Combinatorial Deformations of Algebras: Twisting and Perturbations
The framework used to prove the multiplicative law deformation of the algebra of Feynman-Bender diagrams is a \textit{twisted shifted dual law} (in fact, twice). We give here a clear interpretation of its two parameters. The crossing parameter is a d…
Authors: ** - G. H. E. Duchamp (LIPN – Université Paris 13) - C. Tollu (Laboratoire de Physique Théorique de la Matière Condensée – Université Pierre et Marie Curie) - K. A. Penson (idem) - G. Koshevoy (Central Institute of Economics, Mathematics, Russian Academy of Sciences) **
Com binatorial Deformations of Alge bras: Twisting and P erturbatio n s G. H. E. Duchamp, C. Tollu, ∗ , K. A. Penson † and G. Koshev o y ‡ , Con ten ts 1 In tro duction 2 2 The deformed al gebra LDIA G ( q c , q s ) 3 2.1 Review of the construction of the algebra . . . . . . . . . . . . . . . . . . . 3 2.2 Co ding and the recursiv e definition . . . . . . . . . . . . . . . . . . . . . . 5 3 Colour factors and pro ducts 7 4 Sp ecial classes of laws 8 4.1 Dual la ws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.1.1 Algebras and coalgebras in duality . . . . . . . . . . . . . . . . . . 8 4.1.2 Dualit y b etw een grouplik e elemen ts a nd unities . . . . . . . . . . . 9 4.2 Deformed la ws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.3 Shifted la ws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5 Application t o the struct ure of LDIAG ( q c , q s ) 11 5.1 Asso ciativit y of LDI AG ( q c , q s ) using the previous to ols . . . . . . . . . . . 11 5.2 Structure of LDIA G ( q c , q s ) . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6 Conclusion 13 Abstract The framew ork u s ed to pro v e the multiplicat ive deformation of the algebra of F eynman-Bender diagrams is a twiste d shifte d dual law (in fact, t wisted t wice). W e giv e here a clea r interpretation of its t wo parameters. The crossin g p arameter is a deformation of the tensor structur e whereas the su p erp osition p arameter is a p er- turbation of the s h uffle coprod uct whic h, in turn , can b e in terpreted as the diagonal restriction of a sup erp r o duct. Here, we systematically detail these constructions. ∗ LIPN - UMR 7030 CNRS - Universit´ e P aris 13 F-9343 0 Villetaneus e, F r ance † Lab oratoir e de Physique Th´ eoriq ue de la Mati` ere Condens´ ee Universit´ e Pie r re et Mar ie C ur ie, CNRS UMR 7600 T o ur 24 - 2i` eme ´ etage, 4 place Jussieu, F 752 52 P aris cedex 0 5 ‡ Cent ra l Institute of Economics and Mathematics (CEMI) Russian Academy o f Sciences 1 1 In tro ductio n In [1], Bender, Bro dy , and Meister introduced a sp ecial field theory , then called “Quan tum Field Theory of P artitions”. This theory is based on a bilinear pro duct fo rm ula which reads H ( F , G ) = F z d dx G ( x ) x =0 . (1) If one dev elops this form ula in the case when F and G are free exp onentials, one obtains a summation o v er all the (finite) bipartite 1 graphs with multiple edges a nd no isolated p oin t [6] (the set of these diagra ms will b e called diag ), a data structure whic h is equiv a lent to classes of pac k ed matrices [8] under p ermutations of ro ws and columns. So, one has a F eynman-t yp e ex pansion of the pro duct f o rm ula H exp( ∞ X n =1 L n z n n ! ) , exp( ∞ X n =1 V n z n n ! ) ! = X n ≥ 0 z n n ! X d ∈ diag | d | = n mul t ( d ) L α ( d ) V β ( d ) (2) where mul t ( d ) is the num ber of pairs ( P 1 , P 2 ) of ( ordered) set partit ions of { 1 , . . . , n } whic h corresp ond to a diagram d , | d | the n um b er of edges in d and L α ( d ) = L α 1 1 L α 2 2 · · · ; V β ( d ) = V β 1 1 V β 2 2 · · · (3) is the m ultiindex notation for the monomials in L ∪ V where α i = α i ( d ) (resp. β j = β j ( d )) is the n um b er of white (resp. blac k) sp ots of degree i (resp. j ) in d . The set diag endo w ed with disjoint receiv e the structure of a monoid suc h that the arrow d 7→ L α ( d ) V β ( d ) is a morphism (of monoids) and then, b y linear ex tension, one deduces a morphism of algebras C [ diag ] → Pol( C ; L ∪ V ) . (4) where Pol( C ; L ∪ V ) is t he Hopf algebra o f (comm utative) p olynomials with complex co efficien ts generated by the alphab et L ∪ V . F or at least thr ee mo dels of Phys ics, one can sp ecialize L so that the canonical Hopf algebra structure of Pol( C ; L ∪ V ) can be lifted, through (4). The resulting Hopf algebra (based on C [ diag ]) has b een denoted DIAG . T o our gr eat surprise, this Hopf algebra structure could b e lifted at the (noncomm utativ e) lev el of the ob jects themselv e s instead o f classes , resulting in the construction o f a Hopf algebra on (linear com binations of ) “ lab elled diagrams” (the monoid ldiag , see [6]). As these “lab elled diagrams” are in one-to-one correspo ndence with the pac k ed matrices of MQSym , w e get on the v ector space C [ ldiag ] t w o (com binatoria lly natural) structures of algebra (and co-a lgebra) a nd one could raise the question of the existence of a con tinuous deformation b et w een the t w o. The answ er is p ositiv e and can b e p erformed through a three-parameter (t w o formal, or con tinuous and o ne b o o lean) Hopf deformation 2 of LDIA G called LDIAG ( q c , q s , q t ) [6] suc h that LDIA G (0 , 0 , 0) ≃ LDIAG ; LDIA G (1 , 1 , 1) ≃ MQSym . (5) 1 The (bi)-partitio n of the vertices is understo o d o r dered. In this case, the term bic olour e d can also b e found in the litera ture. 2 This algebra deformation has r e ceived recently another realisatio n in terms of bi-w ords [9]. 2 The r ˆ ole of the tw o para meters q c , q s (algebra para meters, whereas q t is a coalge- bra parameter) w as disco v ered just coun ting crossings and sup erp o sitions in the t wisted lab elled diagrams (see [6] for details). This simple stat istics (counting crossings and su- p erp ositions) yields an asso ciat ive pro duct on the diagrams. The first pro o f giv en for the asso ciativit y w as mainly computational a nd it w as a surprise that ev en the a sso ciativit y held. This raised the need to understand this phenomenon in a deep er w a y and the ques- tion whether the t w o parameters ( q c and q s ) w ould b e of differen t nature. The aim o f this pap er is to answ er this ques tion and giv e a concep tual pro of of associativity b y de- v eloping four building blo c ks whic h are general a nd separately easy to test: addition of a group-lik e elemen t to a co-asso ciativ e coa lgebra, shifting lemma, co diagonal deformation of a semigroup and extension of a colour factor to w ords. The essen tial ingredien t in the tw o last op erations is what has b ecome no w adays a useful to ol, the coloured pro duct of algebras, fo r whic h w e giv e some new results. A cknow ledge ments : T he authors are pleased to ac kno wledge the hospitalit y of institutions in Mosco w a nd New Y ork. Sp ecial thanks are due to Catherine Bo r g en for ha ving created a fertile atmosphere in Exeter (UK) where the first and last parts of this man uscript we re prepared. W e tak e adv a n tage of these lines to ac kno wledge supp ort fr o m the F renc h Ministry of Science a nd Hig her Education under Gr an t ANR Ph ysCom b. W e are also gra teful to Jim Sta sheff for ha ving raised the question of the different natures of the parameters q c and q s . 2 The deformed algebra L DIA G ( q c , q s ) 2.1 Review of the construction of the algebra The complete story of t he alg ebra of F eynman-Bender diagrams whic h arose in Combina- torial Ph ysics in (2 005) can b e found in [6] and a fragment of it, as w ell as a realization with an alternativ e dat a structure, in [9 ]. Recall that (classical) sh uffle pro ducts (of w ords) can b e expressed in t w o w ay s a) recursion b) summation on (and b y means of ) some permutations. Here, we will trace bac k the construction of the deformed pro duct b etw een tw o dia- grams, starting from an analog of (b) (using how ev er the symmetric semigroup instead of the sym metric group, see b elow) and going gradually to (a ) following in that the first description of the deformed case whic h w as graphical (and was disco vere d as suc h [6]). The diagra ms on whic h the pro duct has to b e p erformed are plane bipartite g raphs (v er- tices b eing called black and white sp ots) with multiple ordered edges ; they lo ok as fo llows. 3 ♠ ♠ ♠ ♠ ⑥ ⑥ ⑥ 1 2 3 4 1 2 3 Fig 1 . — L ab el le d diagr am of format 3 × 4 . One can define more formally this data structure using the equiv alent notion of a w eigh t f unction. Here, it is a function ω : N + × N + → N (as in [9]) with support supp ( ω ) = { ( i, j ) ∈ N + × N + | w ( i, j ) 6 = 0 } (6) ha ving pro j ections of the form pr 1 ( supp ( ω )) = [1 . . . p ]; pr 2 ( supp ( ω )) = [1 . . . q ] for s ome p, q ∈ N + . This last prescription can b e rephrased without pr i remarking t ha t p (resp. q ) is the last i suc h that ( ∃ j ∈ N + )( ω ( i, j ) 6 = 0) (resp. q is the last j suc h that ( ∃ i ∈ N + )( ω ( i, j ) 6 = 0)). In this wa y our gra phs a r e in one-to- one corresp ondence with suc h w eigh t functions. j 2 3 1 2 3 3 4 i 1 1 2 2 2 3 3 ω ( i, j ) 2 1 1 1 3 1 2 Fig 2 . — The weight function (when not 0 ) of the diagr am in Fig 1. He r e p = 3 and q = 4 . W e are now in the p osition of describing the ( defo r med) pro duct of our diagrams b y means of the symme tric semigroup (whereas the symm etric g roup w ould only pro vide crossings as it o ccurs with the sh uffle pro duct). The symmetric semigroup on a finite set F (denoted here S S G F ) is the set of endo- functions F → F . In order to preserv e the requiremen t that blac k sp ots kept on b eing lab elled from 1 to some integer, w e ha ve to ask that the mapping acting on the diagram d with n black sp o ts had its image of the ty p e [1 . . . m ] for some m ≤ n . The result noted d.f has m black sp o ts suc h that the blac k spot of ( f ormer) lab el “ i ” b ears the new lab el f ( i ). If w e consider any on to mapping [1 . . . p ] → [1 . . . r ], the diagram d.f = d ′ has the following w eigh t f unction ω ′ ω ′ ( i, k ) = X f ( j )= i ω ( j, k ) . (7) whic h can b e easily chec k ed to b e admissible in our con text. 4 Before giving the expression of the deformed pro duct, w e m ust define lo cal partial degrees. F or a black sp ot with lab el “ l ” , w e denote by bk s ( d , l ) its degree (nu mber of adjacen t edges). Then, for d 1 (resp. d 2 ) with p (resp. q ) blac k sp ots, the pro duct reads [ d 1 | d 2 ] L ( q c ,q s ) = X f ∈ S hs ( p,q ) Y if ( j ) q bk s ( d,i ) .bks ( d,j ) c Y i 0) is reducible iff there exists 0 < k < l suc h that indices ( Alph ( m 1 .m 2 . · · · .m k ))) ≺ ( indices ( Alph ( m k +1 .m k +2 . · · · .m l )) (42) where, for t w o nonempt y subsets X , Y ⊂ N + , one notes ≺ the relatio n of ma joration i.e., ( ∀ ( x, y ) ∈ X × Y )( x < y ) . (43) One c hec ks at once tha t the monoid M c is generated by the subalphab et ir r ( M ) ∩ M c and therefore is free. No w, w e need a classical to ol of general algebra (see [4], c hapter I I I for details). Let ( A , µ ) be a n algebra endow ed with an increasing exhaustiv e filtrat ion ( A n ) n ∈ N (i.e., t w o-sided ideals suc h that A n ⊂ A n +1 and ∪ n ∈ N A n = A ). It is classic al to construct the asso ciated graded a lg ebra Gr ( A ) = ⊕ n ≥ 0 A n / A n − 1 b y passing the la w to quotien ts i.e., ¯ µ p,q : A p / A p − 1 ⊗ A q / A q − 1 → A p + q / A p + q − 1 (one sets A − 1 = { 0 } ) . A classical lemma (and easy exercise ) states that, if the asso ciated graded algebra is free, so is A . No w, returning to ( K h MON + ( X ) i , ¯ ↑ ) ( ¯ ↑ is the shifted deformed la w), one constructs a filtration by t he num b er of irreducible comp onen ts of a w ord o f monomials (call it l ( w ) for w ∈ MON + ( X )). F rom (15), one gets, w 1 ¯ ↑ w 2 = w 1 ¯ ⋆w 2 + X l ( w )
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