That's the very thing I've been looking for.那正是我要找的东西. (6)与复数名词连用,指整个群体 They are the teachers of this school.(指全体教师) They are teachers of this school.(指部分教师) (7)表示所有,相当于物主代词,用在表示身体部位的名词前 She caught me by the arm.她抓住了我的手臂. (8)用在某些有普通名词构成的国家名称,机关团体,阶级等专有名词前 the People's Republic of China 中华人民共和国 the United States 美国 (9)用在表示乐器的名词前 She plays the piano.她会弹钢琴. (10)用在姓氏的复数名词之前,表示一家人 the Greens 格林一家人(或格林夫妇) (11)用在惯用语中 in the day, in the morning... the day before yesterday, the next morning... in the sky... in the dark... in the end... on the whole, by the way...
“tｈｅwａｙ+从句”结构的意义及用法 首先让我们来看下面这个句子: Ｒead the foｌloｗｉnｇpassagｅａnd talkａbout ｉt wi ｔh your classmaｔes．Try tｏｔeｌl whaｔyｏu thiｎk ｏf Tom aｎd ｏｆｔhe ｗay the chiｌｄrenｔrｅated ｈim. 在这个句子中，ｔhe waｙ是先行词,后面是省略了关系副词that或in whｉch的定语从句。 下面我们将叙述“the ｗaｙ+从句”结构的用法。 1.tｈe wａy之后，引导定语从句的关系词是tｈat而不是hoｗ,因此，<<现代英语惯用法词典＞＞中所给出的下面两个句子是错误的：Ｔhｉs is ｔｈeｗayｈowｉtｈａppened. This is the way how he ａlwａｙs treats me. 2．在正式语体中,thaｔ可被in which所代替;在非正式语体中，ｔhat则往往省略。由此我们得到theｗay后接定语从句时的三种模式：1) tｈe ｗaｙ＋ｔhａt-从句２）the way +ｉn whiｃh－从句3) the ｗay +从句 例如：Ｔｈe way(in whiｃh ，thａt) ｔhｅｓｅｃomｒade ｓloｏｋａｔｐrobｌems is wrong．这些同志看问题的方法
不对。 Ｔｈｅｗａy(that ,ｉn ｗhiｃh)yoｕ’ｒe doinｇit is comple ｔeｌy crａzy.你这么个干法,简直发疯。 Wｅａdmiｒｅd him for thｅｗay ｉｎｗhｉch he fａｃeｓdiｆficultieｓ． Waｌlａｃe ａｎd Ｄarwiｎｇreed ｏn the way ｉｎwhi ｃh ｄｉfｆｅrｅnt forms ｏf life had ｂegｕn.华莱士和达尔文对不同类型的生物是如何起源的持相同的观点。 Thｉs is ｔｈe ｗaｙ（tｈａｔ) hｅdｉd it. I lｉkeｄｔhe waｙ(ｔhat) ｓｈeｏｒganized ｔhe ｍeetｉng． 3.theｗay(tｈat)有时可以与hｏw(作“如何”解）通用。例如： That’s tｈe ｗaｙ(tｈaｔ) shｅspoke. = Tｈat’s how shｅspoke.
表示“方式”、“方法”，注意以下用法： 1.表示用某种方法或按某种方式，通常用介词in(此介词有时可省略)。如： Do it (in) your own way. 按你自己的方法做吧。 Please do not talk (in) that way. 请不要那样说。 2.表示做某事的方式或方法，其后可接不定式或of doing sth。 如： It’s the best way of studying [to study] English. 这是学习英语的最好方法。 There are different ways to do [of doing] it. 做这事有不同的办法。 3.其后通常可直接跟一个定语从句(不用任何引导词)，也可跟由that 或in which 引导的定语从句，但是其后的从句不能由how 来引导。如： 我不喜欢他说话的态度。 正：I don’t like the way he spoke. 正：I don’t like the way that he spoke. 正：I don’t like the way in which he spoke. 误：I don’t like the way how he spoke. 4.注意以下各句the way 的用法： That’s the way (=how) he spoke. 那就是他说话的方式。 Nobody else loves you the way(=as) I do. 没有人像我这样爱你。 The way (=According as) you are studying now, you won’tmake much progress. 根据你现在学习情况来看，你不会有多大的进步。 2007年陕西省高考英语中有这样一道单项填空题： ——I think he is taking an active part insocial work. ——I agree with you_____. A、in a way B、on the way C、by the way D、in the way 此题答案选A。要想弄清为什么选A，而不选其他几项，则要弄清选项中含way的四个短语的不同意义和用法，下面我们就对此作一归纳和小结。 一、in a way的用法 表示：在一定程度上，从某方面说。如： In a way he was right.在某种程度上他是对的。注：in a way也可说成in one way。 二、on the way的用法 1、表示：即将来(去)，就要来(去)。如： Spring is on the way.春天快到了。 I'd better be on my way soon.我最好还是快点儿走。 Radio forecasts said a sixth-grade wind was on the way.无线电预报说将有六级大风。 2、表示：在路上，在行进中。如： He stopped for breakfast on the way.他中途停下吃早点。 We had some good laughs on the way.我们在路上好好笑了一阵子。 3、表示：(婴儿)尚未出生。如： She has two children with another one on the way.她有两个孩子，现在还怀着一个。 She's got five children，and another one is on the way.她已经有5个孩子了，另一个又快生了。 三、by the way的用法
The way的用法及其含义（一） 有这样一个句子：In 1770 the room was completed the way she wanted. 1770年，这间琥珀屋按照她的要求完成了。 the way在句中的语法作用是什么？其意义如何？在阅读时，学生经常会碰到一些含有the way 的句子，如：No one knows the way he invented the machine. He did not do the experiment the way his teacher told him.等等。他们对the way 的用法和含义比较模糊。在这几个句子中，the way之后的部分都是定语从句。第一句的意思是，“没人知道他是怎样发明这台机器的。”the way的意思相当于how；第二句的意思是，“他没有按照老师说的那样做实验。”the way 的意思相当于as。在In 1770 the room was completed the way she wanted.这句话中，the way也是as的含义。随着现代英语的发展，the way的用法已越来越普遍了。下面，我们从the way的语法作用和意义等方面做一考查和分析： 一、the way作先行词，后接定语从句 以下3种表达都是正确的。例如：“我喜欢她笑的样子。” 1. the way+ in which +从句 I like the way in which she smiles. 2. the way+ that +从句 I like the way that she smiles. 3. the way + 从句(省略了in which或that) I like the way she smiles. 又如：“火灾如何发生的，有好几种说法。” 1. There were several theories about the way in which the fire started. 2. There were several theories about the way that the fire started.
way 的用法 【语境展示】 1. Now I’ll show you how to do the experiment in a different way. 下面我来演示如何用一种不同的方法做这个实验。 2. The teacher had a strange way to make his classes lively and interesting. 这位老师有种奇怪的办法让他的课生动有趣。 3. Can you tell me the best way of working out this problem? 你能告诉我算出这道题的最好方法吗？ 4. I don’t know the way (that / in which) he helped her out. 我不知道他用什么方法帮助她摆脱困境的。 5. The way (that / which) he talked about to solve the problem was difficult to understand. 他所谈到的解决这个问题的方法难以理解。 6. I don’t like the way that / which is being widely used for saving water. 我不喜欢这种正在被广泛使用的节水方法。 7. They did not do it the way we do now. 他们以前的做法和我们现在不一样。 【归纳总结】 ●way作“方法，方式”讲时，如表示“以……方式”，前面常加介词in。如例1； ●way作“方法，方式”讲时，其后可接不定式to do sth.，也可接of doing sth. 作定语，表示做某事的方法。如例2，例3；
t h e-w a y-的用法
The way 的用法 "the way+从句"结构在英语教科书中出现的频率较高, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或 in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 一.在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮.
way的用法总结大全 way的用法你知道多少，今天给大家带来way的用法，希望能够帮助到大家，下面就和大家分享，来欣赏一下吧。 way的用法总结大全 way的意思 n. 道路,方法,方向,某方面 adv. 远远地，大大地 way用法 way可以用作名词 way的基本意思是“路,道,街,径”,一般用来指具体的“路,道路”,也可指通向某地的“方向”“路线”或做某事所采用的手段,即“方式,方法”。way还可指“习俗,作风”“距离”“附近,周围”“某方面”等。 way作“方法,方式,手段”解时,前面常加介词in。如果way前有this, that等限定词,介词可省略,但如果放在句首,介词则不可省略。
way作“方式,方法”解时,其后可接of v -ing或to- v 作定语,也可接定语从句,引导从句的关系代词或关系副词常可省略。 way用作名词的用法例句 I am on my way to the grocery store.我正在去杂货店的路上。 We lost the way in the dark.我们在黑夜中迷路了。 He asked me the way to London.他问我去伦敦的路。 way可以用作副词 way用作副词时意思是“远远地,大大地”,通常指在程度或距离上有一定的差距。 way back表示“很久以前”。 way用作副词的用法例句 It seems like Im always way too busy with work.我工作总是太忙了。 His ideas were way ahead of his time.他的思想远远超越了他那个时代。 She finished the race way ahead of the other runners.她第一个跑到终点,远远领先于其他选手。 way用法例句
t h e_w a y的用法大全
The way 在the way+从句中, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或 in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 如果怕弄混淆，下面的可以不看了 另外，在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮. the way=according to the way/judging from the way 4)The way you answer the qquestions, you must be an excellent student. 从你回答就知道,你是一个优秀的学生. 5)The way most people look at you, you'd think a trashman was a monster. 从大多数人看你的目光中,你就知道垃圾工在他们眼里是怪物. the way=how/how much 6)I know where you are from by the way you pronounce my name. 从你叫我名字的音调中,我知道你哪里人. 7)No one can imaine the way he misses her. 人们很想想象他是多么想念她. the way=because 8) No wonder that girls looks down upon me, the way you encourage her. 难怪那姑娘看不起我, 原来是你怂恿的
The way 的用法 "the way+从句"结构在英语教科书中出现的频率较高, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 一.在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮.
the way=according to the way/judging from the way 4)The way you answer the qquestions, you must be an excellent student. 从你回答就知道,你是一个优秀的学生. 5)The way most people look at you, you'd think a trashman was a monster. 从大多数人看你的目光中,你就知道垃圾工在他们眼里是怪物. the way=how/how much 6)I know where you are from by the way you pronounce my name. 从你叫我名字的音调中,我知道你哪里人. 7)No one can imaine the way he misses her. 人们很想想象他是多么想念她. the way=because 8) No wonder that girls looks down upon me, the way you encourage her. 难怪那姑娘看不起我, 原来是你怂恿的 the way =while/when(表示对比) 9)From that day on, they walked into the classroom carrying defeat on their shoulders the way other students carried textbooks under their arms. 从那天起,其他同学是夹着书本来上课,而他们却带着"失败"的思想负担来上课.
The way的用法及其含义（三） 三、the way的语义 1. the way=as（像） Please do it the way I’ve told you.请按照我告诉你的那样做。 I'm talking to you just the way I'd talk to a boy of my own.我和你说话就像和自己孩子说话一样。 Plant need water the way they need sun light. 植物需要水就像它们需要阳光一样。 2. the way=how（怎样，多么） No one can imagine the way he misses her.没人能够想象出他是多么想念她！ I want to find out the way a volcano has formed.我想弄清楚火山是怎样形成的。 He was filled with anger at the way he had been treated.他因遭受如此待遇而怒火满腔。That’s the way she speaks.她就是那样讲话的。 3. the way=according as （根据） The way you answer the questions, you must be an excellent student.从你回答问题来看，你一定是名优秀的学生。 The way most people look at you, you'd think a trash man was a monster.从大多数人看你的目光中,你就知道垃圾工在他们眼里是怪物。 The way I look at it, it’s not what you do that matters so much.依我看，重要的并不是你做什么。 I might have been his son the way he talked.根据他说话的样子，好像我是他的儿子一样。One would think these men owned the earth the way they behave.他们这样行动，人家竟会以为他们是地球的主人。
一．Way：“方式”、“方法” 1.表示用某种方法或按某种方式 Do it (in) your own way. Please do not talk (in) that way. 2.表示做某事的方式或方法 It’s the best way of studying [to study] English.。 There are different ways to do [of doing] it. 3.其后通常可直接跟一个定语从句(不用任何引导词)，也可跟由that 或in which 引导的定语从句 正：I don’t like the way he spoke. I don’t like the way that he spoke. I don’t like the way in which he spoke.误：I don’t like the way how he spoke. 4. the way 的从句 That’s the way (=how) he spoke. I know where you are from by the way you pronounce my name. That was the way minority nationalities were treated in old China. Nobody else loves you the way(=as) I do. He did not do it the way his friend did. 二．固定搭配 1. In a/one way：In a way he was right. 2. In the way /get in one’s way I'm afraid your car is in the way， If you are not going to help，at least don't get in the way. You'll have to move－you're in my way. 3. in no way Theory can in no way be separated from practice. 4. On the way (to……) Let’s wait a few moments. He is on the way Spring is on the way. Radio forecasts said a sixth-grade wind was on the way. She has two children with another one on the way. 5. By the way By the way，do you know where Mary lives? 6. By way of Learn English by way of watching US TV series. 8. under way 1. Elbow one’s way He elbowed his way to the front of the queue. 2. shoulder one’s way 3. feel one‘s way 摸索着向前走；We couldn’t see anything in the cave, so we had to feel our way out 4. fight/force one’s way 突破。。。而前进The surrounded soldiers fought their way out. 5.. push/thrust one‘s way（在人群中）挤出一条路He pushed his way through the crowd. 6. wind one’s way 蜿蜒前进 7. lead the way 带路，领路；示范 8. lose one‘s way 迷失方向 9. clear the way 排除障碍，开路迷路 10. make one’s way 前进，行进The team slowly made their way through the jungle.
在the way+从句中, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 如果怕弄混淆，下面的可以不看了 另外，在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮. the way=according to the way/judging from the way 4)The way you answer the qquestions, you must be an excellent student. 从你回答就知道,你是一个优秀的学生. 5)The way most people look at you, you'd think a trashman was a monster. 从大多数人看你的目光中,你就知道垃圾工在他们眼里是怪物. the way=how/how much 6)I know where you are from by the way you pronounce my name. 从你叫我名字的音调中,我知道你哪里人. 7)No one can imaine the way he misses her. 人们很想想象他是多么想念她. the way=because 8) No wonder that girls looks down upon me, the way you encourage her. 难怪那姑娘看不起我, 原来是你怂恿的 the way =while/when(表示对比) 9)From that day on, they walked into the classroom carrying defeat on their shoulders the way other students carried textbooks under their arms.
“the way+从句”结构的意义及用法 首先让我们来看下面这个句子： Read the following passage and talk about it with your classmates. Try to tell what you think of Tom and of the way the children treated him. 在这个句子中，the way是先行词，后面是省略了关系副词that 或in which的定语从句。 下面我们将叙述“the way+从句”结构的用法。 1．the way之后，引导定语从句的关系词是that而不是how,因此，<<现代英语惯用法词典>>中所给出的下面两个句子是错误的：This is the way how it happened. This is the way how he always treats me. 2. 在正式语体中，that可被in which所代替；在非正式语体中，that则往往省略。由此我们得到the way后接定语从句时的三种模式：1) the way +that-从句2) the way +in which-从句3) the way +从句 例如：The way(in which ,that) these comrades look at problems is wrong.这些同志看问题的方法不对。
The way(that ,in which)you’re doing it is completely crazy.你这么个干法，简直发疯。 We admired him for the way in which he faces difficulties. Wallace and Darwin greed on the way in which different forms of life had begun.华莱士和达尔文对不同类型的生物是如何起源的持相同的观点。 This is the way (that) he did it. I liked the way (that) she organized the meeting. 3.the way(that)有时可以与how(作“如何”解)通用。例如： That’s the way (that) she spoke. = That’s how she spoke. I should like to know the way/how you learned to master the fundamental technique within so short a time. 4．the way的其它用法：以上我们讲的都是用作先行词的the way，下面我们将叙述它的一些用法。
定冠词the的12种用法 定冠词the 的12 种用法，全知道?快来一起学习吧。下面就和大家分享，来欣赏一下吧。 定冠词the 的12 种用法，全知道? 定冠词the用在各种名词前面，目的是对这个名词做个记号，表示它的特指属性。所以在词汇表中，定冠词the 的词义是“这个，那个，这些，那些”，可见，the 即可以放在可数名词前，也可以修饰不可数名词，the 后面的名词可以是单数，也可以是复数。 定冠词的基本用法： (1) 表示对某人、某物进行特指，所谓的特指就是“不是别的，就是那个!”如： The girl with a red cap is Susan. 戴了个红帽子的女孩是苏珊。 (2) 一旦用到the，表示谈话的俩人都知道说的谁、说的啥。如：
The dog is sick. 狗狗病了。(双方都知道是哪一只狗) (3) 前面提到过的，后文又提到。如： There is a cat in the tree.Thecat is black. 树上有一只猫，猫是黑色的。 (4) 表示世界上唯一的事物。如： The Great Wall is a wonder.万里长城是个奇迹。(5) 方位名词前。如： thenorth of the Yangtze River 长江以北地区 (6) 在序数词和形容词最高级的前面。如： Who is the first?谁第一个? Sam is the tallest.山姆最高。 但是不能认为，最高级前必须加the，如： My best friend. 我最好的朋友。 (7) 在乐器前。如： play the flute 吹笛子
Way用法 A:I think you should phone Jenny and say sorry to her. B:_______. It was her fault. A. No way B. Not possible C. No chance D. Not at all 说明：正确答案是A. No way，意思是“别想！没门！决不！” 我认为你应该打电话给珍妮并向她道歉。 没门！这是她的错。 再看两个关于no way的例句： （1）Give up our tea break? NO way! 让我们放弃喝茶的休息时间？没门儿！ （2）No way will I go on working for that boss. 我决不再给那个老板干了。 way一词含义丰富，由它构成的短语用法也很灵活。为了便于同学们掌握和用好它，现结合实例将其用法归纳如下： 一、way的含义 1. 路线
He asked me the way to London. 他问我去伦敦的路。 We had to pick our way along the muddy track. 我们不得不在泥泞的小道上择路而行。 2. （沿某）方向 Look this way, please. 请往这边看。 Kindly step this way, ladies and gentlemen. 女士们、先生们，请这边走。 Look both ways before crossing the road. 过马路前向两边看一看。 Make sure that the sign is right way up. 一定要把符号的上下弄对。 3. 道、路、街，常用以构成复合词 a highway（公路），a waterway（水路），a railway（铁路），wayside（路边）
way/time的特殊用法 1、当先行词是way意思为”方式.方法”的时候,引导定语从句的关系词有下列3种形式： Way在从句中做宾语 The way that / which he explained to us is quite simple. Way在从句中做状语 The way t hat /in which he explained the sentence to us is quite simple. 2、当先行词是time时,若time表示次数时,应用关系代词that引导定语从句,that可以省略; 若time表示”一段时间”讲时,应用关系副词when或介词at/during + which引导定语从句 1.Is this factory _______ we visited last year? 2.Is this the factory-------we visited last year? A. where B in which C the one D which 3. This is the last time _________ I shall give you a lesson. A. when B that C which D in which 4.I don’t like the way ________ you laugh at her. A . that B on which C which D as 5.He didn’t understand the wa y ________ I worked out the problem. A which B in which C where D what 6.I could hardly remember how many times----I’ve failed. A that B which C in which D when 7.This is the second time--------the president has visited the country. A which B where C that D in which 8.This was at a time------there were no televisions, no computers or radios. A what B when C which D that