搜档网
当前位置:搜档网 › Field Algebras

Field Algebras

Field Algebras
Field Algebras

a r

X i

v

:m

a

t h /

2

4

2

8

2

v

2

[

m

a

t h

.

Q

A ]

7

M

a y

2

2

FIELD ALGEBRAS

BOJKO BAKALOV AND VICTOR G.KAC Dedicated to Ernest Borisovich Vinberg on the occasion of his 65th birthday.Abstract.A ?eld algebra is a “non-commutative”generalization of a vertex algebra.In this paper we develop foundations of the theory of ?eld algebras.0.Introduction Roughly speaking,the notion of a vertex algebra [B1]is a generalization of the notion of a unital commutative associative algebra where the multiplication depends on a parameter.(In fact,in [B2]vertex algebras are described as “singular”commutative associative rings in a certain category.)More precisely,an operator of left multiplication on a vertex algebra V by an element a ∈V is a ?eld Y (a,z )= n ∈Z a (n )z ?n ?1,where a (n )∈End V ,and one requires that Y (a,z )b is a Laurent series in z for any two elements a,b ∈V .The role of a unit element of an algebra is played in this context by a vacuum vector |0 ∈V ,satisfying (vacuum axioms )Y (|0 ,z )=I V ,Y (a,z )|0 =e zT a ,where I V ∈End V is the identity operator and T ∈End V .A linear map a →Y (a,z )of a vector space V with a vacuum vector |0 to the space of End V -valued ?elds satisfying (translation invariance )[T,Y (a,z )]=Y (T a,z )=?z Y (a,z ),is called a state–?eld correspondence .This notion is an analogue of a unital algebra.For example,if V is an ordinary algebra with a unit element |0 and T is a derivation of V ,then the formula Y (a,z )b =(e zT a )b,a,b ∈V (0.1)de?nes a state–?eld correspondence (and it is easy to show that all of them with the property that Y (a,z )is a formal power series in z ,are thus obtained).

Furthermore,the associativity property of the algebra V is equivalent to the following property of the state–?eld correspondence (0.1):

Y (Y (a,z )b,?w )c =Y (a,z ?w )Y (b,?w )c ,a,b,c ∈V ,

(0.2)and the commutativity property of

V is equivalent to:Y (a,z )b =e zT

Y (b,?z )a ,a,b ∈V .(0.3)It turns out that for a general vertex algebra V identity (0.2)holds only after one multiplies both sides by (z ?w )N where N is su?ciently large (depending on

2BOJKO BAKALOV AND VICTOR G.KAC

a,b,c):

(0.4)

(z?w)N Y(Y(a,z)b,?w)c=(z?w)N Y(a,z?w)Y(b,?w)c,N?0. One of the equivalent de?nitions of a vertex algebra is that it is a state–?eld corre-spondence satisfying(0.3)and(0.4)(see Theorem6.3).

A?eld algebra is a state–?eld correspondence satisfying only the associativity property(0.4).We believe that this is the right analogue of a unital associative algebra.In the present paper we are making the?rst steps towards a general theory of?eld algebras.

The trivial examples of?eld algebras are provided by state–?eld correspondences (0.1):this is a?eld algebra if and only if the underlying algebra V is associative. The simplest examples of non-trivial?eld algebras are tensor products of?eld algebras(0.1)with vertex algebras.A special case of this is the algebra of matrices with entries in a vertex algebra.Other examples are provided by a smash product of a vertex algebra and a group of its automorphisms.Thus,many important ring-theoretic constructions with vertex algebras become possible in the framework of ?eld algebras.

One of our main results is the construction of a canonical structure of a?eld algebra in a tensor algebra T(R)over a Lie(even Leibniz)conformal algebra R (Theorem5.1).Imposing relation(0.3)on T(R)gives the enveloping vertex algebra U(R)of R(cf.[K,GMS]).

We also establish the?eld algebra analogues of the density and duality theorems in the representation theory of associative algebras(see Theorems8.5and8.6)and discuss the Zhu algebra construction[Z]in the framework of?eld algebras.

Note that the“?eld algebras”considered in[K,Sec.4.11]are de?ned by a stronger than associativity axiom;we call them strong?eld algebras in the present paper.Surprisingly,it turns out that they are almost the same as vertex algebras (see Theorem7.4),although the“trivial”?eld algebras(0.1)are automatically strong?eld algebras.In the present paper the results of[K,Sec.4.11]on?eld algebras are corrected.

The?rst examples of non-trivial?eld algebras(i.e.,di?erent from(0.1))that are not vertex algebras were constructed in[EK].The“quantum vertex algebras”of[EK]are?eld algebras satisfying in addition a certain“braided commutativity”generalizing(0.3).The relation of the present work to the paper[EK],and also to [B2]and[FR],will be discussed in a subsequent paper.

1.State–Field Correspondence

Let V be a vector space(referred to as the space of states).Recall that a (End V-valued)?eld is an expression of the form

a(z)= n∈Z a(n)z?n?1,

where z is an indeterminate,a(n)∈End V,and for each v∈V one has:

a(n)v=0for n?0,

i.e.,a(z)v is a Laurent series in z.

Denote by g?f(V)the space of all End V-valued?elds.For each n∈Z one de?nes the n-th product of?elds a(z)and b(z)by the following formula:

(1.1)

a(z)(n)b(z)=Res x a(x)b(z)i x,z(x?z)n?b(z)a(x)i z,x(x?z)n .

FIELD ALGEBRAS3 Here i x,z(respectively i z,x)stands for the expansion in the domain|x|>|z|(re-spectively|z|>|x|):

i x,z(x?z)n=

j=0 n j x n?j(?z)j,

while

i z,x(x?z)n=

j=0 n j x j(?z)n?j.

It is easy to see that the space of?elds g?f(V)is closed under all n-th products and also under the derivation by the indeterminate.

Recall that

δ(x?z)=(i x,z?i z,x)(x?z)?1= j∈Z x j z?j?1

(1.2)

is the formal delta-function,characterized by the property:

Res x a(x)δ(x?z)=a(z)for a(x)∈g?f(V).

Formula(1.1)is equivalent to the following two formulas for n∈Z+:

a(z)(n)b(z)=Res x[a(x),b(z)](x?z)n,

a(z)(?n?1)b(z)=:?n z a(z)b(z):/n!.

Here::stands for the normal ordered product of?elds de?ned by

:a(z)b(z):=a(z)+b(z)+b(z)a(z)?,

(1.3)

where

a(z)+= j≤?1a(j)z?j?1,a(z)?= j≥0a(j)z?j?1.

De?nition1.1.Let(V,|0 )be a pointed vector space,i.e.,a vector space V with a?xed non-zero vector|0 ∈V,which will be referred to as the vacuum vector.A state–?eld correspondence is a linear map

Y:V→g?f(V),a→Y(a,z)= n∈Z a(n)z?n?1,

such that following axioms hold:

(vacuum axioms)Y(|0 ,z)=I V,Y(a,z)|0 =a+T(a)z+···∈V[[z]], where I V∈End V is the identity operator,T∈End V,

(translation invariance)[T,Y(a,z)]=Y(T a,z)=?z Y(a,z).

The linear operator T on V is called the translation operator.

Example1.2.Let V be a unital algebra with a unit element|0 ,and let T be a derivation of V.Then

Y(a,z)b=(e zT a)b,a,b∈V,

is a state–?eld correspondence.In fact,all state–?eld correspondences for which the?elds Y(a,z)are formal power series in z are obtained in this way.We will call such a state–?eld correspondence trivial.

4BOJKO BAKALOV AND VICTOR G.KAC

Example1.3.Given two state–?eld correspondences(V i,|0 i,Y i),i=1,2,one de?nes their tensor product(V=V1?V2,|0 =|0 1?|0 2,Y),where Y(a1?a2,z)= Y(a1,z)?Y(a2,z).This is again a state–?eld correspondence,the translation operator being T=T1?I+I?T2.

Example1.4.A special case of Example1.3is the tensor product of a state–?eld correspondence(V,|0 ,Y)and a unital algebra(A,1),which is viewed as a trivial state–?eld correspondence with T=0.In particular,the space Mat N(V)of N by N matrices with entries in V has a structure of a state–?eld correspondence. Example1.5.A generalization of Example1.4is the smash product V?Γ,where Γis a group of automorphisms of the state–?eld correspondence(V,|0 ,Y).We de?ne the space of states to be V?C[Γ],the vacuum vector to be|0 ?1,and let Y(a?g,z)(b?h)=Y(a,z)(gb)?gh,a,b∈V,g,h∈Γ.

WhenΓis?nite,one has the following very useful formula(cf.[MS]):

1

VΓ?{e(?1)(v(?1)e)|v∈V?Γ}?V?Γ,where e=

|Γ| g∈Γa?g for a∈VΓ.

One de?nes in the obvious way the notions of subalgebras,ideals and homo-morphisms of state–?eld correspondences.For example,a subalgebra of a state–?eld correspondence(V,|0 ,Y)is a subspace U of V containing|0 and such that a(n)b∈U for all n∈Z if a,b∈U.A left ideal is a T-invariant subspace I of V such that a(n)b∈I for all n∈Z if a∈V,b∈I.If I is a two-sided ideal of V,there is a natural state–?eld correspondence with a space of states V/I.

Proposition1.6.In terms of Fourier coe?cients a(n)of Y,the de?nition of a state–?eld correspondence can be reformulated as follows:

(local?niteness)a(N)b=0for a,b∈V,N?0,

(weak vacuum axiom)a(?1)|0 =|0 (?1)a=a,

(translation invariance)[T,a(n)]=(T a)(n)=?na(n?1)for n∈Z,a∈V. Proof.We have to check that|0 (n)a=0for n=?1,T a=a(?2)|0 ,and a(n)|0 =0 for n≥0,a∈V.First,T a=(T a)(?1)|0 =a(?2)|0 .Next,we have T|0 = T(|0 (?1)|0 )=2T|0 ,hence T|0 =0.Then for every n=0,(T|0 )(n)=?n|0 (n?1)shows that|0 (n?1)=0.Similarly,T(a(n)|0 )=?na(n?1)|0 implies a(n)|0 =0for n≥0,because a(N)|0 =0for N?0.

Proposition1.7.Any state–?eld correspondence Y has the following properties:

(a)Y(a,z)|0 =e zT a,

(b)e wT Y(a,z)e?wT=Y(e wT a,z)=i z,w Y(a,z+w),

(c)(Y(a,z)(n)Y(b,z))|0 =Y(a(n)b,z)|0 .

Proof.See[K,Proposition4.1].

Proposition1.8.Given a state–?eld correspondence Y,de?ne

(1.5)

Y op(a,z)b=e zT Y(b,?z)a.

Then Y op is also a state–?eld correspondence,and(Y op)op=Y.

FIELD ALGEBRAS5 Proof.Straightforward.

De?nition1.9.The state–?eld correspondence Y op de?ned by(1.5)is called the opposite to Y.

At the end of this section,we study the notion of grading for state–?eld corre-spondences.

De?nition1.10.A state–?eld correspondence Y on a pointed vector space(V,|0 ) is called graded if there is a diagonalizable operator H∈End V satisfying

[H,Y(a,z)]=Y(Ha,z)+z?z Y(a,z),a∈V.

(1.6)

It is called Z+-graded if all eigenvalues of H are non-negative integers.The operator H is called a Hamiltonian of V.

For a homogeneous element a∈V,we denote its degree by?a:Ha=?a a.In terms of modes,equation(1.6)is equivalent to:

[H,a(n)]=(?a?1?n)a(n),a∈V,Ha=?a a,n∈Z.

(1.7)

This implies that for homogeneous a,b∈V one has:

?a

(n)b

=?a+?b?n?1,n∈Z.

(1.8)

One can easily show that H|0 =0and[H,T]=T.The latter is equivalent to:?T a=?a+1for homogeneous a∈V(and is a special case of(1.8)since T a=a?2|0 ).

Proposition1.11.If Y is a graded state–?eld correspondence,then its opposite Y op is also graded,with the same Hamiltonian H.

Proof.Easy exercise,using(1.5),(1.6)and He zT=e zT(H+zT).

Remark1.12.A?eldφ(z)∈g?f(V)is said to have conformal dimension?if

[H,φ(z)]=(?+z?z)φ(z).

Then for each homogeneous a∈V,Y(a,z)is a?eld of conformal dimension?a (see(1.6)).

Letφ(z),φ′(z)∈g?f(V)be two?elds of conformal dimensions?and?′,re-spectively.Then?zφ(z)has conformal dimension?+1,and for each n∈Z,the n-th productφ(z)(n)φ′(z)has conformal dimension?+?′?n?1.

If Y is a Z+-graded state–?eld correspondence,we may shift subscripts as follows:

a n=a(n+?

a?1),a(n)=a n??a+1if Ha=?a a,

(1.9)

so that

[H,a n]=?n a n,a∈V,n∈Z.

(1.10)

It is easy to see that:

((T+H)a)0=0for all a∈V.

(1.11)

6BOJKO BAKALOV AND VICTOR G.KAC

2.Locality

A pair of?elds a(z),b(z)∈g?f(V)is called local if

(z?w)N[a(z),b(w)]=0for N?0.

It is called local on v∈V if

(z?w)N[a(z),b(w)]v=0for N?0.

Proposition2.1.A pair of?elds(a(z),b(z))is local if and only if

[a(z),b(w)]= j≥0?nite a(w)(j)b(w) ?j wδ(z?w)/j!.

It is local on v i?

[a(z),b(w)]v= j≥0?nite a(w)(j)b(w) v?j wδ(z?w)/j!.

Proof.Follows from[K,Corollary2.2].

A pair of?elds(a(z),b(z))is called weakly local if

Res z(z?w)N[a(z),b(w)]=0for N?0.

Note that the weak locality of the pair(a(z),b(z))means that a(z)(N)b(z)=0for N?0.

Remark2.2.If a pair of?elds(a(z),b(z))is local,then the pair(b(z),a(z))is also local.However,the weak locality of a pair(a,b)does not imply the weak locality of the pair(b,a).Indeed,let a(z)be the free boson,so that[a(m),a(n)]=mδm,?n, and take b(z)=a(1)z?1.Then[a(z),b(w)]=?w?1;therefore,a(z)(n)b(z)=0, b(z)(n)a(z)=(?z)n for all n≥0.

A collection of?elds is called local(respectively local on v∈V,respectively weakly local)if each pair of?elds from this collection is local(respectively local on v∈V,respectively weakly local).

A state–?eld correspondence Y is called local(respectively weakly local)if the collection of?elds{Y(a,z)}a∈V is local(respectively weakly local).

Example2.3.Let U be a vector space and let V?g?f(U)be a weakly local space of End U-valued?elds in the variable x.Let|0 =I U∈V.Then the following formula de?nes a state–?eld correspondence:

Y(a(x),z)b(x)= n∈Z a(x)(n)b(x)z?n?1,a(x),b(x)∈V,

the translation operator being T=?x.

The following proposition generalizes the Uniqueness theorem from[K,Sec.4.4]. Proposition2.4.Let Y be a state–?eld correspondence on a pointed vector space (V,|0 ).Let B i(z)∈g?f(V),i=1,2,be such that

(i)B1(z)|0 =B2(z)|0 ,

(ii)all pairs(Y(a,z),B i(z))are local on|0 .

Then B1(z)=B2(z).

FIELD ALGEBRAS7 Proof.Let B(z)=B1(z)?B2(z).Then B(z)|0 =0and all pairs(Y(a,z),B(z)) are local on|0 .We have(z?w)N[B(z),Y(a,w)]|0 =0,so(z?w)N B(z)Y(a,w)|0 =0for N?https://www.sodocs.net/doc/dc4992445.html,ing Proposition1.7a,we get(z?w)N B(z)e wT a=0.Letting w=0,we obtain B(z)a=0,a∈V.

The following is a generalization of Dong’s lemma(see[K,Lemma3.2]). Lemma2.5.Let a,b,c∈g?f(V)be three?elds.

(a)If the pair(a,b)is weakly local and the pairs(a,c),(b,c)are local,then

(a(n)b,c)is local for any n∈Z.

(b)If the pairs(a,b)and(a,c)are weakly local and the pair(b,c)is local,then

(a(n)b,c)is local for any n≥0.

(c)If the pairs(a,b),(a,c),(b,c)are weakly local,then(a(n)b,c)is weakly local

for any n≥0.

(d)If the pairs(a,b),(a,c),(b,c)are weakly local,then(a,b(n)c)is weakly local

for any n∈Z.

Proof.Same as in[K,Sec.3.2].

Remark2.6.In general,it is not true that if all pairs(a,b),(a,c),(b,c)are weakly local,then(a(?1)b,c)is weakly local as well.For example,let a(z)be the free boson (see Remark2.2),and take b(z)=a(z),c(z)=a(z)+.Then[a(z)?,a(w)+]= i z,w(z?w)?2and[a(z)+,a(w)+]=0;hence(a,a)and(a,a+)are weakly local. We have:[:a(z)2:,a(w)+]=[a(z)+a(z)+a(z)a(z)?,a(w)+]=2a(z)i z,w(z?w)?2, which shows that the pair(:a2:,a+)is not weakly local.

Lemma2.7.Let X and Y be two state–?eld correspondences on a pointed vector space(V,|0 ),and let a,b,c∈V be such that

(i)(Y(a,z),Y(b,z))is a weakly local pair,

(ii)the pair(Y(a,z),X(c,z))(respectively,(Y(b,z),X(c,z)))is local on|0 and b(respectively,on|0 and a).

Then the pair(Y(a,z)(n)Y(b,z),X(c,z))is local on the vacuum vector for all n∈Z. Proof.Let us write a(z)=Y(a,z),b(z)=Y(b,z),c(z)=X(c,z)for short.By the (weak)locality,there exists a number N≥0such that for all r≥N:

Res z

1

(z1?z2)r[a(z1),b(z2)]=0,

(z1?z3)r[a(z1),c(z3)]v=0,v=|0 or b,

(z2?z3)r[b(z2),c(z3)]v=0,v=|0 or a.

Let us apply the di?erential operator e?z2(?z1+?z3)to the left-hand side of the second equation for v=b.By Taylor’s formula,the result is:

(z1?z3)r i z

1,z2i z

3,z2

[a(z1?z2),c(z3?z2)]b.

Using Proposition1.7,this is equal to:

(z1?z3)r[e?z2T a(z1)e z2T,e?z2T c(z3)e z2T]b

=(z1?z3)r e?z2T[a(z1),c(z3)]e z2T b

=(z1?z3)r e?z2T[a(z1),c(z3)]b(z2)|0 .

8BOJKO BAKALOV AND VICTOR G.KAC

Applying e z2T to this,we obtain:

(z1?z3)r[a(z1),c(z3)]b(z2)|0 =0,r≥N.

Similarly,we have:

(z2?z3)r[b(z2),c(z3)]a(z1)|0 =0,r≥N.

The rest of the proof is as in[K,Lemma3.2].

In the proof of Lemma2.7,we proved the following result which will be needed later.

Lemma2.8.Let X and Y be two state–?eld correspondences,and let a,b,c∈V be such that

(z?w)N[Y(a,z),X(c,z)]b=0

for some N≥0.Then

(z?w)N[Y(a,z),X(c,z)]T b=0.

Proposition2.9.Let X and Y be two state–?eld correspondences for(V,|0 ),such that

(i)Y is weakly local,

(ii)all pairs(Y(a,z),X(b,z))are local on any v∈V.

Then Y(a,z)(n)Y(b,z)=Y(a(n)b,z)for all n∈Z,a,b∈V.

Proof.Let B1(z)=Y(a,z)(n)Y(b,z),B2(z)=Y(a(n)b,z).Due to Proposition1.7c, the?elds B i(z)satisfy condition(i)of Proposition2.4.Due to Lemma2.7,condi-tion(ii)of Proposition2.4is satis?ed as well.Hence B1(z)=B2(z).

Proposition2.10.Let X and Y be two state–?eld correspondences.Then all pairs (Y(a,z),X(b,z))are local on the vacuum vector if and only if X=Y op.

Proof.Assume that(z?w)N Y(a,z)X(b,w)|0 =(z?w)N X(b,w)Y(a,z)|0 for N?https://www.sodocs.net/doc/dc4992445.html,ing Proposition1.7,this is equivalent to:(z?w)N e wT i z,w Y(a,z?w)b= (z?w)N e zT i w,z X(b,w?z)a.For a su?ciently large N,both sides contain only non-negative powers of z?w,and hence only non-negative powers of z and w. Putting z=0,we obtain X(b,w)a=e wT Y(a,?w)b.

Conversely,if X=Y op,the above calculations show that[Y(a,z),X(b,w)]|0 =0 for all a,b∈V.

3.Field Algebras

Let(V,|0 )be a pointed vector space and let Y be a state–?eld correspondence. We say that Y satis?es the n-th product axiom if for all a,b∈V and n∈Z

(3.1)

Y(a(n)b,z)=Y(a,z)(n)Y(b,z).

We say that Y satis?es the associativity axiom if for all a,b,c∈V

(3.2)

(z?w)N Y(Y(a,z)b,?w)c=(z?w)N i z,w Y(a,z?w)Y(b,?w)c,N?0. Proposition3.1.Let(V,|0 )be a pointed vector space and let Y be a state–?eld correspondence.Then:

FIELD ALGEBRAS 9

(a)Y satis?es the n -th product axiom (3.1)i?for all a,b ∈V [Y (a,z ),Y op (b,w )]= j ≥0?nite

Y op (a (j )b,w )?j w δ(z ?w )/j !,

(3.3)where Y (a,z )= j ∈Z a (j )z ?j ?1and Y op is the opposite to Y (see (1.5)).

(b)Y satis?es the associativity axiom (3.2)i?all pairs (Y (a,z ),Y op (b,z ))are

local on each v ∈V .In particular,the n -th product axiom implies the asso-ciativity axiom.

Proof.Assume that Y satis?es the n -th product axiom.Replace z with ?w in (3.1),multiply both sides by z ?n ?1,and sum over n ∈Z .Then after some manipulation,we get:

Y (Y (a,z )b,?w )c

=i z,w Y (a,z ?w )Y (b,?w )c ?Y (b,?w ) j ≥0(a (j )c )?j w δ(z ?w )/j !.(3.4)After applying e wT to both sides,the left-hand side becomes Y op (c,w )Y (a,z )b .The ?rst term in the right-hand side becomes (using Proposition 1.7):

i z,w e wT Y (a,z ?w )Y (b,?w )c =Y (a,z )e wT Y (b,?w )c =Y (a,z )Y op (c,w )b,

while the second term becomes: j ≥0

Y op (a (j )c,w )b ?j w δ(z ?w )/j !.

Therefore (3.4)is equivalent to (3.3)(with b replaced with c ).Since (3.4)is the generating function of all n -th products,we get statement (a).

Notice that,by the above argument,the locality of the pair (Y (a,z ),Y op (c,z ))on b is equivalent to the associativity property (3.2).This completes the proof.

Corollary 3.2.Let Y be a state–?eld correspondence for (V,|0 ).Then for three elements a,b,c ∈V ,the collection of identities Y (a (n )b,z )c = Y (a,z )(n )Y (b,z ) c,n ∈Z

(3.5)implies the locality of the pair (Y (a,z ),Y op (c,z ))on b ,which in turn is equivalent to the associativity property (3.2).If (3.5)holds for ?xed a,c and all b ∈V ,then the pair (Y (a,z ),Y op (c,z ))is local.

De?nition 3.3.Let (V,|0 )be a pointed vector space.A ?eld algebra (V,|0 ,Y )is a state–?eld correspondence Y for (V,|0 )satisfying the associativity axiom (3.2).A strong ?eld algebra (V,|0 ,Y )is a state–?eld correspondence Y satisfying the n -th product axiom (3.1).

Note that,by Proposition 3.1b,any strong ?eld algebra is a ?eld algebra.

Example 3.4.Let V be an associative algebra with a unit element |0 and let T be a derivation of V .Let Y (a,z )b =(e zT a )b .Then (V,|0 ,Y )is a strong ?eld algebra.All ?eld algebras for which all Y (a,z )are formal power series in z are obtained in this way.We call such ?eld algebras trivial .

10BOJKO BAKALOV AND VICTOR G.KAC

Example3.5.The tensor product of two?eld algebras(cf.Example1.3)is again a?eld algebra.However,the tensor product of two strong?eld algebras is not necessarily a strong?eld algebra.For example,the tensor product of a non-trivial strong?eld algebra and a trivial?eld algebra(see Example3.4),where the under-lying associative algebra is non-commutative,is not a strong?eld algebra. Example3.6.The smash product of a?eld algebra V and a groupΓof its auto-morphisms(see Example1.5)is a?eld algebra.It is not a strong?eld algebra if V is non-trivial and the action ofΓon V is non-trivial.

Theorem3.7.(a)A?eld algebra(V,|0 ,Y)is the same as a state–?eld corre-spondence Y for(V,|0 )such that there exists a state–?eld correspondence X for(V,|0 ),having the property that all pairs(Y(a,z),X(b,z))are local on each v∈V.In this case X=Y op.

(b)A strong?eld algebra is the same as a?eld algebra(V,|0 ,Y)for which the

state–?eld correspondence Y is weakly local.

Proof.(a)Let(V,|0 ,Y)be a?eld algebra.Then,by Proposition3.1b,Y op is local with Y on each v∈V.

Conversely,let X be a state–?eld correspondence which is local with Y on every vector.In particular,it is local on the vacuum.Then,Proposition2.10implies that X=Y op,and again by Proposition3.1b,(V,|0 ,Y)is a?eld algebra.

(b)If(V,|0 ,Y)is a strong?eld algebra,then it is also a?eld algebra.The weak locality of Y follows from(3.1),because a(n)b=0for n?0.

Conversely,if(V,|0 ,Y)is a?eld algebra with a weakly local Y,the n-th product axiom(3.1)follows from part(a)and Proposition2.9.

Remark3.8.If(V,|0 ,Y)is a?eld algebra,then(V,|0 ,Y op)is also a?eld algebra, called the opposite?eld algebra.

Corollary3.9.Let(V,|0 ,Y)be a strong?eld algebra.Then for all a,b∈V the pair(Y(a,z),Y op(b,z))is local and

Y(a,z)(n)Y op(b,z)=Y op(a(n)b,z),n≥0.

Proof.This follows from Proposition3.1a.

Proposition3.10.A strong?eld algebra is a vector space V with a given vector |0 and a linear map Y:V→g?f(V),a→Y(a,z)= n∈Z a(n)z?n?1,satisfying for all n∈Z,a,b∈V the n-th product axiom(3.1)and the weak vacuum axioms

|0 (n)a=δn,?1a,a(?1)|0 =a.

(3.6)

Proof.The only thing that has to be checked is that such a map Y is a state–?eld correspondence.Let T a=a(?2)|0 .Letting b=|0 in n-th product axioms for n≥1,we see that the vacuum axioms hold.Notice that,taking coe?cient of w?k?1,the n-th product axiom gives the following relation:

(a(n)b)(k)c=

j=0(?1)j n j a(n?j)(b(k+j)c)?(?1)n b(n+k?j)(a(j)c) .

(3.7)

Letting k=?2and c=|0 in(3.7)gives[T,a(n)]=?n a(n?1).Letting n=?2 and b=|0 in(3.7),we get(T a)(k)=?k a(k?1).Therefore,Y is a state–?eld correspondence.

FIELD ALGEBRAS11 De?nition3.11.A(strong)?eld algebra(V,|0 ,Y)is called graded,respectively Z+-graded,if the state–?eld correspondence Y is(see De?nition1.10).

Remark3.12.Let(V,|0 ,Y)be a Z+-graded strong?eld algebra,and let a,b,c∈V be homogeneous elements.Then Y(a,z)(n)Y(b,z)=0for n≥?a+?b,and the associativity relation(3.2)holds for N≥?a+?c.This follows from the proof of Proposition3.1and(1.8).

Question3.13.Is it true that a weakly local subspace V of g?f(U),containing I U,?x-invariant and closed under all n-th products,is a strong?eld algebra?(See Example2.3.)

4.Conformal Algebras and Field Algebras

Let(V,|0 )be a pointed vector space and let Y be a state–?eld correspondence. For a,b∈V,we de?ne theirλ-product by the formula

aλb=Res z eλz Y(a,z)b= n≥0?niteλn a(n)b/n!.

We also have the(?1)-st product on V,which we denote as

a.b=Res z z?1Y(a,z)b=a(?1)

b.

The vacuum axioms for Y imply:

(4.1)

|0 .a=a=a.|0 ,

while the translation invariance axiom shows that:

(4.2)

T(a.b)=(T a).b+a.(T b)

and

(4.3)

T(aλb)=(T a)λb+aλ(T b),(T a)λb=?λaλb

for all a,b∈V.

Conversely,if we are given a linear operator T,aλ-product and a.-product on V,satisfying the above properties(4.1)–(4.3),we can reconstruct the state–?eld correspondence Y by the formulas:

(4.4)

Y(a,z)+b=(e zT a).b,Y(a,z)?b=(a??z b)(z?1),

where Y(a,z)=Y(a,z)++Y(a,z)?.Notice that equations(4.1)–(4.3)imply T|0 =0and|0 λa=0=aλ|0 for a∈V(cf.Proposition1.6).

A C[T]-module V,equipped with a linear map V?V→C[λ]?V,a?b→aλb, satisfying(4.3)is called a(C[T]-)conformal algebra(cf.[K,Sec.2.7]).On the other hand,with respect to the.-product,V is a(C[T]-)di?erential algebra(i.e., an algebra with a derivation T)with a unit|0 .

We summarize the above discussion in the following lemma.

Lemma4.1.Giving a state–?eld correspondence on a pointed vector space(V,|0 ) is equivalent to providing V with a structure of a C[T]-conformal algebra and a structure of a C[T]-di?erential algebra with a unit|0 .

Next,we translate the n-th product axioms in terms of theλ-and.-products.

12BOJKO BAKALOV AND VICTOR G.KAC

Lemma4.2.Let(V,|0 )be a pointed vector space and let Y be a state–?eld corre-spondence.Fix a,b,c∈V.Then the collection of n-th product identities(3.5)for n≥0implies

(aλb)λ+μc=aλ(bμc)?bμ(aλc),

(4.5)

aλ(b.c)=(aλb).c+b.(aλc)+ λ0(aλb)μc dμ.

(4.6)

The(?1)-st product identity Y(a(?1)b,z)c= Y(a,z)(?1)Y(b,z) c implies

(a.b)λc=(e T?λa).(bλc)+(e T?λb).(aλc)+ λ0bμ(aλ?μc)dμ,

(4.7)

(a.b).c?a.(b.c)= T0dλa .(bλc)+ T0dλb .(aλc).

(4.8)

Proof.The collection of n-th product identities(3.5)for n≥0is equivalent to: Y(aλb,w)c=Res z eλ(z?w)[Y(a,z),Y(b,w)]c=e?λw[aλ,Y(b,w)]c. (4.9)

Taking Res w e(λ+μ)w,we obtain(4.5).Taking Res w w?1,and using that e?λw w?1= w?1+ ?λ0eμw dμ,we get:

(aλb).c=aλ(b.c)?b.(aλc)+ ?λ0[aλ,bμ]c dμ.

This,together with(4.5),implies(4.6)(after the substitutionμ′=λ+μin the integral).

Due to(1.3)and(4.4),the(?1)-st product identity is equivalent to:

Y(a.b,z)c=(e zT a).(Y(b,z)c)+Y(b,z)(a??z c)(z?1).

Taking Res z eλz and using integration by parts,we get:

(a.b)λc=Res z(e T?λeλz a).(Y(b,z)c)+Res z Y(b,z)(aλ??z c)(eλz z?1)

=(e T?λa).(bλc)+Res z Y(b,z)(aλ??z c) z?1+ λ0eμz dμ

=(e T?λa).(bλc)+Res z e?z?λY(b,z) (aλc)z?1+ λ0Res z Y(b,z)(aλ?μc)eμz dμ, which implies(4.7),using translation invariance.

On the other hand,taking Res z z?1of the(?1)-st product identity,we get: (a.b).c=Res z z?1 Y(a,z)+Y(b,z)+c+Y(a,z)+Y(b,z)?c+Y(b,z)+Y(a,z)?c

=a.(b.c)+Res z z?1((e zT?1)a).(Y(b,z)?c)+Res z z?1((e zT?1)b).(Y(a,z)?c) =a.(b.c)+Res z T0eλz dλa .(Y(b,z)?c)+Res z T0eλz dλb .(Y(a,z)?c), which proves(4.8).

Remark4.3.Equation(4.9)is equivalent to the commutator formula

[a(m),b(n)]c=

j=0 m j (a(j)b)(m+n?j)c,m∈Z+,n∈Z.

(4.10)

FIELD ALGEBRAS13 Equation(4.5)is equivalent to the same formula for m,n∈Z+.

Identity(4.5)is called the(left)Jacobi identity(cf.[K,Sec.2.7]).A conformal algebra satisfying this identity for all a,b,c∈V is called a(left)Leibniz conformal algebra.Equation(4.6)is known as the“non-commutative”Wick formula(cf.[K, (3.3.12)]),while(4.8)is called the quasi-associativity formula(cf.[K,(4.8.5)]).

Notice that the right-hand side of(4.8)is symmetric with respect to a and b, hence(4.8)implies

(4.11)

a.(

b.c)?b.(a.c)=(a.b?b.a).

c.

An algebra satisfying(4.11)for all a,b,c∈V is called left-symmetric.For such an algebra a.b?b.a is a Lie algebra bracket.

Theorem4.4.Giving a strong?eld algebra structure on a pointed vector space (V,|0 )is the same as providing V with a structure of a Leibniz C[T]-conformal algebra and a structure of a C[T]-di?erential algebra with a unit|0 ,satisfying (4.6)–(4.8).

Proof.If(V,|0 ,Y)is a strong?eld algebra,then by the above discussion we can de?ne aλ-product and a.-product on V satisfying all the requirements.

Conversely,given aλ-product and a.-product,we de?ne a state–?eld corre-spondence Y by(4.4).In the proof of Lemma4.2,we have seen that equations (4.5)–(4.8)are equivalent to the identities

Res z Y(a(n)b,z)?Y(a,z)(n)Y(b,z) F=0,a,b∈V,n≥?1,F=eλz or z?1.

Using the translation invariance of Y and integration by parts,we see that this identity holds also with F replaced with?z F.Hence it holds for all F=z k,k<0. For F=eλz,taking coe?cients at powers ofλshows that it is satis?ed also for F=z k,k≥0.This implies the n-th product axioms for n≥?1.

Replacing a with T a and using translation invariance shows that,for n=0, the n-th product axiom implies the(n?1)-st product axiom.Therefore,the n-th product axioms hold for all n∈Z,and(V,|0 ,Y)is a strong?eld algebra.

Corollary4.5.Let(V,|0 )be a pointed vector space,let Y be a state–?eld cor-respondence,and de?ne theλ-product and.-product on V as above.Let U be a T-invariant subspace of V.

(a)If(4.5)and(4.6)hold for all a,b∈U,c∈V,then Y(a,z)(n)Y(b,z)=

Y(a(n)b,z)for all n≥0,a,b∈U.

(b)If(4.7)and(4.8)hold for all a,b∈U,c∈V,then Y(a,z)(n)Y(b,z)=

Y(a(n)b,z)for all n<0,a,b∈U.

5.Tensor Algebra over a Leibniz Conformal Algebra

In this section,we are going to use the notation of the previous one.Let R be a Leibniz C[T]-conformal algebra(see Section4),and let V=T(R)= ∞m=0R?m be the tensor algebra over R(viewed as a vector space over C).We extend the action of T on R to V so that it is a derivation of the tensor product.Let|0 be the element1∈C≡R?0?V.

14BOJKO BAKALOV AND VICTOR G.KAC

Theorem5.1.There exists a unique structure of a?eld algebra on V=T(R) such that the restriction of theλ-product to R×R coincides with theλ-product of R,the restriction of the.-product to R×V coincides with the tensor product,and the n-th product axioms(3.1)hold for a∈R,b∈V.

For a∈R,C∈V,we set a.C=a?C.Next,we de?ne aλC inductively using the Wick formula(4.6)and starting from aλ|0 =0:

aλ(c?C)=(aλc)?C+c?(aλC)+ λ0(aλc)μC dμ,a,c∈R,C∈V.

(5.1)

Lemma5.2.Formula(5.1)de?nes a representation of the Leibniz conformal al-gebra R on V,whose restriction to R coincides with the adjoint representation. Proof.For a,b∈R,C∈V,we have to check the translation invariance:

(T a)λC=?λaλC,aλ(T C)=(λ+T)(aλC),

and the Jacobi identity(cf.(4.5)):

(aλb)λ+μC=aλ(bμC)?bμ(aλC).

Note that these identities hold when C∈R by de?nition.Assuming that they hold for a?xed C∈V and all a,b∈R,we are going to prove them for a,b,c?C for any a,b,c∈R.

The translation invariance is immediate.The Jacobi identity reduces to checking that

λ+μ

0 (aλb)λ+μc νC dν

= λ0 aλ(bμc) νC dν+ λ0(aλc)ν(bμC)dν+ μ0aλ (bμc)νC dν

? μ0 bμ(aλc) νC dν? μ0(bμc)ν(aλC)dν? λ0bμ (aλc)νC dν.

By Jacobi identity,the third and?fth terms in the right-hand side combine to μ0 aλ(bμc) λ+νC dν= λ+μλ aλ(bμc) νC dν,which together with the?rst term gives λ+μ0 aλ(bμc) νC dν.Similarly,the sum of the other three terms is ? λ+μ0 bμ(aλc) νC dν.Now the statement follows from the Jacobi identity for a,b,c.

We have de?ned aλ-product aλC and a.-product a.C=a?C for a∈R,C∈V, satisfying the translation invariance,the Jacobi identity,and the Wick formula.By the results of Section4(see Lemma4.1and Corollary4.5),this gives a linear map from R to g?f(V),a→Y(a,z),such that[T,Y(a,z)]=Y(T a,z)=?z Y(a,z)and Y(a,z)(n)Y(b,z)=Y(a(n)b,z)for all n≥0,a,b∈R.

Next,we set Y(|0 ,z)=I V,and de?ne?elds Y(A,z)∈g?f(V)inductively by the formula

Y(a?A,z)=:Y(a,z)Y(A,z):for a∈R,A∈V.

(5.2)

It is easy to check that Y is a state–?eld correspondence.

Lemma5.3.We have Y(a(n)B,z)=Y(a,z)(n)Y(B,z)for all a∈R,B∈V, n∈Z.In particular,all pairs(Y(a,z),Y(B,z))are weakly local for a∈R,B∈V.

FIELD ALGEBRAS15 Proof.By de?nition,Y(a(?1)B,z)=Y(a?B,z)=:Y(a,z)Y(B,z):,so the state-ment holds for n=?1and hence for all n<0.Thus,it su?ces to show that

Y(aλB,z)=[Y(a,z)λY(B,z)]=Res x eλ(x?z)[Y(a,x),Y(B,z)],a∈R,B∈V. We have already proved this when B∈R,while for B=|0 both sides are trivial.

Assuming the above identity is true for B∈V and all a∈R,consider it for b?B where b∈https://www.sodocs.net/doc/dc4992445.html,ing this assumption and(5.1),we?nd:

Y aλ(b?B),z = Y(a,z)λY(b,z) (?1)Y(B,z)+Y(b,z)(?1) Y(a,z)λY(B,z)

+ λ0 Y(a,z)λY(b,z) μY(B,z) dμ.

But the right-hand side is equal to[Y(a,z)λ(Y(b,z)(?1)Y(B,z))]by the Wick for-mula for arbitrary?elds[K,(3.3.12)].This completes the proof.

Lemma5.4.All pairs(Y(A,z),Y op(B,z))are local for A,B∈V.

Proof.It follows from Lemma5.3and Corollary3.2that all pairs(Y(a,z),Y op(B,z)) are local for a∈R,B∈V.By induction,assume that(Y(A,z),Y op(B,z)) is local,and consider the pair(Y(a?A,z),Y op(B,z))for a∈R.Recall that by de?nition Y(a?A,z)=Y(a,z)(?1)Y(A,z).Since,by Lemma5.3,the pair (Y(a,z),Y(A,z))is weakly local,we can apply Lemma2.5a to conclude that the pair(Y(a,z)(?1)Y(A,z),Y op(B,z))is local.

Now Theorem3.7a and Lemma5.4imply that the so de?ned(V,|0 ,Y)is a ?eld algebra.Uniqueness is clear by construction.This completes the proof of Theorem5.1.

The?eld algebra V=T(R)has the following universality property.Let W be a strong?eld algebra and let f:R→W be a homomorphism of conformal algebras (i.e.,f(a(n)b)=f(a)(n)f(b)and f(T a)=T f(a)for all a,b∈R,n∈Z+).Then there is a unique homomorphism of?eld algebras f:V→W such that f= f?i, where i is the embedding of R in V.

Remark5.5.Although all pairs(Y(a,z),Y(B,z))are weakly local for a∈R,B∈V (see Lemma5.3),in general(V,|0 ,Y)is not a strong?eld algebra.This follows from the results of Section7.

Remark5.6.A conformal algebra R is called graded if there is a diagonalizable operator H∈End R satisfying[H,T]=T and(1.7)for n∈Z+and homogeneous a∈R.Then the tensor algebra T(R)over a Leibniz conformal algebra R is graded if R is graded.Moreover,T(R)is Z+-graded if R is Z+-graded.

6.Field Algebras and Vertex Algebras

Recall that a vertex algebra(V,|0 ,Y)is a pointed vector space(V,|0 )together with a local state–?eld correspondence Y(see e.g.[K]).In particular,any vertex algebra is a strong?eld algebra with Y=Y op(see Theorem3.7).

Example6.1.The state–?eld correspondence given by Example1.2is a vertex algebra i?the algebra V is commutative and associative.

16BOJKO BAKALOV AND VICTOR G.KAC

Example 6.2.Let V be a subspace of g ?f (U )which contains I U ,is ?x -invariant and is closed under all n -th products.Then the state–?eld correspondence de?ned in Example 2.3gives V a structure of a vertex algebra i?V is a local collection.This follows from [K,Proposition 3.2].

Theorem 6.3.A vertex algebra is the same as a ?eld algebra (V,|0 ,Y )for which Y =Y op .

Proof.Let (V,|0 ,Y )be a ?eld algebra with Y =Y op .Recall that (see Proposi-tion 3.1)the associativity relation (3.2)is equivalent to:

(z ?w )N [Y (a,z ),Y op (c,w )]b =0.

By Lemma 2.8,this relation also holds after replacing b with T b .Replacing b with e uT b in (3.2)and using Proposition 1.7b,we get:

(z ?w )N i z,u i w,u Y (Y (a,z ?u )b,u ?w )c

=(z ?w )N i z,w i w,u Y (a,z ?w )Y (b,u ?w )c.

(6.1)There exists P ∈Z +such that b (k )c =0for k ≥P .Hence the right-hand side of (6.1)multiplied by (u ?w )P contains only non-negative powers of u ?w ,hence only non-negative powers of w .Therefore,multiplying (6.1)by (u ?w )P and putting w =0gives:

Y (a,z )Y (b,u )c =z ?N u ?P (z ?w )N (u ?w )P i z,u i w,u Y (Y (a,z ?u )b,u ?w )c w =0

.(6.2)Using again Proposition 1.7b and Y =Y op ,we compute:

i z,u Y (a,z ?u )b =Y (e ?uT a,z )b =e zT Y (b,?z )e ?uT a =i z,u e (z ?u )T Y (b,u ?z )a.Therefore,

i z,u i w,u Y (Y (a,z ?u )b,u ?w )=i z,u i w,z Y (Y (b,u ?z )a,z ?w ).

(6.3)We may take N =P ?0in (6.2).Comparing (6.2)and (6.3),we see that Y (b,u )Y (a,z )c is given by the right-hand side of (6.2)where i z,u is replaced with i u,z .Hence,if K ∈Z +is such that a (k )b =0for k ≥K ,we have

(z ?u )K Y (a,z )Y (b,u )c =(z ?u )K Y (b,u )Y (a,z )c ,

which is locality of Y (a,z )and Y (b,z ).This proves that (V,|0 ,Y )is a vertex algebra.

Corollary 6.4.A state–?eld correspondence is local i?it is local on every vector.Remark 6.5.It follows from (6.2),(6.3)that for N ?0and any n ∈Z :

Y (a,z )Y (b,u )i z,u (z ?u )n ?Y (b,u )Y (a,z )i u,z (z ?u )n

=z ?N u ?N (z ?w )N (u ?w )N i w,u (i z,u ?i u,z )(z ?u )n

×Y (Y (a,z ?u )b,u ?w )c w =0

.(6.4)Using (1.2),we ?nd (i z,u ?i u,z )(z ?u )n Y (a,z ?u )b =

∞ m =0

(a (m +n )b )?m u δ(z ?u )/m !.

FIELD ALGEBRAS 17

Note that this sum is ?nite,because a (m )b =0for m ?0.Therefore,for large enough N ,we can put w =0in (6.4),and obtain:

Y (a,z )Y (b,u )i z,u (z ?u )n ?Y (b,u )Y (a,z )i u,z (z ?u )n

=∞ m =0

Y (a (m +n )b,u )?m u δ(z ?u )/m !.

(6.5)The collection of these identities for n ∈Z is equivalent to the Borcherds identity

[K,(4.8.1)].In particular,taking Res z of both sides of (6.5),we obtain the n -th product identity (3.1).

Remark 6.6.Note that formula (6.1)holds for any ?eld algebra (V,|0 ,Y ).Let K ∈Z +be such that a (k )b =0for k ≥K .After multiplication by (z ?u )K ,the left-hand side of (6.1)contains only non-negative powers of z ?u ,hence only non-negative powers of z .Putting z =0and replacing u with ?u ,we obtain:

i w,u Y (Y (a,u )b,?u ?w )c

=(?w )?N u ?K (z ?w )N (z +u )K i z,w i w,u Y (a,z ?w )Y (b,?u ?w )c z =0.

Since in both sides ?u ?w is expanded in non-negative powers of u ,we can replace w with ?u ?w and use Taylor’s formula to get:

Y (Y (a,u )b,w )c

=i z,u +w i w,u (u +w )?N u ?K (z +u +w )N (z +u )K

×Y (a,z +(u +w ))Y (b,w )c z =0.(6.6)Remark 6.7.Let V be a ?eld algebra.Given a,b ∈V ,denote by ab the C -span of all elements a (n )b (n ∈Z ).More generally,if A and B are two subspaces of V ,we de?ne AB as the C -span of all elements a (n )b (n ∈Z ,a ∈A ,b ∈B ).Formulas (6.2)and (6.6)imply associativity of this product:

(ab )c =a (bc ),a,b,c ∈V .

(6.7)This property allows one to apply many arguments of the theory of associative algebras to ?eld algebras.Notice that (6.7)holds also for a,b ∈V and c ∈M ,where M is a module over the ?eld algebra V (see De?nition 8.1below).

Remark 6.8.Letting a ?b =C [T ](ab ),we still have associativity of the product ?for an arbitrary ?eld algebra V .In addition,|0 ?a =a ?|0 =C [T ]a for all a ∈V .If V is a vertex algebra,the product ?is also commutative.These remarks allow one to use arguments of commutative algebra to study vertex algebras (this will be pursued in a future publication).

For the opposite state–?eld correspondence Y op ,we can consider

a op λ

b =Res z e λz Y op (a,z )b,a op .b =Res z z ?1Y op (a,z )b,

and express these operations in terms of the λ-product and .-product de?ned for Y .We have:

a op λ

b =Res z e λz e zT Y (b,?z )a =?Res z e

?(λ+T )z Y (b,z )a =?b ?λ?T a,(6.8)

18BOJKO BAKALOV AND VICTOR G.KAC

and,using z ?1e zT =z ?1?

?T 0e ?λz d λ,a op .b =Res z z ?1e zT Y (b,?z )a =b.a + ?T 0

b λa d λ.

(6.9)If (V,|0 ,Y )is a vertex algebra,then the λ-product satis?es the skewsymmetry relation a λb =?b ?λ?T a for a,b ∈V .A conformal algebra satisfying the Ja-cobi identity and the skewsymmetry relation is called a Lie conformal algebra [K,Sec.2.7].Theorem 6.9.Giving a vertex algebra structure on a pointed vector space (V,|0 )is the same as providing V with the structures of a Lie C [T ]-conformal algebra and a left-symmetric C [T ]-di?erential algebra with a unit |0 (see (4.11)),satisfying the Wick formula (4.6)and

a.b ?

b.a = 0

?T

a λ

b d λ,a,b ∈V.

(6.10)Proof.By Lemma 4.1,these data de?ne a state–?eld correspondence Y .By (6.8)and (6.9),we have Y =Y op ;therefore,we only need to prove that V is a ?eld algebra (see Theorem 6.3).Due to Theorem 4.4,it remains to show that (4.7)and (4.8)follow from equations (4.5),(4.6),(4.11),(6.10)and the skewsymmetry of the λ-product.

First,using (6.10),rewrite the Wick formula (4.6)as

a λ(b.c )=c.(a λ

b )+b.(a λ

c )+ λ

?T

(a λb )μc d μ.

Then replace λwith ?λin this equation and apply e T ?λto both https://www.sodocs.net/doc/dc4992445.html,ing Taylor’s formula and the fact that T is a derivation,we obtain:

a ?λ?T (b.c )=(e T ?λc ).(a ?λ?T

b )+(e T ?λb ).(a ?λ?T

c )+

?λ?T

?T e T ?λ (a ?λb )μc d μ.

Using (4.3)and the skewsymmetry of the λ-product,the last term can be rewritten as follows:

?λ?T ?T

e T ?λ (a ?λb )μc d μ=? ?λ?T ?T e T ?λ c ?μ?T (a ?λb ) d μ

= λ

0e T ?λ

c μ(a ?λb )

d μ= λ

0c μ(a ?λ+μ?T b )d μ.

Applying the skewsymmetry relation again,we obtain (4.7).

In order to prove (4.8),we manipulate its left-hand side using twice (6.10):

(a.b ).c ?a.(b.c )=c.(a.b )?a.(c.b )+ 0

?T d λ(a.b )λc ?a. 0

?T d λb λc

.

FIELD ALGEBRAS 19

Due to (4.11)and (6.10),the ?rst two terms in the right-hand side of this equation give:

c.(a.b )?a.(c.b )=?(a.c ?c.a ).b =? 0?T d λa λc .b

=?b. 0

?T d λa λc +

0?T d μb μ

0?T d λa λc .

The double integral is equal to: 0?T d μb μ 0?T d λa λc = 0

?T d μ 0?T ?μd λb μ(a λc )

= 0?T d μ μ?T d λb μ(a λ?μc )= 0?T d λ 0

λ

d μb μ(a λ?μc ).

(6.11)Therefore,applying (4.7),which we already proved,we obtain:

(a.b ).c ?a.(b.c )= 0?T d λ(e T ?λa ).(b λc )?a. 0?T d λb λc

+ 0

?T d λ(e T ?λb ).(a λc )?b. 0

?T d λa λc .

From here,it is easy to deduce (4.8).

This completes the proof of the theorem.

Proposition 6.10.If R is a Leibniz C [T ]-conformal algebra with a λ-product a λb ,then the bracket [a,b ]= 0

?T

a λ

b d λ,a,b ∈R

(6.12)de?nes the structure of a Leibniz C [T ]-di?erential algebra on R .If R is a Lie conformal algebra,then this is a Lie bracket.

Proof.It is clear that T is a derivation of the bracket (6.12),because it is a deriva-tion of the λ-product a λb .From relation (6.11),we have:

[b,[a,c ]]= 0?T d λ 0λd μb μ(a λ?μc )= 0?T d λ 0

λ

d μb λ?μ(a μc ),

and similarly,

[a,[b,c ]]=

0?T d λ 0λ

d μa μ(b λ?μc ).On th

e other hand,

[[a,b ],c ]= 0?T d λ

0?T d μa μb

λc = 0

?T d λ

d μ(a μb )λc.

Therefore,the Jacobi identity for the bracket (6.12)follows from the Jacobi identity (4.5)for the λ-product.Finally,it is easy to see that [a,b ]=?[b,a ]if a λb =?b ?λ?T a .

20BOJKO BAKALOV AND VICTOR G.KAC

Remark6.11.Let R be a Lie C[T]-conformal algebra with aλ-product aλb= m≥0λm a(m)b/m!.To it one can associate a Lie algebra Lie R as follows(see [K,Sec.2.7]).As a vector space,Lie R=R[t,t?1]/(T+?t)R[t,t?1].The Lie bracket in Lie R is given by the formula(cf.(4.10)):

[a[m],b[n]]=

j=0 m j (a(j)b)[m+n?j],a,b∈R,m,n∈Z,

where a[m]is the image of at m in Lie R.Note that T acts on Lie R as a derivation: T(a[m])=?m a[m?1].

Let(Lie R)?(respectively,(Lie R)+)be the C-span of all a[m]for a∈R,m≥0 (respectively,m<0).These are subalgebras of Lie R.Let R Lie be R considered as a Lie algebra with respect to the bracket(6.12).Then the map a→a[?1]is an isomorphism of C[T]-di?erential Lie algebras R Lie~??→(Lie R)+.

Theorem6.12.Let R be a Lie C[T]-conformal algebra with aλ-product aλb.Let R Lie be R considered as a Lie algebra with respect to the bracket(6.12),and let V= U(R Lie)be its universal enveloping algebra.Then there exists a unique structure of a vertex algebra on V such that the restriction of theλ-product to R Lie×R Lie coincides with theλ-product of R and the restriction of the.-product to R Lie×V coincides with the product in U(R Lie).

Proof.We have R Lie?(Lie R)+(see Remark6.11).By construction,

V?U((Lie R)+)?Ind Lie R

(Lie R)?

C

(where(Lie R)?acts trivially on C)coincides with the universal vertex algebra associated to Lie R(see[K,Sec.4.7]).

The vertex algebra V from Theorem6.12has the following universality prop-erty.Let W be another vertex algebra and let f:R→W be a homomorphism of conformal algebras(i.e.,f(a(n)b)=f(a)(n)f(b)and f(T a)=T f(a)for all a,b∈R, n∈Z+).Then there is a unique homomorphism of vertex algebras f:V→W such that f= f?i,where i is the embedding of R in V.For this reason,V is called the universal vertex algebra associated to R and is denoted by U(R).

Remark6.13.U(R)is the quotient of the?eld algebra T(R),constructed in Sec-tion5,by the two-sided ideal generated by a?b?b?a? 0?T aλb dλ(a,b∈R).A di?erent proof of Theorem6.12was given in[GMS].

7.Strong Field Algebras and Vertex Algebras

Let(V,|0 ,Y)be a strong?eld algebra.We are going to use theλ-product and .-product on V de?ned in Section4.

SCI收录期刊投稿全过程英文信件模板一览

SCI收录期刊投稿全过程英文信件模板一览 一、最初投稿Cover letter Dear Editors: We would like to submit the enclosed manuscript entitled “Paper Title”, which we wish to be considered for publication in “Journal Name”. No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication. I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed. In this work, we evaluated …… (简要介绍一下论文的创新性). I hope this paper is suitable for “Journal Name”. The following is a list of possible reviewers for your consideration: 1) Name A E-mail: ××××@×××× 2) Name B E-mail: ××××@×××× We deeply appreciate your consideration of our manuscript, and we look forward to receiving comments from the reviewers. If you have any queries, please don’t hesitate to contact me at the address below. Thank you and best regards. Yours sincerely, ×××××× Corresponding author: Name: ××× E-mail: ××××@×××× 二、催稿信 Dear Prof. ×××: Sorry for disturbing you. I am not sure if it is the right time to contact you to inquire about the status of my submitted manuscript titled “Paper Title”. (ID: 文章稿号), although the status of “With Editor” has been lasting for more than two months, since submitted to journal three months ago. I am just wondering that my manuscript has been sent to reviewers or not? I would be greatly appreciated if you could spend some of your time check the status for us. I am very pleased to hear from you on the reviewer’s comments. Thank you very much for your consideration. Best regards! Yours sincerely, ×××××× Corresponding author: Name: ×××

点面结合 场面描写

点面结合场面描写 场面描写指的是在某一特定时间和特定地点范围内以人物活动为中心的生活画面的描写。场面描写一般由“人”、“事”、“境”构成,它是叙事性作品的基本构成单位,是刻画人物、展开情节、表现主题的主要手段。常见的有劳动场面、战斗场面、运动场面以及各种会议场面等。那么如何写好场面呢?点面结合是进行场面描写最好的选择。 所谓“点”,指的是最能显示人事景物场面的形象状态特征的详细描写;所谓“面”,指的是对人事景物场面的叙述或概括性描写。点面结合就是“点”的详细描写和“面”的叙述或概括性描写的有机结合。点面结合一般有以下三种形式: 一、视角笔触横向化,就是要把观察的视线向横的方向展开。要看到整个场面在同一个时间里所发生的事,不能只集中看一点。如下面一段场面描写: “王励勤,加油,中国队,雄起!”随着观众此起彼伏的呐喊声,中国对韩国的世界杯乒乓赛决赛被王励勤与韩国柳承敏的几个大力远拉推向高潮,场内翻滚着一股热浪,坐在电视机前的我们,也目不转睛地看着电视,我、爸爸、哥哥戴着头巾,挥舞着乒乓拍,用力捶着茶几当起场外拉拉队来,王励勤又胜一局,在加油声中一路高歌,这时,对方柳承敏奋起反击,几个短摆,直线,反手对拉,利用王励勤侧身过多,迎头赶上,观众的叫声更响亮了,震耳欲聋,把电视机前的观众的心深深地震撼了。我们一家也急得直跺脚,索性脱掉衣服

在此挥舞,终于,王励勤不负众望,在掌声与欢呼中尽显他的王者风范,一声大叫,一个手势,又使他崛起赢得了比赛,我们也抑制不住兴奋之情,相互拥抱起来。 二、一面带多点,就是要有整体的概括,又有重点的具体描写。一般采用先总述再分述的方法。这也是我们进行场面描写时所常用的手法。如《十里长街送总理》第一段的描写。 天灰蒙蒙的,又阴又冷。长安街两旁的人行道上挤满了男女老少。路那样长,人那样多,向东望不见头,向西望不见尾。人们臂上都缠着黑纱,胸前都佩着白花,眼睛都望着周总理的灵车将要开来的方向。一位满头银发的老奶奶拄着拐杖,背靠着一棵洋槐树,焦急而又耐心地等待着。一对青年夫妇,丈夫抱着小女儿,妻子领着六七岁的儿子,他们挤下了人行道,探着身子张望。一群泪痕满面的红领巾,相互扶着肩,踮着脚望着,望着…… 作者为了把送总理的人很多这一特点写出来就用了“点面结合”的写法:其中“长安街两旁的人行道上挤满了男女老少。路那样长,人那样多,向东望不见头,向西望不见尾。”这是面的描写;“一位满头银发的老奶奶拄着拐杖,背靠着一棵洋槐树,焦急而又耐心地等待着。一对青年夫妇,丈夫抱着小女儿,妻子领着六七岁的儿子,他们挤下了人行道,探着身子张望。一群泪痕满面的红领巾,相互扶着肩,踮着脚望着,望着……”这是点的描写。这样,我们一读文章就能够深刻感受到来总理的自发群众是那样的多。细读之后,我们更能够感受到人民对总理的爱戴和无限悲思。难能可贵的是在这一段中,没有

sci期刊论文格式要求-sci论文格式要求

sci期刊论文格式要求:sci论文格式要求 SCI是目前国际上最具权威性的、用于基础研究和应用基础研究成果的重要评价体系。它的论文格式是怎么样的呢?下面是小编精心推荐的一些sci期刊论文格式要求,希望你能有所感触! sci期刊论文格式要求 1、题目:应简洁、明确、有概括性,字数不宜超过20个字。 2、摘要:要有高度的概括力,语言精练、明确,中文摘要约100200字; 3、关键词:从论文标题或正文中挑选3~5个最能表达主要内容的词作为关键词。 4、目录:写出目录,标明页码。 5、正文: 论文正文字数一般应在3000字以上。 论文正文:包括前言、本论、结论三个部分。 前言(引言)是论文的开头部分,主要说明论文写作的目的、现实意义、对所研究问题的认识,并提出论文的中心论点等。前言要写得简明扼要,篇幅不要太长。 本论是论文的主体,包括研究内容与方法、实验材料、实验结果与分析(讨论)等。在本部分要运用各方面的研究方法和实验结果,分析问题,论证观点,尽量反映出自己的科研能力和学术水平。 结论是论文的收尾部分,是围绕本论所作的结束语。其基本的要点就是总结全文,加深

题意。 6、谢辞:简述自己通过做论文的体会,并应对指导教师和协助完成论文的有关人员表示谢意。 7、参考文献:在论文末尾要列出在论文中参考过的专著、论文及其他资料,所列参考文献应按文中参考或引证的先后顺序排列。 8、注释:在论文写作过程中,有些问题需要在正文之外加以阐述和说明。 9、附录:对于一些不宜放在正文中,但有参考价值的内容,可编入附录中。 关于sci的论文范文 美国《SCI》收录温州医学院论文分析 【摘要】目的: 了解温州医学院(以下简称:温医)作者论文被SCI收录情况。方法:根据SCI-E数据库检索统计1998-2007年温医作者SCI产文情况。结果:1998-2007年温医作者发表SCI论文总数为304篇,其中2005-2007年为247篇,占81.2%;论文类型:论著259篇,占85.2%;发表论文最多的学科是眼科,为32篇,占10.5%,其次生物化学为30篇,占9.8%;发表SCI论文最多的前3位作者为瞿佳、吕帆、李校;被引用的有143篇,被引率为47%,总被引用次数为732次,篇均被引频次为2.41次,单篇论文引用频率最高的为77次,其中引用10次以上的论文17篇,占总被引文章数的11.9%。结论:2005年以来SCI收录温医论文逐年增加,收录的论文涉及各个学科,其中以眼科和生物化学为主。 【关键词】SCI;论文;温州医学院 Abstract: Objective: To know the situation of the authors’papers of Wenzhou Medical College embodied by AmericaSCI. Methods: Papers written by authors of Wenzhou Medical College embodied by AmericaSCIpublished from 1998 to 2007 were counted based upon the researching of SCI-E database. Results: The total number of papers published by authors of Wenzhou Medical College embodied by AmericaSCIin 1998-2007 was 304 pieces of paper. Of which,247 papers were published in 2005-2007,accounting for 81.2%. Paper type: 259 papers were treatise,accounting for 85.2%. The discipline published the most papers was ophthalmology. They published 32 papers,accounting for 10.5%;the second one was biochemistry,they published 30 ones,accounting for 9.8%. The top authors of publishing SCI papers were QU Jia,LV Fan

中外文核心期刊介绍及投稿导引

中外文核心期刊介绍 及投稿导引

主要内容
一、中外文核心期刊介绍
?理性关注核心期刊 ?什么是核心期刊 ?中国科研评价常用核心期刊
二、核心期刊投稿导引
?有的放矢投稿 ?快乐轻松写稿 ?投稿审稿流程
2

为什么关注核心期刊 ?为了毕业拿学位 EI、SCI收录,高影响因子期刊 ?为了提职称 同行或相关 ?希望研究结果能与同行共享 人员能看到 ?希望研究成果能为社会带来效益 您的文章
核心期刊
3

理性关注中外文核心期刊 核心期刊是期刊中学术水平较高的刊物,是 我国学术评价体系的一个重要组成部分, 但不是全部!
在信息爆炸的年代 ? 在核心期刊中检索并阅读相关文献,可以用最少 的精力获得最大的信息量。 ? 将科研成果发表在核心期刊上可以增加自己的研 究成果被同行看到的几率。
4

什么是核心期刊 ?布拉德福定律——核心期刊的起源
对某一主题而言,如果将科学期刊按其刊载某个学科领域 的论文数量以递减顺序排列起来,就可以在所有这些期刊 中区分出载文量最多的‘核心’区和包含着与核心区同等数 量论文的随后几个区,这时核心区和后继各区中所含的期 刊数成 1:a:a2 …… 的关系( a>1 )。
常用表述:某个学科领域的论文绝大多数集 中在少数专业期刊之中。
5

什么是核心期刊 ?加菲尔德文献集中定律:测定核心期刊的另一种方式
1963年Science Citation index (简称SCI)出版问世,加 菲尔德以SCI数据为统计对象,在1971年对2000种期刊中的10 万参考文献统计后发现,24%的引用频次高的文章出自25种期 刊,50%的引用频次较高的文章出自152种期刊,75%出自767 种期刊,而其余25%的被引文章分布在数量更多的期刊中。
通常表述:论文被引用频次高的期刊也有一个比 较集中的核心区域和一个比较分散的相关区 域。
6

场景描绘案例分享

简单、好用的“场景描绘” ——帮助顾客实现梦想 让我们先来看看买花的女孩是如何使用“场景描绘”的: 卖花女孩向一位路过的小伙子兜售鲜花。小伙子说,你的鲜花太贵了。卖花女郎说,送给女孩子最好的礼物 就是鲜花,而且要在众人面前送给她最漂亮的花!假如你捧着一束花去见她,你的女朋友会是什么样呢?我想她一 定会含情脉脉地看着你,脸上洋溢着幸福的笑容,会在众人羡慕的眼光中给你一个最热烈的拥抱的。听到这里, 小伙子立即掏出钱包了。 场景描绘,就是运用一些生动形象的语言给客户描绘一幅使用产品后带来好处的图像,激起顾客对这幅图的向往,有效刺激顾客的购买欲望。试想一下,听到这样一段有诱惑力的话,哪个顾客能不动心呢? 正是因为: 1.带感情色彩的语句能够在顾客心里产生震荡 人是有感情的,富有感情色彩的语句和平淡的语句在顾客心里产生的震荡肯定不一样。每一位顾客都有其特定的经历、经验,从而形成对事物的独特见解、看法和态度。富有感情色彩的语句可以使顾客把你的介绍和他的亲身经验结合起来,从而使得你所推销的物品(饰品)与顾客对于未来的期待和向往合二为一,这样顾客对此物品(饰品)就会产生一种依赖感和依存感,购买的兴趣就会大增。 2.让顾客感到自己的选择合情合理 顾客的购买行为主要是由感情力量引起的,他们仍然会感到有必要为自己的行为找到合理的依据,可能是为了必要时给别人一个解释,或者仅是为了让自己满意地感到自己是理智的。因此,你必须充分意识到这一点,并随时准备提供理由,满足顾客的需要,证明他们的行动是合理的。其实,对于你所提出的理由,顾客也绝对不会认真追问,因为这些论点是他们所需要的。这时候的顾客非常容易相信你的话,如果你能够用一些带有感情色彩的话来说服顾客的话,多半都会成功。 珠宝消费不同于售卖花朵,不是凭一两句场景描绘就足以让顾客“冲动”而消费的,因为它是高额消费品。正是因为它价格昂贵,所以受到重视,顾客对拥有首饰后能达到其目的的期望值会更高!因此“场景描绘”在整个销售过程中起着举足轻重“煸情”的作用,“心动而产生行动”。 那在我们的销售工作中,又要如何来进行呢? 描述时方法有两种: 1.第一种:你可以尝试使用以下三个句型,激发顾客想象力。 “您有没有感觉到/您看……?” “当……时候……” “……像……一样” 例如:“您有没有感觉到这件衣服的布料很柔软,也很保暖?当天冷的时候穿上它,肯定会像睡在羽绒被子里一样舒服和暖和的。” “您看,这枚戒指款式非常简洁,两股线条缠绕于指尖,您有没有感觉到它非常适合您的手型?不炫丽但看上去很舒服,就像您的先生一样,一直会温柔体贴的陪伴在身边,很幸福啊。” 2、第二种:向顾客直接描绘未来 例如:在商品快介绍结束时,推销员向准备为女儿买钢琴的母亲说:“我想用不了多久,您女儿一定能在学校的表演厅里为大家演奏曲子了。” 翡翠饰品快介绍完毕时,“您老公佩戴着这枚挂件,出门在外一定会平平安安的,摸到翡翠就会想起您的叮咛的。”

科技期刊在线投稿及审稿系统

随着多年以来计算机网络技术的发展普及,网络已然逐渐成为我们获取信息和文化资源的主要方法。在过去的二十年来,互联网技术已经被应用于各个领域,成为当今应用中使用最广泛,最具影响力的技术之一。设计科技期刊在线投稿及审稿系统可以方便地管理稿件的信息。本文将介绍科技期刊在线投稿及审稿系统的设计方案与实现过程。 科技期刊在线投稿和审稿系统分为前端系统和后端数据库两个部分。后端数据库主要包括:一般用户信息,专家信息,稿件的信息,基本费用的信息,编辑,首席信息和评级信息。前端系统模块有三种不同类型的用户:作者、专家、主编。作者首次使用系统需要注册一个账号,通过成功注册的该账号登录系统后才能进行修改个人密码、上传个人稿件和管理个人已上传的稿件等操作。专家登录系统后可以修改注册时填写的个人信息以及对稿件进行审核,并可以对该稿件给出审核意见。主编登录系统后可以修个自己的登录密码以及对注册用户信息、专家信息、稿件信息、稿费信息进行管理。 科技期刊在线投稿及审稿系统的开发对提升期刊专家和主编工作的效率和工作的质量有着重要的意义。充分利用计网(计算机网络)功能,可以实现投稿及审稿工作的全程非人工管理,将作者、审稿专家和主编从繁琐的手工书写、邮寄、批改,返回结果操作中释放出来,使投、审稿工作更加规范化和现代化。 关键词:期刊在线投稿及审稿系统;JSP技术;MySQL数据库管理系统;Tomcat应用服务器

Over the years, With the development and popularization of computer technology and the Internet,the network has gradually become the main way for us to obtain information and cultural resources. In the past two decades, Internet technology has been used in various fields, and has become one of the most widely used and influential technologies in today's application. The online contribution and review system of sci-tech periodicals can easily manage the basic information of manuscripts. This paper will introduce the design and implementation process of the system. The online contribution and evaluation system of sci-tech periodicals is divided into two parts: front-end system and back-end database system. The back-end database mainly includes: general user information, expert information, manuscript information, basic cost information, editing, chief information and rating information. There are three different types of users in the front-end system module: author, expert, editor-in-chief. For the first time, the author needs to register an account through the successfully registered account before he can modify his personal password, upload personal manuscripts and manage personal uploaded manuscripts. After experts log on to the system, they can modify their personal information and review their manuscripts. Give the audit opinion. After the editor-in-chief logs in the system, he can fix his own login password and manage the registered user information, expert information, manuscript information and manuscript fee information. The development of online contribution and review system of sci-tech periodicals based on JSP is of great significance to improve the efficiency and quality of periodical editing. Making full use of the computer network function (hereinafter referred to as the network function) can realize the non-manual management of the whole process of contribution and review, and release the author, the reviewer and the editor-in-chief from the tedious manual operation. Make the submission and examination work more standardized and modern. Key words:Online Journal Submission and Review System;MySQL Database Management System;Tomcat

初中作文指导:细节描写案例【精品】

细节描写案例 教学目标:A知识技能目标:明确细节描写的内涵,学习细节描写的方法,初步学会运用细节描写。 B思想情感目标:养成良好的观察习惯,积累对百态人生的体验,做一个生活的有心人。 教学重点:细节描写的方法及运用 教学难点:通过细节描写表达真情实感 教学手段:多媒体课件 教学方法:体验法、启发式(教学过程中可以讨论法和演练法为主,充分发挥学生学习的主动性,培养学生的探究能力。)教学过程: 一、导入 著名文艺批评家兰色姆指出,使文学成为文学的东西不在于文学作品的框架结构、中心逻辑,而在于作品的细节描写,只有细节才属于艺术,也只有细节的表现力最强。其实,现在世界中细节的力量又何尝可以小觑?每增加一厘米的倾斜,都有可能导致比萨斜塔的倾覆;使人疲惫的不是远方的高山,而是鞋子里的一粒沙子。 今天,我们就一起来探讨一下细节描写对作文的影响作用。(多媒体课件展示课题,学生齐读课题:《一枝一叶总关情——作文指导之细节描写》。) 二、明确定义 细节描写是指抓住生活中的细微而又具体的典型情节,加以生动细致的描绘,它具体渗透在对人物、景物或场面描写之中。它是最生动、最有表现力的手法,是作者精心的设置和安排,不能随意取代。一篇文章,恰到好处地运用细节描写,能起到烘托环境气氛、刻画人物性格和揭示主题思想的作用。(多媒体课件展示课文片段 三、回顾我们课文中的细节描写 (多媒体课件展示课文片段) 1.(阿Q)要画圆圈了,那手捏着笔只是抖,于是那人将纸铺在地上,阿Q伏下去,使尽平生的力画圆圈。他生怕被人笑话,立志要画得圆,但这可恶的笔不但很沉重,并且不听话,刚刚一抖一抖的几乎合缝,却又向外一耸,成了瓜子模样了。”(鲁迅《阿Q正

SCI投稿状态

SCI投稿状态自己查-投稿术语名词解释 1. Submitted to journal 刚提交的状态 2. Manuscript received by Editorial Office 就是你的文章到了编辑手里了,证明投稿成功。 3. with editor 如果在投稿的时候没有要求选择编辑,就先到主编那,主编会分派给别的编辑。这当中就会有另两个状态: 3.1 A waiting Editor Assignment指派责任编辑 Editor assignment 是把你的文章分给另一个编辑处理了。 3.2 technical check in progress 检查你的文章符合不符合期刊的投稿要求 3.3 Editor Declined Invitation 如果编辑接收处理了就会邀请审稿人了。 4. 随后也会有2种状态 4.1 Decision Letter Being Prepared 就是编辑没找到审稿人就自己决定了,那根据一般经验,对学生来说估计就会挂了1)英文太差,编辑让修改。2)内容太差,要拒了。除非大牛们直接被接收。 4.2 Review(s) invited 找到审稿人了,就开始审稿 5 Under review 这应该是一个漫长的等待。当然前面各步骤也可能很慢的,要看编辑的处理情况。如果被邀请审稿人不想审,就会decline,编辑会重新邀请别的审稿人。 6. Required Reviews Completed 审稿人的意见已经上传,审稿结束,等待编辑决定 7. Evaluating Recommendation 评估审稿人的意见,随后你将受到编辑给你的decision 8. Minor revision/ Major revision 这个时候可以稍微庆祝一下了,问题不大了,因为有修改就有可能。具体怎么改就不多说了,谦虚谨慎是不可少的。 9. Revision submitted to journal 又开始了一个循环 10 Accepted 恭喜了 11. Transfer copyright form 签版权协议 12. Uncorrected proof 等待你校对样稿 13. In press, corrected proof 文章在印刷中,且该清样已经过作者校对 14. Manuscript sent to production 排版 15. in production 出版中

计算机科学期刊介绍--各种杂志投稿方式与评价

计算机科学期刊介绍--各种杂志投稿方式与评价 一、计算机科学期刊[/B]介绍 计算机科学的publication最大特点在于:极度重视会议[/B],而期刊[/B]则通常只用来做republication。大部分期刊[/B]文章都是会议[/B]论文的扩展版,首发就在期刊[/B]上的相对较少。也正因为如此,计算机期刊[/B]的影响因子都低到惊人的程度,顶级刊物往往也只有1到2左右----被引的通常都是会议[/B]版论文,而不是很久以后才出版的期刊[/B]版。因此,要讨论计算机科学的publication,首先必须强调的一点是totally forget about IF (IF指影响因子)。 另外一点要强调的是,计算机科学的绝大多数期刊[/B]和大部分的“好”会议[/B]都规模非常有限。很多好的期刊[/B]一期只登十来篇甚至三四篇论文,有的还是季刊或双月刊。很多好的会议[/B]每年只录用三四十篇甚至二十篇左右的论文。所以,当你发现计算机的每个领域都有好几种顶级刊物和好几个顶级会议[/B],不必惊讶。 整个计算机科学中最好的期刊[/B]为Journal of the ACM(JACM)。此刊物为ACM的官方学刊,受到最广泛的尊敬。但由于该刊宣称它只刊登那些对计算机科学有长远影响的论文,因此其不可避免地具有理论歧视(theory bias)。事实上确实如此:尽管JACM征稿范围包括了计算机的绝大部分领域,然而其刊登的论文大部分都是算法、复杂度、图论、组合数学等纯粹理论的东西,其它领域的论文要想进入则难如登天。 另外一份在计算机科学领域有重大影响的刊物为Communications of the ACM (CACM)。从某种意义上来说,CACM比JACM要像Nature/Science很多。JACM上登的全是长篇大论,满纸的定义、定理和证明,别说一般读者没法看,就连很相近的领域的专家都未必能看懂。而CACM则是magazine,既登高水平的学术论文和综述,也登各种科普性质的文章和新闻。即便是论文,CACM也要求文章必须通俗易懂,不追求数学上的严格证明,而追求易于理解的直觉描述。在十几二十年前,CACM的文章几乎都是经典。但最近几年,由于CACM 进一步通俗化,其学术质量稍有下降。 IEEE Transaction on Computers为IEEE在计算机方面最好的刊物。但由于IEEE的特点,其更注重computer engineering而非computer science。换句话说,IEEE Transaction on Computers主要登载systems, architecture, hardware等领域的东西,尽管它的范围已经比大部分刊物要广泛。 就刊物的质量而言,ACM Transactions系列总体来讲都高于IEEE Transactions系列,不过也不可一概而论。大部分ACM Transactions都是本领域最好的刊物或最好的刊物之一。大部分IEEE Transactions都是本领域很好的刊物,但也有最好的或者一般的。非ACM/IEEE 的刊物中,也有好的甚至最好的。例如,SIAM Journal on Computing被认为是理论方面最好的期刊[/B]之一。 计算机科学方面的会议[/B]论文事实上起着比刊物论文更大的作用。大部分会议[/B]都是每年一次,偶尔也有隔年一次的。正规的会议[/B]论文需要经过2-4个甚至更多个审稿人的双向或单向匿名评审,并且所有被接收的论文会被结集正式出版。 大部分ACM的会议[/B]都是本领域顶级的或很好的会议[/B]。大部分IEEE的会议[/B]都是本领域很好的会议[/B],但也有顶级的或者一般的。 会议[/B]的档次通常可以通过论文录用率表现出来。顶级会议[/B]通常在20%左右或更低,有时能达到10%左右。我所知道的最低的录用率为7%。很好的会议[/B]通常在30%左右。达到40%以上时,会议[/B]的名声就很一般了。60%以上的会议[/B]通常很难受到尊敬。 但也有例外。大名鼎鼎的STOC(ACM Symposium on Theory of Computing)录用率就达到30%以上,但它毫无疑问是理论方面最好的会议[/B]。造成这样的情形,主要是因为理论方

记叙文细节描写篇分析

高三语文写作指导与训练系列二编号 GS YW XZ--002 记叙文写作之细节描写篇导学案 编写:徐明生审定:张明玺时间:2014-08-26 班级:组别:组名:姓名: 知识与技能:理解细节描写的概念及分类,掌握细节描写的方法,学会运用细节描写表现人物细腻的情感,为文章增色。 过程与方法:1.创设情境,赏析富有情趣的细节描写。2.例文引导,掌握细节描写的方法。 3.写作训练,将细节描写融入作文中。 情感态度与价值观:培养观察力、想象力,引导学生发现生活中的细节之美,形成热爱生活、积极的人生态度。 教学方法:揣摩、讨论、归纳、操练 课前准备:1.了解细节描写的相关知识点; 2.回顾所学过的记叙文(小说),初步体会细节描写的作用。 课时:2课时 导学过程:一、导语众所周知,记叙文的灵魂在于细节描写。同学们在作文里缺少的不是把某件事写完整的能力,而是缺少细节捕捉、描写的能力。这些缺失细节描写的文章读来生涩呆板,缺乏感染力。而高考作文评分在发展等级项中有明确的规定,即:“材料丰富,形象丰满,意境深远”,这“形象丰满”很大程度上依赖于考生对细节的描摹与刻画。细节描写之重要,不仅可以在“生动”上“出彩”加分,而且可以使作品形象丰满,使整个文章升格。所以作文缺少感人的细节实在是一个制约记叙文成绩提高的瓶颈。 一沙一世界,一叶一菩提。其实,纷繁的生活,是由无数小的细节构成的。细节虽小,却是美的源泉,情的聚焦。生活中的细节之美,看在眼里,便是风景;握在掌心,便是花朵;拥进怀中,便是温暖;写在笔端,便是精彩。今天这节课,我们来一起用双眼发现细节,用心灵感悟细节,用文字展现细节,让我们作文中的人与事如生活中一般于细处见情,微处见妙! 二、了解概念、分类,感受作用、魅力 (1)什么是细节描写? 细节描写就是某些细小的能很好表现中心的环节和情节,如对人物的一个动作、一种表情、一句话等以及事物发展的具体环节、环境中的细小物体,通过准确、生动、细致的描绘,使读者“如见其人”“如睹其物”。 细节描写如电影中的特写镜头,具体渗透在对人物、景物或场面描写之中。它将我们身边

sci论文写作插图一般要求

sci论文写作插图一般要求 sci论文写作的图表一般是重要研究结果的展示,插图质量的好坏往往也会直接影响着科研论文的发表。越来越多的研究者认识到了高质量的SCI论文插图对于sci论文写作发表的重要性。 以下将介绍一般SCI杂志对插图的各种要求,并说明如何在实际科研工作中做好原始数据和图片的采集工作,希望能从根本上帮助科研工作者减少这类问题的发生。 1. SCI论文插图一般要求: 1)尺寸符合杂志社的要求(宽度8.3~17.6厘米,高度一般不超过20厘米); 2)字体符合杂志社的要求(Times New Roman/Arial); 3)同类型文字的字号保持一致(Font size ≥8 pt,字体太小印刷版看不清楚); 4)线条粗细保持一致(Line weight; 0.25~1 pt); 5)准确、清楚、有条理的图片标记,插图上所有元素对位整齐; 6)插图内容应占据整张插图的90%以上空间,四周不能留太多空白区域; 7)颜色模式符合杂志社的要求(RGB, CMYK); 8)图片分辨率超过杂志社的最低要求(彩图≥300 dpi;线条图≥1000 dpi;灰度图≥600 dpi;组合图≥500 dpi); 9)格式符合规范(位图,TIFF,矢量图,PDF/EPS); 10)大小合适(每张插图最好不超过10M,推荐保存为TIFF格式并选择LZW无损压缩模式); 2.sci论文写作如何获取高质量的原始素材? 大家在收集原始数据和图片时,应特别注意获取高质量的原始文件,并长期保存。 (1)照相机拍摄类照片 拍摄时应注意如下要点:1)注意摄入参照物。如需比较拍摄物尺寸大小的,应辅以

SCI投稿全攻略

SCI投稿过程总结、投稿状态解析、拒稿后处理对策及接受后期相关问答综合荟萃目录 (重点是一、二、四、五、六): (一)投稿前准备工作和需要注意的事项、投稿过程相关经验总结 (二)SCI期刊投稿各种状态详解及实例综合(学习各种投稿状态+投稿经历总结) (三)问答综合篇(是否催稿、如何撤稿、一稿两投及学术不端相关内容等) (四)如何处理审稿意见(回复意见、补实验、润色、重整数据、作图及调整、申辩及其他)(五)Reject 或者Reject and resubmit后的对策和处理 (六)稿件接受后期相关问题(作者信息、地址版权、单行本、彩图费、版面费、如何汇款、清样相关等) (七)进阶篇(如何选投SCI杂志、各专业方向期刊选择、SCI写作经验) (一)投稿前准备工作和需要注意的事项、投稿过程相关经验总结 1)第一作者和通信作者的区别: 通信作者(Corresponding author)通常是实际统筹处理投稿和承担答复审稿意见等工作的主导者,也常是稿件所涉及研究工作的负责人。 通信作者的姓名多位列于论文作者名单的最后(使用符号来标识说明是Corresponding author),但其贡献不亚于论文的第一作者。 通讯作者往往指课题的总负责人,负责与编辑部的一切通信联系和接受读者的咨询等。 文章的成果是属于通讯作者的,说明思路是通讯作者的,而不是第一作者。 第一作者仅代表是你做的,且是最主要的参与者! 通信作者标注名称:Corresponding author,To whom correspondence should be addressed,或The person to whom inquiries regarding the paper should be addressed.若两个以上的作者在地位上是相同的,可以采取“共同第一作者”(joint first author)的署名方式,并说明These authors contributed equally to the work。 2)作者地址的标署: 尽可能地给出详细通讯地址,邮政编码。有二位或多位作者,则每一不同的地址应按之中出现的先后顺序列出,并以相应上标符号的形式列出与相应作者的关系。 如果第一作者不是通讯作者,作者应该按期刊的相关规定表达,并提前告诉编辑。期刊大部分以星号(*)、脚注或者致谢形式标注通讯联系人。 3)挑选审稿人的几个途径: 很多SCI杂志都需要作者自己提出该篇论文的和您研究领域相关的审稿人,比较常见的是三名左右,也有的杂志要求5-8人。 介绍几个方法: ①利用SCI、SSCI、A&HCI、ISTP检索和您研究相关的科学家; ②文章中的参考文献; ③相关期刊编委或学术会议的主席、委员; ④以前发表的类似文章的审稿人; ⑤询问比较熟识的一些专业人士; ⑥交叉审稿,邀请以前的作者; ⑦若是团队序贯研究,斟酌考虑自建期刊审稿人专家库。 PS: 如果有熟悉的同领域的专家,可以推荐一两位为宜(若你全部推荐熟人也无可厚非,但编辑基本不会全部考虑,可能对你还有点特殊“眼色”了)。考虑推荐自己文章的参考文献作者较为常用,当然,如果你是负面引用的话,务必慎重了。

中国核心期刊投稿简介及要求

在中国核心期刊中发表论文,如今是很多领域展开工作和提升自己的时候都需要重视的一个问题,如果可以在这个过程中有着不错的表现,自然就是可以让自己提高和进步的。事实上,在这些年的发展中,我们必须要搞清楚的一个事实就是,中国核心期刊的发表都是有着自己的要求的,对于稿件的类型,选题和格式等都是有着明确的需求的,大家是要注意的。 任何时候,我们想要提高自己在中国核心期刊中发表的成功率的话,这些地方都是要事先注意的。很多人之所以投稿失败就是因为之前没有搞清楚这些方面的东西,自然就是无法获得提高的。投稿联系编辑扣10883103而今天,我们就举一个简单的例子来让大家知道中国核心期刊到底是有什么投稿的要求。可以肯定的是,各个期刊肯定在细节上是有一定区别的,但是大体上是差距不大的。 我们就以《管理世界》为例子,这是我国金融和管理领域一个非常重要的期刊,在很多的大学中都是绝对优秀的核心期刊,不少人在投稿的时候都会选择它。它对于投稿的文章要求主要围绕中国的现实经济管理问题展开分析,力求资料详实、行文规范。文章应对已有文献进行学理性梳理,并在正文中明确说明其对本学科的学术贡献。我们不鼓励简单以中国数据重复国外已有研究的文章。 不少期刊对于研究方法是有一定的倾向性的,如果是一定创赢意

义的研究方法肯定是会得到一定倾向的。比如《管理世界》欢迎任何形式的研究方法,包括定性与定量研究方法,但所有的研究方法都须遵循严谨的学术规范。如果是实验研究,我们一般不接受以本科生做为实验对象的研究,除非该实验样本相对于其研究问题是合适的。 最后,我们就要谈到在给《管理世界》投稿的时候有关于数据的问题了,很多人是不喜欢公开自己的实验数据的。但是在这里编辑部在审稿过程中,若向作者索取文章所用的研究数据,作者应提供这些数据。为了促进知识传播,在文章发表后,若读者向作者索取文章所用的研究数据,作者也应配合提供这些数据;若作者出于合理原因无法提供研究数据时,应详细说明获取这些数据的途径。国内大部分的期刊对于数据都是这样规定的。

《科技风》期刊介绍 投稿要求

《科技风》期刊介绍 刊名:科技风 Technology Wind 主办:河北省科技咨询服务中心 周期:半月 出版地:河北省石家庄市 语种:中文; 开本:16开 ISSN:1671-7341 CN:13-1322/N 历史沿革: 现用刊名:科技风 创刊时间:1988 科技风杂志社1988年3月批准成立,经过几十年发展,现设有北京、石家庄两个运营中心,业务范围涵盖期刊出版、网络传播、公关咨询、讲座培训、出版服务等多个领域,聚合了大批相关领域的专家学者及一线专业工作人员,特别是在教育科技、科学学科和汽车科技领域有着广泛影响。 《科技风》是经国家新闻出版总署批准,河北省科学技术协会主管、河北省科技咨询服务中心主办的国内公开发行的大型综合类科技期刊。国内统一刊号:CN13-1322/N;国际标准刊号:ISSN1671-7341,邮发代号:18-38,半月刊标准大16K开本。 《科技风》的办刊宗旨: 《科技风》杂志以“把脉科技创新、引领发展风尚”为办刊宗旨,以第一线科技、教育工作者为读者对象,追踪科研、教育工作动态,反映科教工作者的言论和呼声,关注基层科教工作者在其理论领域内的探索、创新实践活动和学术研究成果,致力于发展研究改革开放,经济发展过程中出现的各种科学技术、经济问题以及教育教学等方面具有较高水平的理论文章,是广大科教工作者阐释观点和理论的平台。 《科技风》主要栏目: 《科技风》杂志主要栏目涵盖:行业高新技术、工程和建筑科学、IT和信息技术等各类应用科技理论及教育教学、党建政工、文化艺术等方面的内容。 《科技风》征稿范围: 本刊向全国广大教育工作者、财务人员、工程、农业技术人员、医务工作者、科技管理人员、经济管理人员社会科学工作者等人士征集符合以上内容等各方面的创新工作经验及学术论文。在本刊发表的论文均符合中、高级职称的评审要求。

作文技巧:围绕中心写场面

作文技巧:围绕中心写场面 场面描写,那么什么是场面呢?场面一般是指集体活动中某一段时间内,呈现在人们面前的情景。就如同我们在话剧中看到的某一场、某一幕一样。生活中,有哪些场面是我们参加过或者看到过的呢?下面举一些例子供参考。劳动场面。我们在学校经常参加校办工厂、农场或服务性劳动,劳动中大家的热、干劲以及劳动中出现的好人好事是很多的。我们也经常去工厂、农村、商店参观、问,看到叔叔阿姨们是怎样劳动的,这些就是劳动场面。游戏场面。同学们在家里或校园里经常在一起做游戏,体育活动时老师也组织大家开展各种活动。这些游戏场面常常非常活跃,非常有趣,也是同学们比较熟悉的娱乐场面。在我们丰富多彩的课余生活中,娱乐活动是一项很重要的内容。文艺演出,联欢活动,观看影视,都有许多生动的场面值得我们记叙下来。典礼场面。每学期开学时,我们都要举行开学典礼。你所在的城市、村镇、有什么新的建筑落成,也常常举行典礼仪式。这些典礼的场面有意义、有教育性,应该认真观察,记录下这一值得纪念的时刻。运动场面。运动会上,运动员你追我赶,奋力拼搏,争取创造优异成绩;观众们为运动员鼓掌加油,欢呼助兴,这些激动人心的场面是令人难忘的。生活场面。在日常生活中有许多场面也是很有特点的。例如购买货物,家人团聚,乘车坐船,求医治病,盖房修墙,;,布置居室等等,都具有浓厚的生活气息。会议场面。班会、队会、校会,;,和英雄模范人物会见,参加各种庆祝会,纪念会,那会议的场面也往往给我们留下深采刻的印象。总之,练习场面描写,可以提高苛我们

观察认识能力、记叙描写能力。我们在记叙场面时要注意以下三点.1.认真观察。比如进行会议场面的描写,要仔细观察会场的环境:主席台在哪儿,会标是怎吞样写的,台上是怎样布置的,有哪些人在台上就座,他们的神态、动作、讲话、|内容是怎样的;会场有多大,有多少人参加,会场的气氛怎样;会议的议程怎样,怎样开始,哪时达到高潮,会议是怎样结束的等等,都应如实具体观察,认真积累写作的材料。2.抓住重点3;把人和事物写具体。场面描写常常离不开人物和事物,这是场面描写的主要内;:l容。人物的外貌、神情、语言、动作、心理,事物的起因、经过、高潮和结果,以及物体的i l形状、颜色、变化、活动情况等等,都应细致具体地加以描述,使读者如身临其境,这才是成功的场面描写。当然,场面描写也应围绕一个中心,不要想到什么写什么,这样就不集中了,这也是要注意的问题。

相关主题