Much ado about 248

Muc h ado ab out 248 M.C. Nucci and P .G.L. Leach ∗ Diparti men t o di Matema tica e Info rmati ca, Universit` a di Perugia, 0612 3 P er u g ia, Italy Abstract In this note w e present t h ree represen tations of a 248-dimensional L ie algebra, namely the al gebra of Lie p oin t symmetries adm itted b y a system of fiv e trivial ordinary differen tial equations eac h of order fort y-four, that admitted b y a system of sev en trivial ord inary differen tial equations eac h of order tw en ty- eight and that a d mitted b y one trivial ordinary differential equation of order t wo h undr ed and fort y-four. 1 In tro duction A system o f n ordinary differential equations each of order M > 1, u ( M ) k = f k ( u ( s ) j , t ) , j, k = 1 , n, s = 0 , M − 1 , (1) has a v ariable n um b er o f Lie p o int symme tr ies dep ending up on the structure of the functions f k . The maximal dimension D of the alg ebra of admitted Lie p oint symmetries can b e obtained b y the formulæ [9] M = 2 = ⇒ D = n 2 + 4 n + 3 (2) M > 2 = ⇒ D = n 2 + M n + 3 . (3) Some explicit n um b ers are giv en in T able 1. Recen tly the elab oration of the elemen ts of the L ie algebra, E 8, of order 248 has b een v ariously anno unced [3, 7, 13 , 17, 16] in the serious p opular media. The au- thoritativ e source is the A tlas of Lie Groups and Represen tations [2] whic h is funded b y the National Science F oundation through the American Institute of Mathematics [1]. The results of the E8 computation w ere announced in a talk at MIT by Da vid V og an on Monda y , Marc h 19, 2 0 07, and the details may be found at [1 5]. The A tla s of Lie Groups and Represen tations is a pro ject to make a v ailable info rmation a b out represen tations of semisimple Lie g roups o ve r real and p -adic fields. Of particular imp ortance is the problem of the unitary dual, ie the classification of all of the ir- reducible unitary represen tations of a give n L ie g r o up. The goal of the Atlas of Lie ∗ per manent a ddress: School of Mathematical Sciences, W estville Campus, Univ ersity of KwaZulu-Natal, Durban 4000 , Republic of South Africa 1 T able 1 : : The maximal dimension o f the algebra of admitted L ie p oin t symmetries for systems of equations of v arying order (horizontal) and n umber (v ertical). n \ M 2 3 4 5 6 7 8 9 10 1 8 7 8 9 10 11 12 13 1 4 2 15 1 3 15 17 19 21 23 25 27 3 24 2 1 24 27 30 33 36 39 42 4 35 3 1 35 39 43 47 51 55 59 5 48 4 3 48 53 58 63 68 73 78 6 63 5 7 63 69 75 81 87 93 99 7 80 7 3 80 87 94 101 108 115 122 8 99 9 1 99 1 07 115 123 131 139 147 9 120 111 120 129 138 147 156 165 174 10 143 133 1 43 153 163 173 183 193 203 Groups a nd Represen tations is to classify t he unitary dua l of a real Lie group, G , b y computer. A step in this direction is to compute the admissible represen tations of G including their Ka zhdan-Lusztig-V og an p olynomials. The computation for E 8 w as an imp ortant test of t he tec hnology . While the computation is an impressiv e ac hiev emen t, it is only a small step to w ar ds the unitary dual and should not b e rank ed as imp ortant as the original w ork of Kazhdan, Lusztig, V ogan, Beilinson, Bernstein et al . (See fo r example [4, 5, 6, 11, 12, 14, 18 , 8].) Nev ertheless the result w as regarded as b eing suitable for a concerted campaig n of publicit y to heighten a warene ss of Mathematics in the comm unit y at large: “Symmetrie ist m¨ oglic herw eise das erfolgreic hste Prinzip der Ph ysik ¨ uberhaupt” [7]. “Un g r oup e de cherc heurs am´ ericains et europ ´ eens, parmi lesquels o n trouve deux F ran¸ cais, est parven u ` a d´ eco der une des structures les plus v astes de l’histoire des math ´ ematiques” [13]. “It ma y b e that some day this calculation can help phy sicists to understand the univ erse” [1 7]. “Eigh teen mathematicians sp en t f o ur y ears and 77 hours o f sup ercomputer compu- tation to describ e this structure” [16]. In this note w e demonstrate three represen tations of a Lie algebra of dimension 248. The tw o of us sp ent four hours and 77 seconds o f p o ck et-calculator computation to describ e these t hree structures. 2 Three simple sys tems F or D = 248 for mula (2) do es not hav e in tegra l solutions and so there is no system of second-order ordinary differential equations of ma ximal symmetry p o ssessing a 248-dimensional algebra of it s Lie p oint symm etries 1 . Ab out formula (3) the factors of 248-3=245 are 1, 5 and 7 (49 is out o f que stion b ecause 49 2 > 245). Consequen tly 1 Is this another instance of the intrinsically uniqueness of Classical Mec hanics? 2 p ossible v alues o f n are 1, 5 and 7. The corresp onding v alues of M are 244, 44 and 28, resp ectiv ely . The systems of maximal symmetry are easily obtained a s one simply puts f k = 0 ∀ k . Th us the systems w e construct are the simplest represen tatio ns of the eq uiv alence class under p oin t transformation of systems of equations o f maximal symmetry . Firstly w e consider the following system: u (44) k = 0 , k = 1 , 5 . (4) It is easy to sho w that this simple system admits a 24 8 -dimensional algebra of its Lie p o int symmetries since 5 2 + 5 · 44 + 3 = 248. The algebra is generated b y the op erators Γ 1 = t 2 ∂ t + 43 t P 5 i =1 u i ∂ u i , Γ 2 = t∂ t , Γ 3 = ∂ t , Γ i,k = u k ∂ u i , k = 1 , 5 , i = 1 , 5 Γ i +5 ,s = t s ∂ u i , s = 0 , 43 , i = 1 , 5 . (5) Secondly w e consider the system u (28) r = 0 , r = 1 , 7 . (6) This equally simple system admits a 248 -dimensional algebra (7 2 + 7 · 28 + 3 = 248) of it s Lie p o in t symmetries generated b y Γ 1 = t 2 ∂ t + 27 t P 7 j =1 u j ∂ u j , Γ 2 = t∂ t , Γ 3 = ∂ t , Γ j,r = u r ∂ u j , r = 1 , 7 , j = 1 , 7 Γ j +7 , n = t n ∂ u j , n = 0 , 27 , j = 1 , 7 . (7) Thirdly and finally the scalar equation, u (244) = 0 , (8) admits a 248-dimensional Lie algebra (1 2 + 1 · 244 + 3 = 248) of its p oin t symmetries generated b y the op erato rs Γ 1 = t 2 ∂ t + 243 tu∂ u , Γ 2 = t∂ t , Γ 3 = ∂ t , Γ 4 = u∂ u , Γ n +5 = t n ∂ u , n = 0 , 243 . (9) 3 3 Conclus ion W e hav e demonstrated three represen tatio ns of Lie algebras of dimension 248 whic h is the dimension o f E 8 . Although the algebras we pr esen t are not simple, their metho d of construction is. The reason for this simplicity is that we used represen- tations for systems of equations of maximal symmetry . W e do not den y that larger systems , b e that in order or nu mber, of less than maximal symmetry could p ossibly ha v e an alg ebra of dimension 248, but ev en on the assumption that suc h systems b e linear the complexit y of the calculation b ecomes immense [10] and defeats the purp ose of the presen t note. Note that w e hav e used the simplest f o rms for the g enerators of t he algebras of the three systems , (4) , (6) and (8), for our primary in terest is the demonstration of the existence o f the algebras. Normally one w ould use comb ina t ions which reflect subalgebraic structures. F or example in the case o f (8 ) fo r whic h the algebra is ob viously sl (250 , I R ) one w ould replace Γ 2 with ˜ Γ 2 = 2 t∂ t + 243 u∂ u to underline the subalgebraic structure { sl (2 , I R ) ⊕ A 1 } ⊕ s 244 A 1 , where Γ 1 , ˜ Γ 2 and Γ 3 constitute a represen tation of sl (2 , I R ), Γ 4 reflects the homogeneit y of the equation in the dep en- den t v ariable and the 2 4 4-elemen t ab elian subalgebra is comp osed of the solution symmetries , so called b ecause the co efficien t functions are solutions of (8). Ac kno wle dgemen ts PGLL tha nks the Univers ity of Kw azulu-Natal for it s contin ued supp ort. 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