The Fundamental Basis Theorem of Geometry from an algebraic point of view

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS The Fundaental Basis Theore of Geoetry fro an algebraic point of view To cite this article: U Bekbaev 2017 J Phys: Conf Ser View the article online for updates and enhanceents Related content - THE ALGEBRAIC ANALOG OF COMPACT COMPLEX SPACES WITH A SUFFICIENTLYLARGE FIELD OF MEROMORPHIC FUNCTIONS II B G Mošezon - New Geoetry with All Killing Vectors Spanning the Poincaré Algebra Huang Chao-Guang Tian Yu Wu Xiao- Ning et al - THE ALGEBRAIC ANALOG OF COMPACT COMPLEX SPACES WITH A SUFFICIENTLYLARGE FIELD OF MEROMORPHIC FUNCTIONS I B G Mošezon This content was downloaded fro IP address on 05/07/2018 at 04:46

2 37th International Conference on Quantu Probability and Related Topics QP37 IOP Conf Series: Journal of Physics: Conf Series doi:101088/ /819/1/ International Conference on Recent Trends in Physics 2016 ICRTP2016 Journal of Physics: Conference Series doi:101088/ /755/1/ The Fundaental Basis Theore of Geoetry fro an algebraic point of view U Bekbaev Departent of Science in Engineering Faculty of Engineering International Islaic University Malaysia PO Box Kuala Lupur Malaysia E-ail: bekbaev@iiueduy Abstract An algebraic analog of the Fundaental Basis Theore of geoetry is offered with a pure algebraic proof involving the faous Waring s proble for polynoials Unlike the geoetry case the offered syste of invariant differential operators is couting which is a new result even in the classical geoetry of surfaces Moreover the algebraic analog works in ore general settings then does the Fundaental Basis Theore of geoetry 1 Introduction Let n be any fixed natural nubers H be any subgroup of the general linear group GLn R B R be an open unit ball and xt x 1 t 1 t 2 t x 2 t 1 t 2 t x n t 1 t 2 t be infinitely sooth -paraetric variable surface in R n where R is the field of real nubers The surface xt is assued to be written in the row for Definition 11 An infinitely sooth function f xt of xt x 1 t x 2 t x n t and its finite nuber of derivatives with respect to t 1 t 2 t is said to be Ĝ H-invariant of the surface if the equality f δ xsth f xt holds true for any h H t B and s Ĝ where Ĝ DifB stands for the group of diffeoorphiss of B stands for t 1 t 2 t st s 1 t s 2 t s t and δ i s i t i 1 2 One of the iportant probles of differential geoetry is the description of all such invariant functions Rx ĜH It is called the set of differential invariant functions of -diensional surfaces with respect to the otion group H The Fundaental Basis Theore of geoetry first forulated by Lie states that there exist Ĝ H-invariant differential operators δ1 δ 2 δ and a finite syste of the Ĝ H-invariant functions such that locally any Ĝ H-invariant function is a function of this finite syste of eleents and their finite nuber of derivatives with respect to the δ 1 δ 2 δ Note that the Fundaental Basis Theore does not state the existence of such couting syste of invariant differential operators But it states that the coutators δ i δ j δ j δ i i j 1 2 can be represented as linear cobinations of δ 1 δ 2 δ For the Content fro this work ay be used under the ters of the Creative Coons Attribution 30 licence Any further distribution of this work ust aintain attribution to the authors and the title of the work journal citation and DOI Published under licence by Ltd 1

3 37th International Conference on Quantu Probability and Related Topics QP37 IOP Conf Series: Journal of Physics: Conf Series doi:101088/ /819/1/ ore detailed inforation about the Fundaental Basis Theore of geoetry one can see [1 2] We note also that the ain ethod of geoetry in dealing with differential invariant functions and invariant differentials are the The Moving Frae Method and its generalizations [3] In the current paper an algebraic analog of the Fundaental Basis Theore of geoetry is offered with a pure algebraic proof involving the faous Waring s proble for polynoials Moreover it is shown that the syste of invariant differential operators δ 1 δ 2 δ can be chosen couting which is a new result even in classical differential geoetry of surfaces In addition our approach works in ore general settings than does The Moving Frae Method To forulate an algebraic analog of the Fundaental Basis Theore note that the above change of paraeters t t 1 t can be presented in the following for t i j1 s j t t i s i ie δ g 1 where g is atrix with the eleents gj i s jt t i i j 1 and δ is the colun vector with the coordinates t 1 t 2 t respectively s 1 s 2 s Moreover every infinitely sooth paraeterized surface x : B R n can be regarded as an eleent of differential odule F n ; 1 2 with the coordinate-wise action of i t i on eleents of F n where F C B is the differential ring of infinitely sooth functions relative to differential operators t 1 t 1 t If eleents of this odule are written in the row for the above considered transforations look like x x 1 x 2 x n xh g 1 where g is a atrix with eleents g i j s jt t i i j 1 2 h H s Ĝ Therefore the following algebraic analogue of the above proble is natural Let F ; be any characteristic zero differential field where is a colun-vector of couting syste of differential operators 1 2 of F C {a F : i a 0 for i 1 2 } be its constant field H be a subgroup of GLn C and GL F {g GL F : i g j k j g i k for i j k 1 2 } One can verify easily that whenever g GL F the syste δ 1 δ 2 δ is also a couting syste of differential operators of F where δ g 1 This is an analogue of gauge transforations change of variables for the abstract differential field F ; In general the set GL F is not a group with respect to the ordinary product of atrices as far as it is not closed with respect to that product But by the use of it a natural groupoid [4] can be constructed with the base {g 1 : g GL F } Further let x 1 x n be differential algebraic independent variables over F x stand for the row vector with coordinates x 1 x 2 x n C x be the field of -differential rational functions in x over C and G GL F Definition 12 An eleent f x C x; is called to be G H- invariant if the equality holds true for any g G h H f g 1 xh f x 2

4 37th International Conference on Quantu Probability and Related Topics QP37 IOP Conf Series: Journal of Physics: Conf Series doi:101088/ /819/1/ We denote by C x; GH the field of all such G H- invariant -differential rational functions Fro the algebraic point of view this field is a natural analog of Rx ĜH and as such it is worthy to be described The organization of the paper is as follows In the next section with its own notations a generalization of classical Faa di Bruno forula is presented via the syetric tensor product of atrices This generalization is used to prove an algebraic analog of the Fundaental Basis Theore of geoetry in the next section In the hypersurface case a siilar result have been obtained in [7] by another algebraic ethod the finite group H case is considered in [8] The used notions of differential algebra can be found in [5] 2 A generalization of Faa di Bruno forula The following classical Faa di Bruno forula on higher order derivatives of the coposite functions is well known: For sooth functions of one variable f fx g gt and natural nuber the following equality d dt fgt f k gt k1 k t! g 1! 1 g t 2 g t holds true where 1 2! 1! 2!! It can be proved by induction on taking into account the additive the Leibniz and chain rule properties of the derivative Of course the fact that eleents g t g t coute with each other is used silently d dgt d dx Note that if one uses notations d : d dt g : g t δ : rule and the above entioned forula can be written in the for d k1 k! g 1! 1 dg 2 d 1 g δk then d gδ the chain In this section we are going to show that the sae forula is true in partial derivatives ultivariate case but to do it we need the syetric tensor product The ain properties of it with proofs can be found in [9 10] Here soe needed definitions and properties are presented without proofs except for soe results which are very closely related to the proof of the generalized Faa di Bruno forula Let n be any positive integer and I n stand for all row n-tuples with nonnegative integer entries with the following linear order: 1 2 n < 1 2 n if and only if < or and 1 > 1 or 1 1 and 2 > 2 etcetera We write! if i i for all i 1 2 n stands for!! We allow n to be zero also and in this case it is accepted that I n {0} Consider any associative algebra A with 1 over a field of zero characteristic and let C stand for the center of A For any nonnegative integer nubers p p let M np p; A Mp p; A stand for all p p size atrices A A p p presents row presents colun and I I n with entries fro A p The ordinary size of such atrix is + 1 p + Over this kind of 1 atrices in addition to the ordinary su and product of atrices we consider the following syetric product as well [9 10]: 3

5 37th International Conference on Quantu Probability and Related Topics QP37 IOP Conf Series: Journal of Physics: Conf Series doi:101088/ /819/1/ Definition 21 If A Mp p; A and B Mq q; A then A B C Mp + q p + q; A such that for any p + q p + q where I I n C A B A B where the su is taken over all I I n for which p p We use the syetric tensor product as well when all coponents of A are linear aps fro A to A and all coponents of B are also linear aps fro A to A or all coponents of it are in A In the first case A B stands for coposition A B and in the second case it stands for A B In future A eans the -th power of atrix A with respect to the product Proposition 22 a A B B A whenever all entries of A or B are in C b λ 1 A + λ 2 B C λ 1 A C + λ 2 B C and A λ 1 B + λ 1 C λ 1 A B + λ 2 A C whenever λ 1 λ 2 C c A B C A B C d AB H AB H for any row H M0 p; A e For any natural v M1 0; C and h M0 1; C one has v 0 v h 0 h where h stands for h 1 1 h 2 2 h n n Let 1 2 n δ δ 1 δ 2 δ n be colun vectors of couting systes of differential operators of the algebra A for which gδ where gj i C k gj i i gj k for all i k 1 2 j 1 2 n In ordinary one variable derivative d case the change of the variable eans a change of d according to d a 1 d where a F In this case to prove the Faa di Bruno forula one needs only additive property of d the Leibniz law and that eleents {d k a} k12 coute with each other Therefore to prove siilar forula in partial ultivariate derivatives case one should have an appropriate product for which the above listed properties of d hold true for the partial derivatives The following result says that for an appropriate product one can take the syetric tensor product Lea 23 The following are true a A B A B + A B the Leibniz law b Ap qbq p AB + 1 q+1 A gδ B whenever δ k B e k k q + 1 δ B k δ B for all k 1 2 n I n q + 1 and p Here AB stands for the ordinary product of atrices A and B all coponents of e i I n are zero except for i-th which is 1 c If δ k B e k k δ B for all k 1 2 n I n q + 1 and p then the atrix δ B also has siilar property: δ k δ B e k k δ δ B for all k 1 2 n I n q + 2 and p d A δ δ A + A g δ +1 +1! the generalized chain rule 4

6 37th International Conference on Quantu Probability and Related Topics QP37 IOP Conf Series: Journal of Physics: Conf Series doi:101088/ /819/1/ Proof We use coponent-wise checking a A B i1 i A B e i n i1 i i1 i A B e i i1 + i1 i A e i B + A B + A A B e i A i B e i i1 i B e i A B A B + A B A B + A B b AB i1 i AB e i i1 i A e i B i1 i A e i B + A e i i B A B + i1 A e i n gjδ i j B j1 Note that A B AB and A e i i1 n j1 g i jδ j B n i1 j1 A e i g i j j δ B+e j n i1 j1 A e i gj i j + 1 q + 1 δ B+e j A g i1 j1 n ej A e i e j g i j n i1 j1 1 q + 1 δ B 1 q + 1 A e i e j gj i 1 q + 1 δ B j A gδ B c Proof of it is siilar to the b case d It is a consequence of the second relation as far as δ has property δ k δ e k for any k 1 2 n I n and + 1 k + 1 δ +1 q + 1 δ B So we have the following properties of : Additive property Leibniz law Lea 23 a the generalized chain rule Lea 23 d and k g k012 coute with each other with respect to In the last case we are using the fact that the center C is invariant with respect to derivatives Therefore without repeating routine induction on as in one variable case one ore tie we can forulate the following generalization of well known classical Faa di Bruno forula [11] 5

7 37th International Conference on Quantu Probability and Related Topics QP37 IOP Conf Series: Journal of Physics: Conf Series doi:101088/ /819/1/ Theore 24 Let A be any associative algebra with 1 over a field of zero characteristic and let C stand for the center of A If 1 2 n δ δ 1 δ 2 δ n are colun vectors of couting syste of differential operators of the algebra A for which gδ where g gj i i12 ;j12n gj i C k gj i i gj k for all i k 1 2 n j 1 2 n then for any natural the following equality is true k1 k 0 1! g 0 0! 1 g 1 1! 1 g 1 δ k 1! where 0 0! g 1 g g One can check directly that in n 1 case whenever k then! g 1! 0 dg 1 d 1 g 1 0 1! g 0 0! 1 g 1 1! 1 g 1 1! at any 1 k where d gδ For it one should take into account that i 1 i! g should be treated as i 1 size atrix though it s ordinary size is 1 1 and notice i 1 i! i 1 i! g i 1 j 1 j! g i 1 di 1 g i 1 i 1! i! g j 1 di 1 g i! i 1 dj 1 g j 1 i 1 + j 1! j! 3 The Fundaental Basis Theore of Geoetry As we have agreed in the introduction F ; stands for the characteristic zero differential field with the given couting syste of differential operators 1 2 of F C is its constant field H be a subgroup of GLn C and GL F {g GL F : i g j k j g i k for i j k 1 2 } Further it is assued that the syste of differential operators 1 2 is linear independent over F Definition 31 An eleent f x C x; A differential operator d x : C x; C x; is said to be G H- invariant if the equality holds true for any g G h H f g 1 xh f x respectively dg 1 xh d x Definition 32 An eleent f x C x; is said to be G H- relative invariant if there exist integer nubers k l such that the equality holds true for any g G h H f g 1 xh detg k deth l f x Proposition 33 If f 1 x f 2 x are two nonzero relative invariants with corresponding k 1 l 1 k 2 l 2 and k 1 0 k 2 0 then for the colun vector M 1 x f 2 x k 2 f 2 x f 1 x k 1 f 1 x one has M 1 x gm δ 1 xh for any g G h H 6

8 37th International Conference on Quantu Probability and Related Topics QP37 IOP Conf Series: Journal of Physics: Conf Series doi:101088/ /819/1/ Proof Application of to the both sides of the equality results in f g 1 i xh detg k i deth l i fi x gδ f δ i y deth l i k i detg k i 1 detgf i x + deth l i detg k i f i x where δ g 1 y xh Hence assuing k i 0 one has g δ f δ i y k i f δ i y detg detg + f i x k i f i x Therefore for the colun vector M1 x f 2 x k 2 f2 x f 1 x k 1 f1 x for any g G h H M g 1 1 xh g 1 M 1 x one has Theore 34 The Fundaental Basis Theore of Geoetry There exists a couting syste of G H- invariant differential operators δ δ 1 x δ 2 x δ x such that C x; GH is a finitely generated δ -differential field over C Proof The proof is constructive in ters of the syste of invariant differential operators For any natural k g G and δ g 1 due to Theore 24 one has the following representation 2 k g g g 0 k 1 g The nuber of equations rows in this equality is k The nuber of coluns of x l is + l Let us show that whenever 1 l k 1 < lk 2 < < lk j k g k δ δ 2 δ k 1 is a sequence of integers such that 31 1 j k i1 + l k i 32 one can construct a nontrivial G H-relative invariant Indeed in this case for h H xh y due to xh p p! x p h p p! p! for any natural p one has x j 1 j 1! x j2 j 2! x jk j k! Diagh j1 j 1! h j2 j 2! h jk j k! y j1 j 1! y j2 j 2! y jk j k! 7

9 37th International Conference on Quantu Probability and Related Topics QP37 IOP Conf Series: Journal of Physics: Conf Series doi:101088/ /819/1/ Due to Lea 23 b application of 31 to this equality results in 2 k x j1 j 1! x j2 x jk j 2! j k! Diagh j1 j 1! h j2 j 2! g g h jk j k! g y j1 j 1! y j2 y jk j 2! j k! k 1 g g k δ k By taking the deterinant of both sides of this equality one has k n + ji 1 i1 deth n where fk x stands for the det f k X 2 X k X x C x; is a relative invariant: δ δ 2 fk x detg + 1 X x j1 j 1! x j 2 f δ k y j 2! x j k j k! which eans that f g 1 k xh detg + 1 k n + ji 1 i1 deth n fk x 33 for any g G and h H Note that to get it we have used the equality n + p 1 det h p p! deth n for any square atrix h M nn 1 1; F and natural p [10] Let us now justify the existence of infinitely any values of k for which equality 32 holds true for soe 1 l1 k < lk 2 < < lk j k + k At any fixed n > 1 the nuber is the value of n 1-order polynoial without a prie divisor p[t] + t t + t + n 2t + 1! at t k The positive solution of Waring s proble for polynoials [6] guarantees the exists of a natural nuber s such that every ie not only of the for 1 natural nuber except finite of the can be represented as a su of values of this polynoial at s positive integers In our case we do not even need j k in 32 to be sae s for different values of k Therefore one can find a sequence 1 k 1 < k 2 < < k +1 of values of k for each of which 32 type equality holds true For each k k i we can construct the corresponding relative invariant 8

10 37th International Conference on Quantu Probability and Related Topics QP37 IOP Conf Series: Journal of Physics: Conf Series doi:101088/ /819/1/ f k i x and have the sequence of G H-relative invariants f k 1 x f k 2 x f k +1 x By the use of the one can construct a atrix M x consisting of coluns M i x f k i+1 x i f k i+1 x + f k 1 x fk 1 x i 1 2 Due to the construction the obtained atrix M x GL C x; is a nonsingular atrix for which the equality M g 1 xh g 1 M x holds true It iplies that the couting syste of differential operators δ δ 1 x δ 2 x δ x M x 1 is G H-invariant Therefore C x; GH is a finitely generated δ- differential field over C as a subfield of the finitely generated δ- differential field C x M x ; δ [5] This theore does not provide a ethod to find a finite syste of generators for the δ- differential field C x; GH over C We hope that the obtained syste of invariant differential operators can be used to reduce this proble to finding a syste of generators of the field of not differential invariants of an action of H as we have done it for patches case in [13] Note also that the Theore 34 holds true even if one changes G to any sub groupoid of GL F Reark 35 It is said that the Waring s proble for polynoials is valid in the following ore stronger for as well: If an integer valued at nonnegative integers polynoial p[t] with positive leading coefficient has nor prie divisor then there exists a natural nuber s such that every natural nuber except finite of the can be represented as a su of values of this polynoial at s distinct positive integers In this case the existence of representation 32 for all k except finite of the is an iediate consequence of it Exaple 36 Now as an application of the above presented results let us consider two diensional surfaces in R 3 that is 2 n 3 case where H ay be any subgroup of GL3 R In this case one has the following two sequences + i 1 k123 i123 { } { } Representing the eleents of the first sequence as the sus of different eleents of the second one when it is possible one has: which iplies that k 3 l1 3 1 l3 2 2 and due to 33 for f 3 x x; x 2 det 2 x; x 2 the equality f g 1 3 xh detg 10 deth 5 f3 x is valid 3 3! x; x which iplies that k 6 l1 6 2 l6 2 5 and due to 33 for x 2 2 x 2 f6 3 x det 3! x 2 4 4! x 2 5 x 2 6 6! x 2 9

11 37th International Conference on Quantu Probability and Related Topics QP37 IOP Conf Series: Journal of Physics: Conf Series doi:101088/ /819/1/ the equality f g 1 6 xh detg 56 deth 39 f6 x is true which iplies that k 8 l1 8 1 l8 2 3 l8 3 6 and due to 33 for f 8 x the equality f g 1 8 xh detg 120 deth 67 f8 x holds true So in this case one has colun vectors M 1 x f 6 x 56f 6 x f 3 x 10f 3 x M 2 x f 8 x 120f 8 x f 3 x 10f 3 x the second order square atrix M x consisting of these two coluns and the couting syste of invariant differential operators δ δ 1 δ 2 defined by δ M x 1 So even in this siple case expressions for couting syste of invariant differential operators δ 1 δ 2 in ters of initial 1 2 and x are quite coplicated nontrivial May be it is the reason that even in 2 n 3 case the classical Fundaental Basis theore didn t guarantee the existence of a couting syste of invariant differential operators with the needed properties 4 Conclusion In the paper a qualitative iproveent of the Fundaental Basis theore is provided by showing that the needed invariant syste of differential operators can be chosen couting A ethod to construct the needed syste of invariant differential operators is presented as well The approach of the paper to the proble is pure algebraic and is applicable in ore general settings than do geoetric ethods 5 Acknowledgeent I would like to thank Greg Martin for inforing e about paper [6] This research is supported by the MOHE Malaysia under grant FRGS References [1] Olver P J 2009 DiffGeoAppl [2] Olver P J 2011 Cotep Math [3] Olver P J and Pohjanpelto J 2008 Canadian J Math [4] Weinstein A Notices of the AMS [5] Kolchin E R 1973 Differential Algebra and Algebraic Groups New York: Acadeic Press [6] Kake E 1921 Math Ann [7] Bekbaev U 1999 In book: Proc Pengintegrasion Teknologi dala Sains Mateatik Penang USM Press [8] Bekbaev U 2005 Bull of Malays Math Sci Soc [9] Bekbaev U 2013 Inter J Pure Appl Math [10] Bekbaev U 2010 arxiv: [11] Bekbaev U 2012 In book: Inter Conf Operator algebras and related topics Tashkent National University of Uzbekistan Press [12] Bekbaev U 2015 arxiv: [13] Bekbaev U 2015 AIP Conf Proc