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Supermanifolds and Supergroups

Mathematics and Its Applications

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M. HAZEWINKEL

Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 570

Supermanifoldsand SupergroupsBasic Theory

Gijs M. TuynmanUniversité de Lille I„Lille, France

KLUWER ACADEMIC PUBLISHERSNEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: 1-4020-2297-2

Print ISBN: 1-4020-2296-4

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Table of Contents

Preface

I.

II.

III.

commutative linear algebra

1.2.3.4.5.6.7.8.

commutative rings and(Multi-) linear mapsDirect sums, free and quotientsTensor productsExterior powersAlgebras and derivationsIdentificationsIsomorphisms

Linear algebra of free graded

1. Our kind of algebra2.3.4.5.6.

Free gradedConstructions of free gradedLinear maps and matricesThe graded trace and the graded determinantThe body of a free graded

Smooth functions and

1.2.3.4.5.

Topology and smooth functionsThe structure of smooth functionsDerivatives and the inverse function theorem

Constructions of

ix

1

27

141724283446

55

565862687480

91

92103112124130

v

vi Table of Contents

IV. Bundles

1.2.3.4.5.6.7.8.

V.

1.2.3.4.5.6.7.8.

VI.

1.2.3.4.5.6.7.8.9.

VII.

1.2.3.4.5.6.7.8.9.

10.

Fiber bundlesConstructions of fiber bundlesVector bundles and sectionsConstructions of vector bundlesOperations on sections and on vector bundlesThe pull-back of a sectionMetrics on vector bundlesBatchelor’s theorem

The tangent space

Derivations and the tangent bundleThe tangent map and some standard applicationsAdvanced properties of the tangent mapIntegration of vector fieldsCommuting flowsFrobenius’ theoremThe exterior derivativede Rham cohomology

141

142151156163173181187196

203

204211219228236242247260

265

266277286292298306311315323

335

336341350354360366374382389395

groups

groups and their algebrasThe exponential mapConvergence and the exponential of matricesSubgroups and subalgebrasHomogeneousPseudo effective actionsCovering spaces and simply connected groupsInvariant vector fields and formsLie’s third theorem

Connections

More about vector valued formsEhresmann connections and FVF connectionsConnections on principal fiber bundlesThe exterior covariant derivative and curvatureFVF connections on associated fiber bundlesThe covariant derivativeMore on covariant derivativesForms with values in a vector bundleThe covariant derivative revisitedPrincipal fiber bundles versus vector bundles

Table of Contents vii

References

Index of Notation

Index

405

409

411

Preface

This book is a self contained introduction to super differential geometry, intended forgraduate students in mathematics and theoretical physics and other people who want tolearn the basics about supermanifolds. It is self contained in that it only requires standardundergraduate knowledge. However, some knowledge of ordinary (non super) differentialgeometry will make this text much easier to read.

Various versions of super differential geometry exist, some of which are equivalentand some of which are not. The version presented here is equivalent to those that aremost widely used: the supermanifolds of DeWitt and the sheaf theoretic approachto supermanifolds of Kostant and Leites. The approach taken here is based on an indexfree formalism using a graded commutative ring containing the usual real numbers aswell as so called anticommuting numbers. Starting with a non-standard definition of adifferentiable function, valid in the real case, in the complex case and in the super case,the theory is developed as if it were ordinary differential geometry. It is shown that mostconstructions and theorems in ordinary differential geometry have a natural generalizationto the super context. Moreover, even the proofs bear more than a superficial resemblanceto their counter parts in ordinary differential geometry. The (equivalent) sheaf-theoreticapproach to supermanifolds makes it manifest that the theory is “independent” of the choiceof but at the same time it hides the more geometric nature of the theory. The approachpresented here can be seen as a theory with a parameter Choosing givesordinary differential geometry, choosing gives super differential geometry,choosing gives the theory of complex manifolds, etc. Of course, in each of thesecases some small but usually superficial changes have to be made, and not all resultsremain true in all cases (e.g., Batchelor’s theorem, which uses partitions of unity, is notvalid for (super) complex manifolds). But the main body of the results is not affected bythe choice of

In Chapter I the general theory of graded linear algebra (graded by an arbitrary abeliangroup) is outlined. This plays the same role in super differential geometry as does linearalgebra in ordinary differential geometry and as does commutative algebra in algebraicgeometry. Since the basic ring is (in principle) not commutative, we have to make adistinction between left and right linear maps. The isomorphism between these two kinds

ix

x Preface

of maps is given by the operator which will later be identified with (super) transpositionof matrices.

In chapter II we specialize to linear algebra and we impose some restrictionson the ring The canonical example of that satisfies all conditions (the ones imposedin chapter II and also other ones imposed later on) is the exterior algebra of an infinitedimensional real vector space: Some of the more important points ofthis chapter are the following. In section 2 it is shown that any (finitely generated) freegraded admits a well defined graded dimension. In section 4 the relationbetween matrices and linear maps is explained. The reader should really pay attentionhere, because there are three different natural ways to associate a matrix to a linear map,and these three different ways imply different ways how to multiply a matrix by an elementof (so as to be compatible with the multiplication of the corresponding linear map bythe element of It is here that we see most clearly the role of the transposition operatorintroduced in chapter I to relate left and right linear maps. In section 5 the graded trace isdefined for any linear map (and thus for any matrix, not only the even ones), as well as itsintegrated version for even maps, the graded determinant or Berezinian. Finally in section6 the body map B is introduced, which provides an “isomorphism” between equivalenceclasses of free graded and direct sums of two real vector spaces. It is this bodymap which gives the link between standard linear algebra and linear algebra.

The heart of this book lies in chapter III, in which the notion of a supermanifold isdeveloped based on a non-standard definition of differentiable functions. The key idea isexpressed by the following formula, valid for functions of class on convex domainsin

If we write this as it is obvious that is of classif and only if the function is of class Moreover, if a with this property exists, itis also easy to see that the derivative of is given by If we nownote that the formula does not involve quotients norlimits, we can apply the same definition to super functions, for which there generally donot exist quotients (because of nilpotent elements in nor does the natural topology(the DeWitt topology) admit unique limits (being non Hausdorff). Based on this idea,smooth functions on super domains with even coordinates and odd coordinatesare defined. It is shown, using the body map B defined in chapter II, that these smoothfunctions are in bijection with ordinary smooth real-valued functions of real variables,multiplied by antisymmetric polynomials in variables. This result is usually taken as thedefinition of smooth super functions; here it is a consequence of a more general definition,a definition which applies as well to ordinary functions as to super functions. The last twosections of chapter III are devoted to copying the standard definition of manifolds in termsof charts and transition functions to the case in which the transition functions are supersmooth functions.

In chapter IV the general theory of fiber and vector bundles is developed. The first twosections deal with general fiber bundles and how to construct new ones out of given ones.The next two sections deal with vector bundles and how to generalize the construction

Preface xi

of new to the setting of vector bundles. In section 5 the behavior of theoperation of taking sections under the various operations one can perform onis considered. In section 6 the exterior algebra of a (dual) bundle is discussed in moredetail, as well as the pull-back of sections. The main purpose of sections 5 and 6 isto provide a rigorous justification for operations everybody performs without thinkingtwice. In section 7 one finds a proof of part of the Serre-Swann theorem that the module ofsections of a vector bundle is a finitely generated projective module over the ring of smoothfunctions on the base manifold. The proof of this result needs the notion of a metric on afree graded a notion whose definition is subtly different from what one wouldexpect. These results are not used elsewhere, but they are needed to complete the proofsof statements given in section 5. The last section in chapter IV on Batchelor’s theoremmerits ample attention. This theorem says that any supermanifold is “isomorphic” to anordinary vector bundle over an ordinary manifold, or, stated differently, for any smoothsupermanifold there exists an atlas in which the transition functions are of the specialform: even coordinates depend on even coordinates only, and odd coordinates depend ina linear way on odd coordinates. The proof is “constructive” in that it provides an explicitalgorithm to compute such an atlas given an arbitrary atlas. The quotes are needed becausethis algorithm requires a partition of unity on the underlying ordinary manifold.

Chapter V treats the standard machinery of differential geometry. In section 1 thetangent bundle is defined and it is shown that sections of it, called vector fields, areequivalent to derivations of the ring of smooth functions. In section 2 the tangent map isdefined, which in turn gives rise to the notions of immersion and embedding. In section3 the relationship between the tangent map and the derivative of a map are studied inmore detail. It turns out that in the super case this is in general not a 1–1 correspondence.Generalizing the notion of the derivative of an function to vector bundle valuedfunctions, a necessary and sufficient condition is given for a vector bundle to be trivial asa vector bundle. Here one also can find an example of a vector bundle which is trivial asfiber bundle, but not as vector bundle. Sections 4 and 5 then concentrate on the notion ofthe flow of a vector field and the well known proposition that two vector fields commuteif and only if their flows commute. For odd vector fields this amounts to saying that anodd vector field is integrable if and only if its auto commutator is zero. Section 6 treatsFrobenius’ theorem on integrability of subbundles of the tangent bundle, the notion ofintegral manifolds and the existence of leaves for a foliation. In section 7 the calculus of(exterior differential) is given, including the definition of the Lie derivative andits relation with the flow of a vector field. Finally in section 8 an elementary proof is givenof the fact that the de Rham cohomology of a supermanifold is the same as that of theunderlying ordinary manifold (its body).

Chapter VI treats the basic facts about super Lie groups and their associated superLie algebras. In section 1 one finds the basic definition of a super Lie group and theconstruction of the associated super Lie algebra. The exponential map from the superLie algebra to the super Lie group is defined in section 2. There one also finds the proofthat it intertwines a homomorphism of super Lie groups and its induced morphism on theassociated super Lie algebras. Section 3 is rather technical and computes the derivative ofthe exponential map. Section 4 deals with the relationship between Lie subgroups and Lie

xii Preface

subalgebras, whereas section 5 treats homogeneous supermanifolds. Section 6 is againtechnical and proves that any smooth action can be transformed into a pseudo effectiveaction. The last section gives a geometric proof that to each finite dimensional super Liealgebra corresponds a super Lie group.

Chapter VII is more advanced and discusses the general concept of a connection on afiber bundle. Sections 1 and 8 are technical and provide the necessary theory of vectorvalued and vector bundle valued differential forms. In section 2 the general concept ofan Ehresmann connection is introduced, as well as the more restrictive notion of FVFconnection, which is an Ehresmann connection determined by the fundamental vectorfields of the structure group on the typical fiber. FVF connections have nice properties:they are defined on any fiber bundle, they include the standard examples of connectionssuch as the (principal) connection on a principal fiber bundle and linear connections onvector bundles, and they always allow parallel transport. In sections 3 and 4 the particularcase of an FVF connection on a principal fiber bundle is studied, which includes thedescription by a connection 1-form, the exterior covariant derivative and a discussionabout the curvature 2-form. In section 5 it is shown that any FVF connection can beseen as induced by an FVF connection on a principal fiber bundle. Sections 6 and 7treat the notion of a covariant derivative on a vector bundle and prove that it is equivalentto an FVF connection. It includes the proof that the covariant derivative measures howfar away a (local) section is from being horizontal. In sections 9 and 10 the covariantderivative on a vector bundle is generalized to vector bundle valued differential formsand it is shown how the exterior covariant derivative (on a principal fiber bundle), theordinary exterior derivative of differential forms and the generalized covariant derivative(on a vector bundle) are intimately related.

This book is written in a logical order, meaning that a proof of a statement never usesfuture results and meaning that related subjects are put together. This is certainly not themost pedagogical way to present the subject, but it avoids the risk of circular arguments.As a consequence, the novice reader should not read this book in a linear order. For a firstreading, one can easily skip sections 7 and 8 of chapter I. From chapter IV one shouldcertainly read sections 1–3, but coming back for sections 4–6 (and then only superficially)just before starting to read section 6 of chapter V. The reader who already has a workingknowledge of ordinary manifold theory need not read all sections with the same attentionand at a first reading (s)he can even skip chapter IV completely.

One final word on terminology: in this introduction I have systematically used theadjective super. On the other hand, in the main text I never use this adjective, but ratherthe prefix The reason to do so is that one should regard this theory not as opposedto ordinary differential geometry (super versus non-super), but more as a theory with aparameter indicating over which ring it is developed.

In preparing chapters I–VI I have relied heavily on the first three chapters of F. Warner’sclassic “Foundations of Differentiable Manifolds and Lie Groups,”, while chapter VII isbased on H. Pijls’ review article “The Yang-Mills equations.” Other sources of inspirationhave been the first volume of M. Spivak’s “A Comprehensive Introduction to DifferentialGeometry” and “Les Tenseurs” of L. Schwartz. During the years it took me to write this

Preface xiii

book, I have benefitted from the hospitality of the following three institutions: MSRI(Berkeley, USA), CPT (Marseille, France) and LNCC (Rio de Janeiro, Brazil). Specialthanks are due to P. Bongaarts for some excellent suggestions concerning chapter I andto V. Thilliez who helped me with [III. 1.12]. Finally, I am convinced I got the idea for[IV.7.3] from a paper by S. Sternberg, but I can no longer find the source.

Lille, january 2004

Chapter I

commutative linear algebra

Linear algebra is concerned with the study of vector spaces over the real numbers (or moregenerally over a field) and linear maps. A standard course on linear algebra more or lessstarts with the introduction of the concept of a basis. Immediately afterwards one usuallyrestricts attention to finite dimensional vector spaces. Next on the list is the concept ofa subspace and with that notion one derives some elementary properties of linear maps.Then one introduces bilinear maps, with a scalar product as the most important example.This gives rise to the notions of orthogonal basis, orthogonal linear map, and orthogonalsubspaces, eventually followed by a classification of quadrics. More advanced coursestreat the notions of multilinear maps, tensor products, and exterior powers. Algebras, andin particular Lie algebras, are usually treated separately.

Besides analysis, these concepts in linear algebra form the basis of differential geom-etry. One could even say that differential geometry is the interplay between analysis andlinear algebra. Algebraic geometry is closely related to differential geometry, but hardlyrelies on analysis; it is mainly concerned with algebraic structures. For that it needsa generalization of linear algebra in which a vector space over a field is replaced by amodule over a commutative ring with unit. Commutative algebra is the theory which playsin algebraic geometry the same role as linear algebra does in differential geometry. Incommutative algebra the notion of basis more or less disappears, but subspaces, tensorproducts, and exterior powers can still be defined.

In supergeometry one replaces the field of real numbers, not by a commutative ring,but by a graded commutative ring. Since such a ring is not commutative, commutativealgebra does not apply. In this context, graded means i.e., the ring and allmodules are a direct sum of two subspaces, the even and odd parts. In this first chapterwe look at an even more general situation. We denote by an arbitrary abelian groupand we denote by an arbitrary commutative ring with unit (i.e., a ringwhich splits as a direct sum of subspaces indexed by and satisfying conditions how thesesubspaces commute). We will show that all concepts of linear algebra that are important

1

2 Chapter I. commutative linear algebra

for differential geometry can be generalized to commutative linear algebra, i.e.,to the theory of

1. COMMUTATIVE RINGS AND

In this first section we give the definitions of the principal objects of this book:commutative algebras and are a special kindof a fact that will greatly facilitate constructions of new

one of which is discussed in this section: the submodule.

1.1 Definition. Given abelian groups and H, a mapis called if for all and for all we have:

1.2 Definition. Let G be an abelian group and let be a family of subgroups. Onewrites if and only if for each element there exist uniqueonly finitely many of them non-zero, such that it is called the (unique)decomposition of into

1.3 Definitions. Let be a ring. A left module over the ring (or a left isan abelian group E equipped with a map that is bi-additive and satisfies

This map is called left multiplication by elements of and (as is usual) we will omit thesymbol if no confusion is possible and just write or for If containsa unit we also require that for all In a similar way, a rightis an abelian group E equipped with a map (right multiplication) that isbi-additive and satisfies And as before, if no confusion ispossible we will just write or for As for left if contains aunit we require that for all Since is in general not commutative,the notions of left and right do not coincide.

An is an abelian group E which is at the same time a left and a rightsuch that the left and right actions commute, i.e., for all

which can also be written asA subset F of a left/right E is called a submodule if F is a subgroup with

respect to the additive structure of E such that It follows that F, with theinduced multiplication of is itself a left/right

§1. commutative rings and

1.4 Convention. Throughout this book we will denote by an arbitrary abelian group.

1.5 Definition. An ring is a ring with the additional property (thethat there exists a family of subgroups (subgroups with respect to the additive(abelian) structure of the ring such that:

(i) and

(ii)The elements of are called homogeneous elements of parity For homogeneouselements the parity map is defined by Note that the paritymap is not defined on the whole of In analogy with the special case elementsof parity 0 will always be called even.

1.6 Remarks. There is an ambiguity in the definition of the parity of the zero elementHowever, we will not avoid this ambiguity since it comes in useful to say that

0 has every parity one wishes. Purists might want to exclude 0 from having a parity, butthen in a lot of proofs one has to treat the zero element as a special case.

Using the parity map, condition (ii) above can be stated as: if are homoge-neous, then

1.7 Lemma. If an ring has a unit then

Proof. Let be the decomposition of the unit into homogeneous components,and let be any homogeneous element. From the equation the uniquedecomposition into homogeneous components and the fact that wededuce that Since multiplication is bi-additive, it follows that for all

In the same way one proves that for all But units, if they exist,are unique and hence

1.8 Definition. An commutative ring is an ring together with asymmetric bi-additive map such that

The above property is called the commutativity of

1.9 Remark. A more general definition of commutativity is possible if is an[6.1] with or In that case one can define

commutativity by the condition

3


where is a “bi-additive” map satisfying(we put bi-additive in quotes because the abelian operation in is multiplication).

For instance, if and we can take for themap with a third root of unity. On the other hand, foror and the only possibilities for are of the formwith

1.10 Convention. Throughout this text we will denote by an arbitrary com-mutative ring with unit From time to time we will impose additional restriction on

but those restrictions will always be stated clearly.

1.11 Definition. An left (respectively right) is a left (respectivelyright) E together with a family of subgroups (subgroups withrespect to the additive (abelian) structure of E) satisfying:

(i) and(ii) (respectively

The following definitions and remarks are as for rings. The elements of arecalled homogeneous elements of parity The parity map is defined by

Note that the parity map is not defined on the whole of E and that the parityof the zero element is ambiguous. Using the parity map, condition (ii) can be statedas: for homogeneous and Elements of parity 0 willbe called even.

1.12 Example. Let be an ring and a natural number, then we canmake into an Addition and left/right multiplication by iscomponent wise, while the is given by For we formallydefine which is trivially an

1.13 Definition. A subset F of an left/right E is called ansubmodule if F is a submodule of the left/right E such that F together with thesubsets is itself an left/right

1.14 Lemma. Let F be a submodule of an left/right E. Then F isan submodule if and only if whereis the unique decomposition of e into homogeneous components in the left/right

E.

§1. commutative rings and

1.15 Lemma. Let be an commutative ring and let be anleft/right Then there exists a unique right/left action of on E

turning E into an right/left with the same subsets such that theleft and right actions of are related by the formula

5

Proof. Let us suppose that has been given, then if exists, bi-additivity impliesthat it must be given by

where are the unique decompositions of these elementsinto homogeneous components. From this formula, the uniqueness of is clear. On theother hand, bi-additivity of proves that defined by this formula is also bi-additive.It thus remains to show that is a right action, i.e.,Since this relation is 3-additive in the variables it suffices to showit for homogeneous elements:

1.17 Remark. The left and right actions of on the are relatedto each other by (1.16).

1.18 Lemma. With the assumptions as in [1.15], the left and right actions of on Ecommute, i.e.,

Proof. The relation is 3-additive in and it thus suffices to show itfor homogeneous elements:

and


1.19 Definitions. By an (without the adjectives left or right) we willalways mean an for which the left and right actions of are relatedby (1.16). Note that [ 1.15] implies that every left and everyright can be turned into such an

A subset F of an E is called an submodule of E if it isan submodule of the left or right E. It follows from [1.20] that thisis a sensible definition and that such an F is automatically an itself.

1.20 Lemma. Let E be an and let F be a submodule of either theleft or the right E. If F is an submodule for

this structure, it is an submodule for the opposite structure. It thus is itself an

Proof. Suppose F is an submodule for the left E. Forand denote by and their decomposition in

homogeneous components. Then Thisbelongs to F because each belongs to F [1.14]. Hence F is also a submodule of theright E. The conclusion then follows again from [1.14].

1.21 Guiding principle. The guiding principle for linear algebra is that in anyformula in which we interchange two homogeneous objects, a signappears. This (additional) sign is already visible in the definition of commuta-tivity and the relation between the left and right actions of on an Inorder to adhere to this principle, we are occasionally led to change notation (e.g., [2.12]).Its advantages will be mostly notational: additional signs in equations will be “obvious”from the order in which one writes the separate terms.

1.22 Definition. Let E be a left or right let S be any subset of E and let beany subset of For a left we define the subset by

for a right one just replaces in this definition by In casewe will drop the subscript and speak of Span(S). This will occur by far the most

frequently, but we will occasionally need proper subsets of The subset Span(S) isobviously a submodule of E, usually called the submodule generated by S.

One special case should be mentioned separately. Suppose is a family ofsubmodules of E, then the submodule Span is usually denoted as andcalled the sum of the submodules This notation is justified because obviously

§2. (Multi-) linear maps 7

Note that is not called the direct sum [3.1 ] of the submodules that is in generala completely different

1.23 Nota Bene. If E is an the notation is ambiguousbecause we have to specify whether we see E as a right or as a left If, ineither view, is an submodule, then it follows from [1.20] that it is an

submodule of the E. However, if the submoduleis not an submodule, then we have to specify whether we view E as a right oras a left [1.26].

1.24 Lemma. If E is an and if is a family ofsubmodules of E, then is an submodule of E.

1.25 Lemma. If E is an and if consists of homogeneouselements only, then Span(S) is an submodule of E.

Proof. According to [1.14] we have to prove that the homogeneous components of anelement are itself in Span (S) . Let be thedecomposition of into homogeneous components, then by assumption ishomogeneous. Regrouping the terms in the (finite) sum according toparity immediately gives the desired result.

1.26 Counter example. To show that the condition of homogeneity is not superfluous,consider the and an element where andare two non-zero homogeneous elements of different parity. In this case the submodule

of E seen as left is not ansubmodule because Note also that, had we interpreted E as a right

the submodule F would (for have been different. This shows thatfor general generating subsets S we have indeed to specify whether we use the left or theright approach.

2. (MULTI-) LINEAR MAPS

After the introduction of in §1, we introduce in this section the notionof a map, of which the more elementary notion of linear map is a special case.We show that the set of all maps not in general an

but that there is a natural subset which isan Two other main points of this section are that left maps

is


should be seen as operating on the right rather than on the left, and that, despite theirdifference, there exists a natural isomorphism between left and right linear morphisms.This section ends with the definition of dual maps.

2.1 Definition. Given and F, then a mapis said to be left if and we have:

(i)

(ii)

The map is called right if condition (ii) is replaced by condition

According to common usage, a 1-linear map will just be called linear, bilinear standsfor 2-linear and trilinear for 3-linear. We denote the set of all left maps by

and the set of all right maps bySince we will often discuss properties of left and right maps at the same time,we will use the notation to denote, during the whole discussion concerned, either

or This should be interpreted as that the discussion is valid for bothand

A map is called of parity if

Note that a map is of parity if and only if is homogeneouswhen all entries are homogeneous and such that By

we denote the subset of all maps offixed parity

In the set we define an addition by

We also define a multiplication by elements in for left maps the right multipli-cation is defined by

and for right maps the left multiplication is defined by

2.2 Lemma. With the above definitions, the set becomes a rightand becomes a left

Proof. What we have to show is in the first place that addition and multiplication arewell defined, i.e., that the result is again left/right When that has been done, theaxioms of a left/right have to be verified. All this is left to the reader.


2.3 Lemma. For and wehave

2.4 Definition. A map is called a (homo)morphismif it is a finite sum of homogeneous maps (see [2.6] for an example in which isan infinite sum). More precisely, we define the set of allhomomorphisms by:

In the context of morphisms, we will also use the name for the set(two names for the same object!). Note that for (homo)morphisms

we use the same notation as for maps: denotes either orbut never both at the same time in a discussion.

In case all coincide, say with the E, wedenote by Two special cases have an alternativenotation. is denoted as its elements are called endomorphismsof E. is denoted as *E and is called the left dual of E; isdenoted as E* and is called the right dual of E.

2.5 Lemma. The left/right together with its subsetsis an If the abelian group is finite, then

Proof. We give the proof for the right linear case; the left linear case is analogous. Fromthe definition of parity it is obvious that the subsets are additivesubgroups that satisfy the condition

It thus remains to show that each element admits a uniquedecomposition in homogeneous parts. That it admits a decomposition in homogeneousparts is immediate from its definition, so remains the uniqueness. Therefore suppose

where the maps are of parity Apply this to homogeneousvectors to obtain But the parities

are all different. Hence for each separately (becauseF is an Since each is for allpossible choices of the not necessarily homogeneous. It follows that i.e., anydecomposition into homogeneous components is unique.

To show the second part of the lemma, choose and definethe maps by:


Since is it follows immediately that which is a finite sumby hypothesis; comparison with the condition for parity also shows immediately that

It thus remains to show that is Since taking homogeneous partsis additive, the will be obvious. To verify conditions and we notethat these are additive in We thus may assume that is homogeneous. But forhomogeneous these relations are obvious after a reparametrization of the appropriatedummy summation variables

2.6 Counter example. The finiteness condition on is not superfluous because a linearmap could be an infinite sum of non-zero homogeneous components, in which case it isnot a morphism in our sense. The following example shows that this can indeed happen.Consider the Z-graded commutative algebra defined by andfor with the trivial bilinear map In this case Z-gradedare nothing more than real vector spaces. Let us consider next the full exterior algebra

where X is an infinite dimensional real vector space; it is a Z-graded with grading for and for In X wechoose an infinite set of independent elements and we define the linearmap by its restrictions toRestricted to each the map augments the parity by but since none of theserestrictions is the zero map, there are infinitely many non-zero maps of different paritiesinvolved in the definition of Hence is not a finite sum of homogeneous maps, i.e.,

2.7 Lemma. For and we have the relationIf is right instead of left we

have the relation

Proof. We prove the left linear case, the right linear case being similar. The relation isadditive in all its entries, i.e., in and but also in We thus may assume that allentries are homogeneous. It follows thatWe then compute:

2.8 Corollary. For we have: ifand only if We are thus allowed to drop the subscript andto write for both sets.


2.9 Definitions. Given two E and F, a map issaid to be invertible if there exists a map such thatand An even invertible map is called an isomorphismbetween E and F. If there exists an isomorphism between E and F, the two

E and F are called isomorphic, denoted by An even invertibleendomorphism of an E is called an automorphism of E; the set of allautomorphisms of E is denoted by Aut(E).

We also introduce the notion of identification as being synonym to even linear map.However, use of the word identification will usually mean that the even linear map con-cerned will not be noted in the sequel. For instance, if is an injective evenlinear map, we may identify E with its image and write forgetting about the actualmap

2.10 Remarks. If a morphism is bijective, it is elementary to showthat its inverse is also a morphism, proving that invertible morphisms are the same asbijective morphisms.

If we equip the (E) with composition of maps as multi-plication, it becomes an ring; its subset Aut(E) becomes a group. This followsimmediately from [2.3].

2.11 Discussion. For right maps [2.7] leads to the following series of relations:

These relations can be summarized by saying that for multiplication by the positionof commas and parentheses is of no importance. The technical way to say the same isthe statement that the evaluation mapis and even. Moreover, one easily verifies that the composition operator

is an even bilinearmap. For left maps the situation might seem to be not so nice. However, a smallchange in notation yields a similar result.

2.12 Notation. For and we introduce an alternativenotation for the evaluation of in the vectors by


In this notation we interpret as an operator on the right rather than on the left. It shouldbe read as substitution of the vectors in the map or as contraction of thevectors with the map a notation well known in differential geometry. Thechoice of the name contraction is explained in [II.5.2] (which refers to [4.10]). In case

we will also use the notation

Parallel to the contraction/evaluation operator we introduce an alternative notation forthe composition of two left linear homomorphisms by

Aficionados of categories might say that is the composition operator in the oppositecategory; we will use it just as a different notation for the composition.

2.13 Discussion. With the alternative notation as introduced above, we obtain for leftmaps the following series of relations:

Thus, here again we see that for multiplication by we can ignore the position ofcommas and parentheses (and of course the symbol As for right maps we canstate this by saying that the evaluation mapis an even map. In the same vein, the alternative composition operatorleads to the statement that is an evenbilinear map. Using the contraction operator and the composition operator at the sametime allows us to write for left linear maps:

As for right linear maps, evaluation of a composite of left linear maps becomes a merequestion of parentheses (and the formal evaluation operator if used).

The definite advantage of our alternative notation for left linear maps is that we canadhere systematically to the guiding principle [ 1.21 ] that interchanging two objects gives anadditional sign With the alternative notation we avoid notational interchangingsthat do not involve additional signs (such as and forleft linear maps).


2.15 Discussion/Definition. So far we have insisted on the difference between left linearand right linear morphisms, and we will continue to do so. Nevertheless there exists anatural isomorphism between these two sets of maps.It is defined by the equation

where and denote the decomposition of and into theirhomogeneous parts. One advantage of interpreting left linear maps as acting on the rightnow becomes obvious: going from the left hand side to the right hand side of (2.16), wehave to interchange and which “explains” the sign

Several verifications have to be made, the first of which is to show that is indeedright linear. Since (2.16) is obviously additive in it is sufficient to show the relation

for homogeneous and If and and thuswe compute:

where we have used that in the the right and leftmultiplication are related to each other by (1.16). In a similar way one proves that itselfis left linear. Since an explicit expression for its inverse is given by

and since obviously preserves parity, we conclude that it indeed is an isomorphism.

2.17 Lemma. If and are homogeneous,

2.18 Definition. We will call the operator the transposition operator, and we will callthe transpose of These names will be justified in However, the reader is

warned that is not its own inverse: is defined on left linear maps and its inverse onright linear maps!

2.19 Discussion. If E is an there exist canonical isomorphismsgiven by The inverse of is given by

We thus find for homogeneous and the relation

We conclude thatIn the special case the maps provide canonical isomorphismsand More precisely and

i.e., and where denotes the multiplication


2.20 Definition. For any we define called theright dual map of by the formula

Similarly we define for any the left dual map bythe formula

One should note that taking the dual map switches sides: the dual of a left linear map isright linear and vice versa.

2.21 Proposition.(i)

(ii)(iii)

(iv)

If is surjective, its dual map is injective.and

The map is even and linear, as is themap

Proof. Properties (i), (ii), and (iii) are elementary. For (iv) one has to realize whichtransposition operators are involved. In on the right hand side they representthe sequence In on the left hand side the first onerepresents the switch and the second one the switch

Once one has this, the proof is elementary.

2.22 Remark. In [2.21-ii] we see again the advantage of the notation for compositionof left linear maps: we do not have to change the order of and in these formulæ.

3. DIRECT SUMS, FREE AND QUOTIENTS

In the previous sections we have seen the construction of the sub-module and morphisms; in this section we provide three new constructions of

In the first place the free on a set G of homo-geneous generators whose parity is given by The next construction is the direct sumof a family of The third construction is that of the quotient of an

by an submodule.

3.1 Construction (direct sums). If is a collection ofwe define their direct sum as the subset of the direct product

§3. Direct sums, free and quotients 15

consisting of those vectors with except for finitely many indices(recall that the direct sum of real vector spaces is defined exactly in this way). By defininga componentwise addition and (left) multiplication by elements of becomes a left

Finally we define the subsets by We leaveit to the reader to verify that with these definitions E becomes an

For each we define maps and by andfor It follows immediately that the are surjective even

linear maps and that the are injective even linear maps, related byWe will usually denote a general element by instead of by

just to stress that it is not an arbitrary element of the direct product, but one with onlyfinitely many non-zero entries. In case the index set I has a finite number of elements,we will write for and an arbitrary element will be denotedby If the spaces are all equal to a given one,the direct sum is also denoted as It is indeed the power of F becausefor a finite index set I the direct sum equals the direct product. And if we define formally

then the equality holds for all

3.2 Remark. One might ask why we do not define direct products ofThere are several reasons. In the first place, if both the index set I and the abelian group

are infinite, one can easily find examples in which the direct product is not anthe failure being that not every element can be written as a finite sum of

homogeneous elements. In the second place, we never need infinite direct products. Andin the third place, a direct product of finitely many is the same asthe direct sum of these spaces.

3.3 Definition. If is a family of submodules of a givenE, we can consider the map defined by Note

that this map is well defined because there are only finitely many non-zero; its imageis One easily verifies that this map is even and linear by definition ofsubmodules. Officially is never a submodule of E (but is); nevertheless,we will write whenever this map is an isomorphism onto E. As for real vectorspaces, this is the case if and only if every element can be written in a unique wayas with and only finitely many of them non-zero. If I contains twoelements, we will write The submodules and will be calledsupplements to each other.

3.4 Construction (free Let be a map from an abstractset G to and define by We define the space as theset of all maps with the property that for all except finitelymany. In we define an addition by and a (left)multiplication by elements of by In this way becomes a left


One usually identifies each element with the map definedby and for It follows that each can be writtenin a unique way as where is defined asand where the sum is actually a finite sum by definition of

To make into an we define by

In words, has parity if and only if the coefficient has parityIn particular the element (map) has parity justifying the

use of the symbol for the abstract map Decomposing the coefficientsof an arbitrary element into homogeneous parts, it follows immediatelythat since by construction is contained in

we conclude that is anThe is usually called the free on

(homogeneous) generators G with parity Using the notion of Span, we can summarizethe construction of by saying Using the notion of directsums, we can write where is the free

on the single generator of parity

3.5 Nota Bene. We have seen that each element admits a unique decompo-sition with Using the induced right action of it follows thatthere also exists a unique representation with the coefficients on the right of the i.e.,

In general the coefficients and are different; only ifcan we be sure that

For any free on a single homogeneous generator wecan define the map by This is a bijective linear map of parity

It is an isomorphism if and only if It follows that for we cannot identify(in the naive and official sense of the word) the withbecause left and right multiplication in the and are notrelated in the same way due to the difference in parity between and

3.6 Corollary. Let be a family of free on generatorsThen where is defined

as

3.7 Construction (quotients). Let E be an and let F be ansubmodule. The quotient with canonical projection is defined inthe sense of abelian groups, i.e., As for abelian groups,the element will also be denoted as mod F. We claim that G can beequipped with the structure of an Addition and (left) multiplication

§4. Tensor products 17

by elements in are defined by The subgroupsare defined by It follows immediately that with

this grading is an even morphism.The only tricky point in proving that G is an is in the proof

that the decomposition in homogeneous components is unique. Therefore, let us supposewith (and of course only finitely many of them non-zero,

which implies By definition of and the projection we may assumethat Since these are homogeneous and F is an submodule, wehave by [1.14] that belongs to F, i.e., This proves that the decompositioninto homogeneous components is unique.

3.8 Lemma. Let E and H be F an submodule of E anda linear map that vanishes on F, i.e., Then there exists a unique

induced map such that If has parity then so has if isa homomorphism, i.e., then so is

3.9 Proposition. Let E and F be and let behomogeneous. Then the following assertions hold (see also [II.3.12]).

(i) and are submodules of E and F respectively.(ii) If is even, there exists a canonical isomorphism

(iii) If is even and if admits a supplement, there exists an isomorphismwhich is completely determined by the choice of the supplement.

Proof. For any linear map it is immediate that and are submodules, so weonly have to check the grading. Therefore, let and decompose intohomogeneous components Since is homogeneous, the

are also homogeneous. It follows that the homogeneous components ofare again in i.e., is a graded submodule. For the reasoning is thesame: if then Since all have different parities, theymust be zero, i.e., the are in The result then follows.

If is even, consider the induced (even!) It is injective byconstruction and hence is an isomorphism onto

If H is an submodule of E, supplement to it follows that therestriction is injective. If is also even, it is an isomorphismWe thus have

4. TENSOR PRODUCTS

In this section we introduce the construction of a tensor product ofWe show that forming tensor products is associative in a very nice way and we prove the


principal property of tensor products: transforming maps into linear maps; inother words, we prove that the tensor product is the solution of a universal problem. Wefinish with the construction of the permutation operator on multiple tensor products.

4.1 Construction (tensor product). Let E and F be and considerthe set i.e., G is the product of allnon-zero homogeneous elements in E and in F. On G we define a parityby which is well defined because and are by assumptionhomogeneous. We thus can consider the free and we recallthat we have identified the abstract elements with the elements Withthis in mind, we define the subset S of as the union of two subsets:with

By construction, all the elements of S are homogeneous and hence Span(S) is ansubmodule [1.25]. With this submodule we then define the

as the quotient This is calledthe tensor product of E and F.

The construction of the tensor product is not complete without the definition of the map

4.2 Lemma. The map is even and bilinear.

Proof. If and are homogeneous, the sum over and in the definition of containsonly one term. It follows immediately that

i.e., is even. The bi-additivity follows easily from the definition of S. Letus show for instance the additivity in

where we used that belongs to

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