- Email: [email protected]
Categorification of Fourier transforms, efficient computation and computing intelligence
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 534 (2004) 324–328 www.elsevier.com/locate/nima
Categorification of Fourier transforms, efficient computation and computing intelligence N.M. Glazunov1 Glushkov Institute of Cybernetics, National Academy of Science, 40, Prospect Glushkova, 03187 Kiev GSP-680, Ukraine Available online 2 August 2004
Abstract We investigate a category theory and homological algebra framework for various types of Fourier transforms and efficient methods for their implementation. r 2004 Elsevier B.V. All rights reserved. PACS: 02.10.Ws; 02.30.Nw; 02.70.+c; 11 Keywords: Fast Fourier transform; Homological algebra; Indecomposable object; Fourier–Mukai
1. Introduction: motivation Evaluation of Feynman diagrams, future experiments in the history of the early Universe and their processing [1], investigations and computations to string theory and some other researches result in many cases in necessity for (i) consideration of many facets of the problem with their interactions and (ii) efficient computations. An overwhelming majority of the fundamental methods for Physics have set theoretic foundations. A category theory framework (by means of a category C or a functor F) for a system S; a process P or a phenomenon F can be thought as a
collection of objects A; B; . . . ; one for each element of S (respectively P; F) which are combined by morphisms f : A ! B; subject to the well-known conditions of associativity and identity [2]. The principle of System Analysis assumes that plant, process or problem are studied as a system with the consideration of all facets of the system with their interactions. If we can represent these facets as categories and interactions as functors we will obtain a mathematical model of the system. Let V be an n-dimensional vector space with a dot-product as a symmetric bilinear form h ; i over a field k: An n-dimensional lattice L in V has a dual lattice L given by L ¼ fy 2 V j hx; yi 2 Zg
E-mail address: [email protected] (N.M. Glazunov). 1 Participation in ACAT’03 is supported by the Local Organizing Committee.
ð1Þ
for all x 2 L: If V ; W are n-dimensional vector spaces over k and there is a perfect pairing between
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.07.082
ARTICLE IN PRESS N.M. Glazunov / Nuclear Instruments and Methods in Physics Research A 534 (2004) 324–328
V and W then the pairing defines the duality between them. There are many generalizations and physical interpretations of dualities. Sometimes researchers say that two theories are dual when they both describe the same physics. A Fourier transform is an example of duality that transforms a difficult mathematical question (for instance the problem of solution a system of differential equations) to an easy problem (of the solution of a system of algebraic equations). Please see the discussion of dualities in Vafa’s paper [3]. One of the useful notions of category theory is the notion of equivalence [2]. Dualities and equivalences are very close [4,5]. Computing intelligence includes processing of information (encoding, decoding, filtering, compression), efficient and reliable computations, digital image processing, ranking data and information, sparse approximation and learning. Cooley and Tukey and other authors have demonstrated that the classical fast Fourier transform has successful applications to all of the problems [6,7]. In the paper, we illustrate the consequences for the category theory treatment of data representation and analysis, scientific computations, models of interaction and computing intelligence. In order to make the paper easy to read we will illustrate definitions of the basic notions by examples.
2. Preliminaries from algebra and category theory Let A be an algebra over a field k and V a vector space. A representation [8,9] of k-algebra A is a homomorphism h from A h : A ! MðV Þ
ð2Þ
to the algebra MðV Þ of linear operators of V : If h is an monomorphism (hðAÞ ’ A) then the representation is the exact. Indecomposable representations are very useful in representation theory [10]. A category C [2] can be thought as a collection of objects A; B; . . . ; which are combined by morphisms f : A ! B; subject to the well-known
325
conditions (i) and (ii) below. Let CðA; BÞ be the class of morphisms from A to B: Then (i) for any three (not necessary distinct) objects A; B or C of C; there is defined a map CðA; BÞ CðB; CÞ ! CðB; CÞ
ð3Þ
called composition which satisfied the associativity axiom. (ii) For every objects A of C; the set CðA; AÞ contains a morphism idA ; called the identity of A: A functor F from a category C to a category K is a function which maps ObðCÞ ! ObðKÞ; and which for each pair A; B of objects of C maps CðA; BÞ ! CðFðAÞ; FðBÞÞ; while satisfying the two well-known conditions [2]. 2.1. Equivalence of Categories Let C be a category [2], Ob C its class of objects, and for a; b 2 Ob C; Cða; bÞ the class of morphisms (arrows) from a to b: An arrow u : a ! b is the equivalence in C if there is an arrow u0 : b ! a such that u0 u ¼ 1a and uu0 ¼ 1b : A functor F from a category C to a category K is a function which maps ObðCÞ ! ObðKÞ; and which for each pair a; b of objects of C maps Cða; bÞ ! KðFðaÞ; FðbÞÞ; while satisfying the two conditions: Fida ¼ idFa for every a 2 ObðCÞ FðfgÞ ¼ Fðf ÞFðgÞ: Let FunctðC; DÞ be the category of functors from C to D with natural transformations as morphisms. An equivalence in FunctðC; DÞ is the equivalence between categories C and D: Example 1. The next result is a particular case of the theorem from [2]. A functor F : C ! D is the equivalence if, and only if (a) F is fully faithful, and (b) any Y 2 Ob D is isomorphic to an object of the form FðX Þ for an object X 2 Ob C: Example 2. Let k be a Q-algebra. The category of formal groups over k is equivalent to the category of finite dimensional Lie k-algebras [11].
ARTICLE IN PRESS N.M. Glazunov / Nuclear Instruments and Methods in Physics Research A 534 (2004) 324–328
326
Task. Compute the product of f and g with complexity Oðn log nÞ complex arithmetic operations.
3. Discrete Fourier transforms and their categorification 3.1. Discrete Fourier transforms Here we follow [7]. Let G be a finite group, #G the number of its elements, CG its group algebra. Theorem 1 (Wedderburn). The group algebra CG of G is isomorphic to an algebra of block diagonal matrices: d j d j CG ’ m j¼1 C
ð4Þ
The number m of blocks equals the number of conjugacy classes of G, and the d j are uniquely determined by G up to permutation. #G ¼ d 21 þ þ d 2m :
ð5Þ
Definition 1. Every isomorphism d j d j F : CG ! m j¼1 C
ð6Þ
of C-algebras is called a discrete Fourier transform (DFT) for CG: Example 3. Discrete Fourier transform and the generation of pseudorandom sequences. Let p be an odd natural number, F2 ¼ Z=2Z the binary field, s ¼ fsi giX0 a binary p-periodic sequence. Task. Compute for all kX0 the minimal m ¼ mðsÞ of a linear recursion X skþm ¼ ci skþi ð7Þ iom
Solution. (Cooley–Tukey). Apply the fast Fourier transform ðors Þr;son for computing the cyclic convolution of the polynomials. 3.2. On categorification of discrete Fourier transforms Let k be a field, Algk the category of algebras over k: Objects of Algk are k-algebras and morphisms are k-homomorphisms of the algebras. Algk contains many subcategories. Let Gk be the category of finite groups, GAlgk the category of group k-algebras. Let Mk the category of all square matrices over k: Objects of Mk are natural numbers and set of morphisms HomðnÞ is the set of all square matrices kn n : We may consider matrices kd 1 d 1 kd m d m
ð8Þ
as block diagonal matrices of the size d 1 þ þ d m:
ð9Þ
For every natural n the set HomðnÞ is a k-algebra, so an object of Algk : For k ¼ C a discrete Fourier transform of an object kG from GAlgk is an isomorphism with an object of the form d j d j m j¼1 C
ð10Þ
d 21 þ þ d 2m ¼ #G:
ð11Þ
4. Varieties and sheaves
that generates a given periodic sequence s: Solution. (E. Berlecamp, J. Massey). The minimal m ¼ mðsÞ of s equals the Hamming weight of its Fourier spectrum. Example 4. Fast multiplication of complex polynomials. Let n be a natural number, f ; g complex polynomials such that the sum of their degrees less then n; o a primitive nth root of unity.
In the section, we follow Refs. [12–14]. At first collect some notions related to ringed spaces. The ringed space is the pair ðX ; OÞ; where X is a topological space and O is a sheaf of rings on X : For a ringed space ðX ; OX Þ and an open U X the restriction of the sheaf OX on U defines the ringed space ðU; OX jU Þ: Let OX be the sheaf of smooth functions on a topological space X : Then any smooth manifold X with the sheaf OX is a ringed space ðX ; OX Þ:
ARTICLE IN PRESS N.M. Glazunov / Nuclear Instruments and Methods in Physics Research A 534 (2004) 324–328
Example 5. Let X be a Hausdorff topological space and OX a sheaf on X : Let it satisfy conditions: (i) OX is the sheaf of algebras over C; (ii) OX is a subsheaf of the sheaf of continuous complex valued functions. Let W be a domain in Cn and Oan the sheaf of analytical functions on W : The ringed space ðX ; OX Þ is called the complex analytical manifold if for any point x 2 X there exists a neighbourhood U 3 x such that ðU; OX jU Þ ’ ðW ; Oan Þ (here ’ denotes the isomorphism of ringed spaces). Let X be a projective algebraic variety over C and CohX the category of coherent sheaves on X : Example 6. If X is the algebraic curve then indecomposable objects of CohX are skyscrapers and indecomposable vector bundles. Recall now the relevant properties of complexes, derived categories, cohomologies and quasimorphisms referring to [13,14] for details and indication of proofs. The cochain complex d
d
d
ðK ; dÞ ¼ fK 0 ! K 1 ! K 2 ! g
ð12Þ
is the sequence of abelian groups and differentials d : K p ! K pþ1 with the condition d d ¼ 0: Let A be an abelian category, KðAÞ the category of complexes over A: Furthermore, there are various full subcategories of KðAÞ whose respective objects are the complexes which are bounded below, bounded above, bounded in both sides. The notions of homotopy morphism, homotopy category, triangulated category are described in Ref. [14]. The bounded derived category Db ðX Þ of coherent sheaves on X has the structure of a triangulated category [14].
327
of the form E : ax3 þ by3 þ cz3 ¼ 0; abca0:
ð13Þ
Its Jacobian is Ref. [16] E^ : abcx3 þ y3 þ z3 ¼ 0:
ð14Þ
The Poincare´ bundle P is a line bundle of degree ^ defined in such a way zero on the product A A; ^ that for all a 2 A the restriction of P on A fag is isomorphic to the line bundle corresponding to the ^ This line bundle is also called the point a 2 A: universal bundle. Let pA : A A^ ! A
ð15Þ
pA^ : A^ A ! A^
ð16Þ
and CA be the category of OA -modules over A; M 2 Ob CA ; ^ SðMÞ ¼ pA; ^ ðP pA MÞ:
ð17Þ
Then, by definition, the Fourier–Mukai transform ^ FM is the derived functor RS^ of the functor S: ^ be bounded derived categories Let Db ðAÞ; Db ðAÞ ^ respectively. of coherent sheaves on A and A; Mukai has proved Ref. [17]. Theorem 2. The derived functor FM ¼ RS^ induces an equivalence of categories between two derived ^ and Db ðAÞ: categories Db ðAÞ Let X and Y be two connected smooth projective varieties. By Orlov’s theorem [18] any derived equivalence F : Db ðY Þ ! Db ðX Þ
ð18Þ
comes from a complex on the product Y X : Equivalences of derived categories 5. Fourier–Mukai transforms
FM : Db ðY Þ ! Db ðX Þ
Let A be an abelian variety and A^ the dual abelian variety which is by definition a moduli space of line bundles of degree zero on A [15].
is called Fourier–Mukai transforms. There are interesting connections between A1 categories and Fourier–Mukai transforms. A1 categories was introduced by Fukaya [19]. These categories was used in Kontsevich’s conjecture on homological mirror symmetry [20].
Example 7. Let E be the elliptic curve over a field k (characteristic ka2; 3) in projective coordinates
ð19Þ
ARTICLE IN PRESS 328
N.M. Glazunov / Nuclear Instruments and Methods in Physics Research A 534 (2004) 324–328
6. Conclusions Categorification of mathematical and physical structures and problems in the frame of homological algebra gives in many cases methods of simplification and linearization of the problems and structures and methods of validated numerics [21–27]. In the framework the decomposition on indecomposable objects reduced the complexity of a problem. Tools from representation theory and homological algebra (differential graded algebras and categories, quivers, derived categories and functors, 2-categories, A1 -categories) give a relevant framework for the simplification and linearization. Acknowledgements I would like to thank the organizers of the ACAT’03 for providing a very pleasant environment during the conference and V. Ilyin for useful discussion as well as to all the participants of the A1 -category seminar at the Institute of Mathematics, Kiev, for attention and discussions, especially to V. Lyubashenko, S. Ovsienko and Yu. Bespalov. References [1] GLC Project, GLC Report Editional Group, KEK, 2003. [2] S. MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer, New York, 1971. [3] C. Vafa, Proceedings of ICM’98, Documenta Math. J. DMV, Vol. 1 (1998). [4] R. Hartshorne, Residues and duality, Lecture Notes in Mathematics, Vol. 20, Springer, New York, 1966.
[5] Y. Kawamata, Equivalences of derived categories of sheaves on smooth stacks, E-print http://arXiv.org/abs/ math.AG/0210439. [6] J.W. Cooley, J.W. Tukey, Math. Comp. 19 (1965). [7] M. Clausen, U. Baum, Fast Fourier Transforms, Wissenschaftsverlag, Mannheim, 1993. [8] Yu.A. Drozd, V.V. Kirichenko, Finite Dimensional Algebras, Vyscha shkola, Kiev, 1980. [9] A.V. Roiter, Representations of marked quivers, E-print http://arXiv.org/abs/math.RT/0112158. [10] I.M. Gel’fand, V.A. Ponomarev, J. Ruos. Math. Surv. 23 (2) (1968) 1; translation from Uspehi Math. Nauk. 140 (1968) 3. [11] J.-P. Serre, Lie algebras and groups, Lectures given at Harvard University, Benjamin, New York, 1965. [12] I.R. Shafarevitch, Foundations of Algebraic Geometry. V.1,2, Nauka, Moscow, 1988. [13] Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978. [14] S.I. Gelfand, Yu.I. Manin, Methods of Homological Algebra, Nauka, Moscow, 1988. [15] D. Mumford, Abelian Varieties, Tata Institute, Bombay, 1968. [16] I.R. Shafarevitch, Docl. AN USSR 114 (2) (1957). [17] S. Mukai, Nagoya Math. J. 81 (1981). [18] D. Orlov, E-print, http://arXiv.org/abs/alg-geom/9606006, 1997. [19] K. Fukaya, in: H.J. Kim (Ed.), The Proceedings of the 1993 GARC Workshop on Geometry and Topology, Seoul National University, 1993. [20] M. Kontsevich, Homological algebra of mirror symmetry, in Proceedings of ICM’94, Zurich, Birkhauser, Basel, 1995. [21] N. Glazunov, E-print, http://arXiv.org/abs/hep-th/ 0311009. [22] N. Glazunov, Problemy Programmirovaniya, No. 3-4, Institute of Software Systems NASU, Kiev, 2002. [23] N. Glazunov, E-print, http://arXiv.org/abs/math.CT/ 0311021. [24] N. Glazunov, Ju. Kapitonova, Cybernetics Syst. Anal. 1 (2003). [25] N. Glazunov, E-print, http://arXiv.org/abs/hep-th/ 0112250. [26] N. Glazunov, Complex Control Systems 109 (1997). [27] N.M. Glazunov, Nucl. Instr. and Meth. Phys. Res. 502 (2–3) (2003).
Nuclear Instruments and Methods in Physics Research A 534 (2004) 324–328 www.elsevier.com/locate/nima
Categorification of Fourier transforms, efficient computation and computing intelligence N.M. Glazunov1 Glushkov Institute of Cybernetics, National Academy of Science, 40, Prospect Glushkova, 03187 Kiev GSP-680, Ukraine Available online 2 August 2004
Abstract We investigate a category theory and homological algebra framework for various types of Fourier transforms and efficient methods for their implementation. r 2004 Elsevier B.V. All rights reserved. PACS: 02.10.Ws; 02.30.Nw; 02.70.+c; 11 Keywords: Fast Fourier transform; Homological algebra; Indecomposable object; Fourier–Mukai
1. Introduction: motivation Evaluation of Feynman diagrams, future experiments in the history of the early Universe and their processing [1], investigations and computations to string theory and some other researches result in many cases in necessity for (i) consideration of many facets of the problem with their interactions and (ii) efficient computations. An overwhelming majority of the fundamental methods for Physics have set theoretic foundations. A category theory framework (by means of a category C or a functor F) for a system S; a process P or a phenomenon F can be thought as a
collection of objects A; B; . . . ; one for each element of S (respectively P; F) which are combined by morphisms f : A ! B; subject to the well-known conditions of associativity and identity [2]. The principle of System Analysis assumes that plant, process or problem are studied as a system with the consideration of all facets of the system with their interactions. If we can represent these facets as categories and interactions as functors we will obtain a mathematical model of the system. Let V be an n-dimensional vector space with a dot-product as a symmetric bilinear form h ; i over a field k: An n-dimensional lattice L in V has a dual lattice L given by L ¼ fy 2 V j hx; yi 2 Zg
E-mail address: [email protected] (N.M. Glazunov). 1 Participation in ACAT’03 is supported by the Local Organizing Committee.
ð1Þ
for all x 2 L: If V ; W are n-dimensional vector spaces over k and there is a perfect pairing between
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.07.082
ARTICLE IN PRESS N.M. Glazunov / Nuclear Instruments and Methods in Physics Research A 534 (2004) 324–328
V and W then the pairing defines the duality between them. There are many generalizations and physical interpretations of dualities. Sometimes researchers say that two theories are dual when they both describe the same physics. A Fourier transform is an example of duality that transforms a difficult mathematical question (for instance the problem of solution a system of differential equations) to an easy problem (of the solution of a system of algebraic equations). Please see the discussion of dualities in Vafa’s paper [3]. One of the useful notions of category theory is the notion of equivalence [2]. Dualities and equivalences are very close [4,5]. Computing intelligence includes processing of information (encoding, decoding, filtering, compression), efficient and reliable computations, digital image processing, ranking data and information, sparse approximation and learning. Cooley and Tukey and other authors have demonstrated that the classical fast Fourier transform has successful applications to all of the problems [6,7]. In the paper, we illustrate the consequences for the category theory treatment of data representation and analysis, scientific computations, models of interaction and computing intelligence. In order to make the paper easy to read we will illustrate definitions of the basic notions by examples.
2. Preliminaries from algebra and category theory Let A be an algebra over a field k and V a vector space. A representation [8,9] of k-algebra A is a homomorphism h from A h : A ! MðV Þ
ð2Þ
to the algebra MðV Þ of linear operators of V : If h is an monomorphism (hðAÞ ’ A) then the representation is the exact. Indecomposable representations are very useful in representation theory [10]. A category C [2] can be thought as a collection of objects A; B; . . . ; which are combined by morphisms f : A ! B; subject to the well-known
325
conditions (i) and (ii) below. Let CðA; BÞ be the class of morphisms from A to B: Then (i) for any three (not necessary distinct) objects A; B or C of C; there is defined a map CðA; BÞ CðB; CÞ ! CðB; CÞ
ð3Þ
called composition which satisfied the associativity axiom. (ii) For every objects A of C; the set CðA; AÞ contains a morphism idA ; called the identity of A: A functor F from a category C to a category K is a function which maps ObðCÞ ! ObðKÞ; and which for each pair A; B of objects of C maps CðA; BÞ ! CðFðAÞ; FðBÞÞ; while satisfying the two well-known conditions [2]. 2.1. Equivalence of Categories Let C be a category [2], Ob C its class of objects, and for a; b 2 Ob C; Cða; bÞ the class of morphisms (arrows) from a to b: An arrow u : a ! b is the equivalence in C if there is an arrow u0 : b ! a such that u0 u ¼ 1a and uu0 ¼ 1b : A functor F from a category C to a category K is a function which maps ObðCÞ ! ObðKÞ; and which for each pair a; b of objects of C maps Cða; bÞ ! KðFðaÞ; FðbÞÞ; while satisfying the two conditions: Fida ¼ idFa for every a 2 ObðCÞ FðfgÞ ¼ Fðf ÞFðgÞ: Let FunctðC; DÞ be the category of functors from C to D with natural transformations as morphisms. An equivalence in FunctðC; DÞ is the equivalence between categories C and D: Example 1. The next result is a particular case of the theorem from [2]. A functor F : C ! D is the equivalence if, and only if (a) F is fully faithful, and (b) any Y 2 Ob D is isomorphic to an object of the form FðX Þ for an object X 2 Ob C: Example 2. Let k be a Q-algebra. The category of formal groups over k is equivalent to the category of finite dimensional Lie k-algebras [11].
ARTICLE IN PRESS N.M. Glazunov / Nuclear Instruments and Methods in Physics Research A 534 (2004) 324–328
326
Task. Compute the product of f and g with complexity Oðn log nÞ complex arithmetic operations.
3. Discrete Fourier transforms and their categorification 3.1. Discrete Fourier transforms Here we follow [7]. Let G be a finite group, #G the number of its elements, CG its group algebra. Theorem 1 (Wedderburn). The group algebra CG of G is isomorphic to an algebra of block diagonal matrices: d j d j CG ’ m j¼1 C
ð4Þ
The number m of blocks equals the number of conjugacy classes of G, and the d j are uniquely determined by G up to permutation. #G ¼ d 21 þ þ d 2m :
ð5Þ
Definition 1. Every isomorphism d j d j F : CG ! m j¼1 C
ð6Þ
of C-algebras is called a discrete Fourier transform (DFT) for CG: Example 3. Discrete Fourier transform and the generation of pseudorandom sequences. Let p be an odd natural number, F2 ¼ Z=2Z the binary field, s ¼ fsi giX0 a binary p-periodic sequence. Task. Compute for all kX0 the minimal m ¼ mðsÞ of a linear recursion X skþm ¼ ci skþi ð7Þ iom
Solution. (Cooley–Tukey). Apply the fast Fourier transform ðors Þr;son for computing the cyclic convolution of the polynomials. 3.2. On categorification of discrete Fourier transforms Let k be a field, Algk the category of algebras over k: Objects of Algk are k-algebras and morphisms are k-homomorphisms of the algebras. Algk contains many subcategories. Let Gk be the category of finite groups, GAlgk the category of group k-algebras. Let Mk the category of all square matrices over k: Objects of Mk are natural numbers and set of morphisms HomðnÞ is the set of all square matrices kn n : We may consider matrices kd 1 d 1 kd m d m
ð8Þ
as block diagonal matrices of the size d 1 þ þ d m:
ð9Þ
For every natural n the set HomðnÞ is a k-algebra, so an object of Algk : For k ¼ C a discrete Fourier transform of an object kG from GAlgk is an isomorphism with an object of the form d j d j m j¼1 C
ð10Þ
d 21 þ þ d 2m ¼ #G:
ð11Þ
4. Varieties and sheaves
that generates a given periodic sequence s: Solution. (E. Berlecamp, J. Massey). The minimal m ¼ mðsÞ of s equals the Hamming weight of its Fourier spectrum. Example 4. Fast multiplication of complex polynomials. Let n be a natural number, f ; g complex polynomials such that the sum of their degrees less then n; o a primitive nth root of unity.
In the section, we follow Refs. [12–14]. At first collect some notions related to ringed spaces. The ringed space is the pair ðX ; OÞ; where X is a topological space and O is a sheaf of rings on X : For a ringed space ðX ; OX Þ and an open U X the restriction of the sheaf OX on U defines the ringed space ðU; OX jU Þ: Let OX be the sheaf of smooth functions on a topological space X : Then any smooth manifold X with the sheaf OX is a ringed space ðX ; OX Þ:
ARTICLE IN PRESS N.M. Glazunov / Nuclear Instruments and Methods in Physics Research A 534 (2004) 324–328
Example 5. Let X be a Hausdorff topological space and OX a sheaf on X : Let it satisfy conditions: (i) OX is the sheaf of algebras over C; (ii) OX is a subsheaf of the sheaf of continuous complex valued functions. Let W be a domain in Cn and Oan the sheaf of analytical functions on W : The ringed space ðX ; OX Þ is called the complex analytical manifold if for any point x 2 X there exists a neighbourhood U 3 x such that ðU; OX jU Þ ’ ðW ; Oan Þ (here ’ denotes the isomorphism of ringed spaces). Let X be a projective algebraic variety over C and CohX the category of coherent sheaves on X : Example 6. If X is the algebraic curve then indecomposable objects of CohX are skyscrapers and indecomposable vector bundles. Recall now the relevant properties of complexes, derived categories, cohomologies and quasimorphisms referring to [13,14] for details and indication of proofs. The cochain complex d
d
d
ðK ; dÞ ¼ fK 0 ! K 1 ! K 2 ! g
ð12Þ
is the sequence of abelian groups and differentials d : K p ! K pþ1 with the condition d d ¼ 0: Let A be an abelian category, KðAÞ the category of complexes over A: Furthermore, there are various full subcategories of KðAÞ whose respective objects are the complexes which are bounded below, bounded above, bounded in both sides. The notions of homotopy morphism, homotopy category, triangulated category are described in Ref. [14]. The bounded derived category Db ðX Þ of coherent sheaves on X has the structure of a triangulated category [14].
327
of the form E : ax3 þ by3 þ cz3 ¼ 0; abca0:
ð13Þ
Its Jacobian is Ref. [16] E^ : abcx3 þ y3 þ z3 ¼ 0:
ð14Þ
The Poincare´ bundle P is a line bundle of degree ^ defined in such a way zero on the product A A; ^ that for all a 2 A the restriction of P on A fag is isomorphic to the line bundle corresponding to the ^ This line bundle is also called the point a 2 A: universal bundle. Let pA : A A^ ! A
ð15Þ
pA^ : A^ A ! A^
ð16Þ
and CA be the category of OA -modules over A; M 2 Ob CA ; ^ SðMÞ ¼ pA; ^ ðP pA MÞ:
ð17Þ
Then, by definition, the Fourier–Mukai transform ^ FM is the derived functor RS^ of the functor S: ^ be bounded derived categories Let Db ðAÞ; Db ðAÞ ^ respectively. of coherent sheaves on A and A; Mukai has proved Ref. [17]. Theorem 2. The derived functor FM ¼ RS^ induces an equivalence of categories between two derived ^ and Db ðAÞ: categories Db ðAÞ Let X and Y be two connected smooth projective varieties. By Orlov’s theorem [18] any derived equivalence F : Db ðY Þ ! Db ðX Þ
ð18Þ
comes from a complex on the product Y X : Equivalences of derived categories 5. Fourier–Mukai transforms
FM : Db ðY Þ ! Db ðX Þ
Let A be an abelian variety and A^ the dual abelian variety which is by definition a moduli space of line bundles of degree zero on A [15].
is called Fourier–Mukai transforms. There are interesting connections between A1 categories and Fourier–Mukai transforms. A1 categories was introduced by Fukaya [19]. These categories was used in Kontsevich’s conjecture on homological mirror symmetry [20].
Example 7. Let E be the elliptic curve over a field k (characteristic ka2; 3) in projective coordinates
ð19Þ
ARTICLE IN PRESS 328
N.M. Glazunov / Nuclear Instruments and Methods in Physics Research A 534 (2004) 324–328
6. Conclusions Categorification of mathematical and physical structures and problems in the frame of homological algebra gives in many cases methods of simplification and linearization of the problems and structures and methods of validated numerics [21–27]. In the framework the decomposition on indecomposable objects reduced the complexity of a problem. Tools from representation theory and homological algebra (differential graded algebras and categories, quivers, derived categories and functors, 2-categories, A1 -categories) give a relevant framework for the simplification and linearization. Acknowledgements I would like to thank the organizers of the ACAT’03 for providing a very pleasant environment during the conference and V. Ilyin for useful discussion as well as to all the participants of the A1 -category seminar at the Institute of Mathematics, Kiev, for attention and discussions, especially to V. Lyubashenko, S. Ovsienko and Yu. Bespalov. References [1] GLC Project, GLC Report Editional Group, KEK, 2003. [2] S. MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer, New York, 1971. [3] C. Vafa, Proceedings of ICM’98, Documenta Math. J. DMV, Vol. 1 (1998). [4] R. Hartshorne, Residues and duality, Lecture Notes in Mathematics, Vol. 20, Springer, New York, 1966.
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