Multiplicity-free Super Vector Spaces [Elektronische Ressource] / Tobias Pecher. Betreuer: Friedrich Knop

Multiplicity-free super vector spacesDer Naturwissenschaftlichen Fakultätder Friedrich-Alexander-Universität Erlangen-NürnbergzurErlangung des Doktorgradesvorgelegt vonTobias Pecheraus NürnbergAlsDissertationgenehmigtvonderNaturwissenschaftlichenFakultätderFriedrich-AlexanderUniversität Erlangen-NürnbergTag der mündlichen Prüfung: 30. Juni 2011Vorsitzender der Promotionskommission: Prof. Dr. R. FinkErstberichterstatter: Prof. Dr. F. KnopZweitberichterstatter: Prof. Dr. P. FiebigiZusammenfassungDie vorliegende Arbeit beschäftigt sich mit der Frage, wann die supersymmetrische Al-gebra P (V ) eines Supervektorraumes V als Modul einer reduktiven Gruppe G multipliz-itätenfrei ist. Diese Klasse von Darstellungen wird auch supermultiplizitätenfrei genannt.DieTatsache, obeingegebenesPaar (G;V )dieseEigenschaftbesitzt, kannvonderGestaltdes Zentrums vonG abhängen. Hierbei treten technische Schwierigkeiten hinsichtlich derKlassifikation supermultiplizitätenfreier Räume auf, welche wir umgehen, indem wir einegewisse Maximalitätsbedingung an das Zentrum voraussetzen. Innerhalb dieses Rahmenserhalten wir eine komplette Klassifikation und verallgemeinern damit bekannte Resultateüber die Multiplizitätenfreiheit symmetrischer und äußerer Algebren.Für supersymmetrische Algebren stehen, im Gegensatz zur reichhaltigen Theorie der sym-metrischen Algebren, kaum für uns nützliche Strukturaussagen bereit.
Publié le : samedi 1 janvier 2011
Lecture(s) : 95
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Source : D-NB.INFO/1015474888/34
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Multiplicity-free super vector spaces
Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades
vorgelegt von Tobias Pecher aus Nürnberg
Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät der Friedrich-Alexander Universität Erlangen-Nürnberg
Tag der mündlichen Prüfung: 30. Juni 2011
Vorsitzender der Promotionskommission: Prof. Dr. R. Fink
Erstberichterstatter: Prof. Dr. F. Knop
Zweitberichterstatter: Prof. Dr. P. Fiebig
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Zusammenfassung Die vorliegende Arbeit beschäftigt sich mit der Frage, wann die supersymmetrische Al-gebraP(V)eines SupervektorraumesVals Modul einer reduktiven GruppeGmultipliz-itätenfrei ist. Diese Klasse von Darstellungen wird auchsupermultiplizitätenfreigenannt. Die Tatsache, ob ein gegebenes Paar(G, V)diese Eigenschaft besitzt, kann von der Gestalt des Zentrums vonG treten technische Schwierigkeiten hinsichtlich derabhängen. Hierbei Klassifikation supermultiplizitätenfreier Räume auf, welche wir umgehen, indem wir eine gewisse Maximalitätsbedingung an das Zentrum voraussetzen. Innerhalb dieses Rahmens erhalten wir eine komplette Klassifikation und verallgemeinern damit bekannte Resultate über die Multiplizitätenfreiheit symmetrischer und äuerer Algebren. Für supersymmetrische Algebren stehen, im Gegensatz zur reichhaltigen Theorie der sym-metrischen Algebren, kaum für uns nützliche Strukturaussagen bereit. Ein wesentlicher Ansatzpunkt besteht deshalb darin, den Kreis potentieller supermultiplizitätenfreier Räume einzugrenzen, indem wir zumindest notwendige Bedingungen für Supermultiplizitätenfrei-heit bestimmen. So zeigen wir, dass - nach einer geeigneten Definition dieses Begriffs - jede Teilstruktur einer supermultiplizitätenfreien Darstellung ebenfalls supermultiplizitätenfrei ist. Ferner existiert für irreduzible DarstellungenVvonGeine bekannte Aussage über die Zerlegung vonVV Diesebezüglich Restriktion auf Levi-Untergruppen. verallgemeinern wir auf beliebige Darstellungen vonG. Hilfreich ist zudem die Tatsache, dass die Su-permultiplizitätenfreiheit von reduziblen SupervektorräumenVdavon unberührt bleibt, wenn einzelne irreduzible Summanden inV Wirdurch ihre Dualen ersetzt werden. zeigen dies unter Verwendung einer Aussage über die Struktur derG-invarianten Differentialop-eratoren auf supermultiplizitätenfreien Räumen, was im wesentlichen eine Übertragung einer aus dem symmetrischen Kontext bekannten Tatsache ist. Gerader und ungerader Anteil eines supermultiplizitätenfreien SupervektorraumsVsind selbst (super)multiplizitätenfrei. Da zu Beginn dieser Arbeit keine komplette Klassi-fikation multiplizitätenfreier äueren Algebren bekannt war, stellt die Vervollständigung dieser Liste den ersten Schritt in unseren Betrachtungen dar. Zum Nachweis der Multipliz-itätenfreiheit in den einzelnen Fällen erweist sich hierbei die Deutung der Gruppenaktion als Restriktion der Operation von allgemeinen linearen Gruppen als hilfreich. Die oben erwähnten Resultate erlauben es dann, die Ergebnisse über symmetrische und antisym-metrische Algebren effektiv zur Behandlung des Allgemeinfalls, d.h. die Untersuchung der Multiplizitätenfreiheit echter Supervektorräume, zu kombinieren und einzusetzen.
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Abstract The current thesis is engaged in the question when the supersymmetric algebraP(V)of a super vector spaceVis multiplicity-free as a module of a reductive groupG. This class of representations is calledsuper multiplicity-free. For a given pair(G, V)this property can depend on the structure of the center ofG. This causes technical difficulties concerning the classification of super multiplicity-free spaces, which we circumvent by assuming a certain maximality condition on the center. Within this framework, we obtain a com-plete classification and thus generalize well-known results on the multiplicity-freeness of symmetric and exterior algebras. In contrast to the comprehensive theory of symmetric algebras, there is hardly any useful statement for superssymmetric algebras. A main approach lies in narrowing down the list of potential super multiplicity-free spaces by determining at least necessary conditions for super multiplicity-freeness. Thus, we show that - given an appropriate definition for this term - every substructure of a super multiplicity-free representation is super multiplicity-free either. In addition, for irreducible representationsVofG, there is a known propositon on the decomposition ofVVwhen restricted to Levi subgroups. As a further useful fact, the super multiplicity-freeness of reducible super vector spacesVis not affected by the exchange of some irreducible summands by their duals. We show this by making use of a statement on the structure ofG-invariant differential operators on super multiplicity-free spaces, which is in turn mainly a translation of a fact that is known in the symmetric context. Even and odd part of a super multiplicity-free super vector space are again (super) multiplicity-free. Since at the beginning of this project there was no complete classifi-cation of multiplicity-free exterior algebras, its finishing is the first step in our consider-ations. The interpretation of group actions as restricted operations from general linear groups proves useful for the verification of the multiplicity-freeness in the indivual cases. The above mentioned conclusions then allow a combination and effective application of the results on symmetric and anti-symmetric algebras for the treatment of the general case, i.e. the investigation of muliplicity-freeness for proper super spaces.
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Danksagungen In diesen Tagen jährt sich zum 175. Mal die erste Fahrt einer deutschen Eisenbahn zwischen Nürnberg und Fürth. Neben seiner landesweiten Bedeutung zeigt dieses Eregnis zudem, wie die Bürger der zueinander in Rivalität stehenden beiden Städte trotz der gegenseitigen Ressentiments zu einer fruchtbaren Zusammenarbeit fähig sind. Auch wenn man es auf keiner Seite offen zugeben möchte, respektiert man doch tief im Inneren den anderen, und der Lohn der gemeinsamen Anstrengung lät viele Schmähungen (die vor allem die Fürther von ihren “groen Nachbarn” erleiden müssen) schnell verblassen. Das Thema dieser Arbeit - und die daraus resultierenden Anstrengungen - wurden an-geregt von Friedrich Knop, dem ich hiermit meinen gröten Dank aussprechen möchte: Dafür, dass er mich die ganze Zeit meiner Promotion hinweg bestens unterstützt und mir viele Dinge verständlich erklärt hat. Auerdem nahm er sich immer wieder die Zeit, meine Ideen und Berechnungen nachzuvollziehen und mich dabei auch verlälich auf ver-steckte Fehler und Ungenauigkeiten hinzuweisen. Vor allem in der Anfangsphase war die Gewiheit, durch seine Leitung stets die Zielrichtung im Blick zu behalten, eine enorme Hilfe. Die Gespräche mit ihm über die Mathematik waren jedesmal äuerst motivierend und inspirierend. Dem Prädikat meines Vorgängers [We] über Friedrich Knop kann ich deshalb nur voll und ganz zustimmen. Auch allen anderen Mitarbeitern des Emmy-Noether-Zentrums möchte ich für die kol-legiale Arbeitsatmosphäre und die nette gemeinsame Zeit danken. Besonders erwähnen möchte ich hierbei Peter Fiebig, nicht nur für seine Bereitschaft, als Zweitberichterstatter zu fungieren. Durch seine weitere Unterstützung in den letzten Monaten stellte er für mich sicher auch eine Art Zweitbetreuer dar. Ein kleiner, aber entscheidender, Teil dieser Arbeit enstand während eines Aufenthalts am “Centro di Ricerca Mathematica Ennio De Giorgi” in Pisa. Die gemeinsamen Diskussionen mit Oksana Yakimova waren äuerst hilfreich und auch ansonsten waren die Tage in Pisa ein sehr schönes Erlebnis. Ich danke ihr an dieser Stelle nochmals herzlich für die Einladung dorthin, sowie dafür, dass Sie groe Teile des Manuskripts las und eine Vielzahl von Verbesserungsvorschlägen lieferte. Desweiteren danke ich Bertfried Fauser, für viele interessante Informationen über sym-metrische Funktionen, Branching rules, sowie etliche andere mathematische und physikalis-che Anekdoten. Zudem kam ich durch ihn in Kontakt mit Ron C. King, von dem ich entscheidende Hinweise bezüglich der Modification rules erhielt. Dies machte mir den Be-weis der Schiefmultiplizitätenfreiheit vonSL3SO2n+1möglich und so schulde ich auch ihm groen Dank. Abschlieend bedanke ich mich bei Guido Pezzini, Petra Weber und Miriam Wegner für mathematische und sprachliche Korrekturen der Arbeit, sowie beim gesamten Team der Rechenanlage des Departments, das mich all die Jahre hinweg mit ausreichend Kaffee versorgt hat.
Contents 1 Introduction 2 Preliminaries 2.1 The category of super vector spaces . . . . . . . . . . . . . . . . . . . . . . 2.2 Invariant Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Characters of the classical groups . . . . . . . . . . . . . . . . . . . . . . . 2.4 Branching rules forSOmandSp2n. . . . . . . . . . . . . . . . . . . . . . . 2.5 Decomposition formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Main Results 3.1 Classification of saturated indecomposable super MF space . . . . . . . . . 3.2 Reduction arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Classification of super MF spaces 4.1 Irreducible skew MF spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Decompositions of reducible super MF spaces . . . . . . . . . . . . . . . . 4.3 Completion of the proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . 4.4 Exhaustiveness of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Exhaustiveness of Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . 5 Open problems References
1 5 5 9 12 14 17 19 19 22 25 25 33 36 41 47 49 52
1 INTRODUCTION
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1 Introduction One of the most famous and important formulas in classical invariant theory is the identity [Ca] found by A. Capelli det [ρ(Eij) +δij(nj)] = det[xij]Ω,(1.1) which he understood as “tracing back theΩoperation to ordinary polarization operations”. In fact, this can be stated, more formally, in representation-theoretic terms. For an arbitrary super vector spaceVwith an action of a reductive groupGwe consider the set PD(V) Ifof polynomial-coefficient differential operators on its supersymmetric algebra. gdenotes the Lie algebra ofG, the group actionρonVgives rise to a homomorphism ρ:Z(g)PD(V)G(1.2) from the center of the enveloping algebra into theG Con--invariant differential operators. cerning the action ofGLnon the space of matrices, the left hand side of (1.1) describes the differential operator on the right hand side by its preimage under (1.2). This in-terpretation led [HU] to formulate the “abstract and concrete Capelli problem”, whether for an arbitrary action (1.2) is always surjective and, if so, how to compute inverse im-ages. They particularly investigated purely even super spacesVand showed that for multiplicity-free actionsPD(V)Ga polynomial algebra with canonical generators and ais canonical basis. For such actions they solved the Capelli problems. IfVis an arbitrary (but multiplicity-free) super vector space,PD(V)Gis in general no polynomial algebra but still has a canonical basis. Thus, also in this case it seems worthwhile to compute such basis elements and try to find generalized Capelli identities. Moreover, the fairly clear description ofPD(V)Gshould make it possible to determine spherical functions and, more generally, develop harmonic analysis on these spaces. Analogues of the classical Capelli identity are not the only interesting aspects of multipli-city-free super spaces. They also played an important role in a systematical reinterpre-tation of classical results from invariant theory done by R. Howe. While historically, invariant theory was mainly concerned with group actions on polynomial algebras, he extended in [Ho2] the scope to supersymmetric algebras. He showed how Hodge decom-position, spherical harmonics, or the cohomology of the unitary group can be described in a uniform way by group operations on supersymmetric algebras and their invariants. In another paper [Ho1] he emphasizes the relevance of multiplicity-free actions in invariant theory. All these results motivate a classification of multiplicity-free super vector spaces.
We shall point out some results which are known in the two special cases whereVis either a purely even or odd space. In the first case we are dealing withS(V)which can be treated as coordinate ringC[V]of the varietyV. Since those spaces can be studied by means of algebraic geometry, they are quite well understood. In fact, we are in a very particular situation: AG-varietyXis calledsphericalif a Borel subgoupBGhas an open orbit in X. It is a well-known fact that the regular functionsC[X]onXdecompose multiplicity-freely as aGmodule if and only ifXis spherical. Thus, the multiplicity-free (purely
1 INTRODUCTION
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even) spacesVare exactly the linear spherical varieties. Beside the above statement onB-orbits, one knows that theU-invariants inC[V] Theform a polynomial algebra. classification of multiplicity-free even vector spaces is completely solved by [Kac], [BR1] and [Lea]. In the first mentioned paper (in which the irreducible spaces were classified) the results were derived from detailed studies of orbit structures of (multiplicity-free) group actions, and also the other two articles made partly use of the geometric properties. The first investigations concerning vector spaces with multiplicity-free exterior algebras were done in [Ho1], where they were calledskew multiplicity-free. There was given a complete list of irreducible skew multiplicity-free modules for simple groups. This was done by observing that skew multiplicity-free spaces have to beweight multiplicity-free, i.e. all their weight spaces have to be one dimensional. So the classification of skew multiplicity-free spaces was derived from an initial classification of weight multiplicity-free spaces. However, there was much need of case-by-case analysis by the absence of further geometric criterions. Unfortunately, also the statement on the highest weight vectors of C[V]has  Nevertheless,no appropriate analogue in the skew symmetric setting. some skew multiplicity-free modules still admit a relatively clear description of their highest weight vectors. [Ho1] calls them(symplectic) geometrically decomposable exists. There an interesting connection between them and the cohomology of nilradicals in simple Lie algebras. If the nilradicalnis abelian,H(n) =Vnsince the coboundary operator is zero. But even forna Heisenberg Lie algebra, the cohomology is essentially given byVn. As the computations in [Ko] show, the decomposition forH(n)under the action of the Levi subalgebra is always multiplicity-free and the instances ofH(n)for abelian (Heisenberg) n also [St3] for a Seeare exactly the (symplectic) geometrically decomposable modules. related survey. The above discussion indicates that the structures of symmetric and skew symmetric multiplicity-free spaces can be, in some sense, very different from each other. Although, in some points there are similarities. The action ofGLn×GLmon its defining representation gives rise to decompositions S(CnCm) =MVλVλ,^(CnCm) =MVλVλt,(1.3) where the direct sum runs over all partitionsλ Since,that “can actually occur”. in both cases, the isotypic components forGLnandGLmstand in bijective correspondence, this is also called(GLn,GLm)(skew) duality. This will also play an important role in our considerations. In principle,(GLn,GLm) corresponds to Itduality is very well-known: the character formulas YnYm1n m i=1j=11xiyj=Xsλ(x)sλ(y),Y Y(1 +xiyj) =λXsλ(x)sλt(y),(1.4) λ i=1j=1 which are commonly called theCauchy identity, resp. itsdual [Ho1] isversion. But (apparently) the first attempt where (1.3) are treated conceptually by giving proofs in terms of the group operation.
1 INTRODUCTION
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Considering arbitrary super vector spaces, basic multilinear algebra tells us that even and odd part of a multiplicity-free space are multiplicity-free as well. This serves as foundation for the general classification by indicating the main steps to be done: By the above results, we have to complete the list of skew multiplicity-free spaces and combine them with the symmetric counterparts. This yields all super vector spaces that are potentially multiplicity-free. Furthermore, we can restrict ourselves toindecomposableactions, i.e. those which are not of the form(G1×G2, V1V2)withGiacting trivially onVjfor i6=jfact that super multiplicity-freeness of an difficulties arise from the . Technical action can depend on the center of the group. If the central torus has enough copies of C, the action is super multiplicity-free if and only if all multihomogeneous components of the supersymmetric algebra are multiplicity-free. Within this framework we obtain a complete classification. A posteriori, we can make a remarkable observation on a certain symmetry shared by the set of multiplicity-free super vector spaces: Up to some counterexamples, which exist only in low dimensions, the super multiplicity-freeness of(G, V)is equivalent the super multiplicity-freeness of(G, VΠ), whereVΠis the super vector space obtained fromVby shifting the parity of some irreducible submodules andGdenotes the group in which those simple factors ofGthat act nontrivially on the corresponding submodules, are, exchanged by their Langlands dual group. This seems to be a relation in the spirit of Koszul dualityof exterior and symmetric algebras due to [Be]. We give a detailed overview of the following chapters: Section 2 is mainly expository and does not contain any striking new results. In Section 2.1 we recall some standard notions from the category of super vector spaces, give the main definitions and introduce the concept of representation diagrams which is a very useful combinatorial description of group actions. By some simple modifications of known results, we prove in Section 2.2 a characterization of super multiplicity-free spaces in terms of theirG-invariant differential operators: They are exactly those spaces for whichPD(V)Gis commutative. Thereby we derive the statement that the super multiplicity-freeness of(G, V)is invariant under replacing some irreducible submodules inVby their duals. Section 2.3 describes a class of symmetric functions that become, after restricting the number of variables, characters of a classical group of appropriate rank. For this reason, they are also called universal characters. The effect on a universal character by passing into an actual one can be specified by so-called modification rules. These are described - in connection with two particular branching rules - in Section 2.4. The preliminaries close with Section 2.5 that lists a couple of decomposition rules for symmetric and antisymmetric plethysms which have applications in further calculations. Section 3 gives a compact overview of our results and also of the main techniques developed for deriving them. Hence, the complete classification of super multiplicity-free spaces is presented in Section 3.1. For convenience, also the classification of symmetric- and skew MF spaces is recalled. Section 3.2 deals with two necessary conditions on super MF spaces that reduce the number of case-by-case calculations tremendously. The first one is a particular result for purely odd super vector spaces and states a relation between skew multiplicity-free spaces of a groupG Originally,and their Levi subgroups. this
1 INTRODUCTION
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statement was published in [Ho1], but only for irreducibleV give a generalization. We to the reducible case. The second argument applies to arbitrary super spaces and is formulated by means of representation diagrams. It says, that every subdiagram in the representation diagram of a super MF space must also yield a super MF space. The necessary calculations for the classification are all carried out in Section 4. First, we consider irreducible skew multiplicity-free spaces in Section 4.1. The main idea is to describe the actions through(GLn,GLm)skew duality via restriction to appropriate subgroups and investigation of the corresponding branching rules. As a consequence, the action of the subgroup is skew multiplicity-free if and only if all the branching rules are multiplicity-free. By far the hardest case is the verification forSL3SO2n+1. The proof heavily relies on the modification rules and their description by removals of strips from Young diagrams which is due to R. C. King. Inside the theory of skew symmetric variables, one has similar difficulties when considering the decomposition forSLnSp4. Initially, we proved skew multiplicity-freeness by using methods due to S. Sundaram [Su1], but realized then that this could be also easily deduced from the symmetric setting. Regarding the exhaustiveness of the list of skew multiplicity-free spaces, we consider several particular decompositions of irreducible representation under Levi subgroups. By the statement on Levi subgroups, this makes it possible to discard many modules without further computations. The classification of proper super vector spaces with a multiplicity-free super symmetric algebra starts in Section 4.2 where the majority of super MF spaces are verified. Sec-tions 4.3 - 4.5 are mainly concerned with proving the completeness of the lists of super multiplicity-free representations for groups with one up to three simple factors respec-tively. A number of calculations here is unavoidable and it should be pointed out, that initially, many of them (as well as some computations of skew plethysms in Section 4.1) were done with the computer algebra systems LIE [LCL] and Schur [Wy]. But most of them are easily checked by hand using the Littlewood-Richardson- or the Pieri Rule (or, for the skew plethysms, by imposing(GLn,GLm)skew duality). Some interpretations and observations on the results are presented in Section 5.
2 PRELIMINARIES
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2 Preliminaries 2.1 The category of super vector spaces Although it can be avoided in principle, it makes sense for our investigations to use the language of super vector spaces. In order to keep this work self-contained we recall some basic terms and constructions of this category (as a reference see e.g. [Sch]). Furthermore, we develop some notation which will be used throughout the text. Definition 2.1A super vector space is a vector space together with a fixed decomposition into subspaces V=V0V1. V0is called the even part andV1the odd part ofV. ForxViwe define the parity|x|=i. Note that the-sign just means thatV, interpreted as usual vector space, is given as direct sum ofV0andV1. Commonly, one uses an Nevertheless,also in the super setting. the modified notation is convenient when considering a (reductive) groupGacting onV. An action ofGon a super vector spaceV0V1is given by the actions ofGon even and odd part ofV. IfV0=U1. . .UkandV1=W1. . .Wlare decompositions into irreducible submodules, we write V=U1. . .UkW1. . .Wl. So by convention, all irreducible submodules ofVthat stand to the left ofbelong to the even part ofVwhile the odd part consists of all submodules to the right of. Given two super vector spacesVandW, the spaceHom(V, W)of all linear maps fromV toWis naturally a super vector space by Hom(V, W)0= Hom(V0, W0)Hom(V1, W1),Hom(V, W)1= Hom(V0, W1)Hom(V1, W0). If we want to speak about the categorySVecof super vector spaces, we have to define the morphisms. Clearly, they should respect the structure of the objects, so HomSVec(V, W) = Hom(V, W)0. Furthermore, there is an auto-equivalenceΠofSVec, also referred to asparity shift, which is defined byΠ(V) = Π(V)0Π(V)1=V1V0andΠ(f) =f. LetV,W following constructions in Thebe super vector spaces.SVecgeneralize well-known objects from the category of usual vector spaces: a)V= (V0)(V1)(Dual super vector space) b)VW= (V0W0)(V1W1)(Direct sum) c)VW= (V0W0)(V1W1)(V0W1)(V1W0)(Tensor product)
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