q-multiple Zeta Values I: from Double Shuffle to Regularisation

Transcription

1 -Multiple Zeta Values I: from Double Shuffle to Regularisation Kurusch Ebrahimi-Fard β Paths to, from and in renormalisation Univ. Potsdam Institut für Mathematik 8-12 February 2016 β Joint work with J. Castillo Medina, D. Manchon, and J. Singer K. Ebrahimi-Fard 9 February 2016

2 Motivation 1. MZVs: regularised shuffle + uasi-shuffle relations 2. Word (uasi-)shuffle Hopf algebras: : H H H (w) := u v uv=w 3. Renormalisation problem for MZVs at non-positive arguments regularisation through a parameter λ: ζ λ : H A Connes Kreimer Birkhoff factorisation for Hopf algebra characters 4. -analogues of MZVs ψ = ψ 1 ψ + ζ BZ (k 1,..., k n ) := (1 ) k 1 + +k n (k 1 1)m 1 + +(k n 1)mn (1 m 1) k 1 (1 m n) k n K. Ebrahimi-Fard 9 February

3 Multiple Zeta Values (MZVs) k := k kn, ω i (t) := ζ(k 1,..., k n ) := = Recall 1>t 1 > >t k >0 1 m k 1 1 mk n n ω 1 (t 1 ) ω k (t k ) 1 t dt, i {k 1, k 1 + k 2,..., k}, and ω i (t) = dt t otherwise ζ(a)ζ(b) = ζ(a, b) + ζ(b, a) + ζ(a + b) ζ(a)ζ(b) = a 1 i=0 ( i + b 1 ) ζ(b + i, a i) + b 1 b 1 j=0 ( j + a 1 ) ζ(a + j, b j) a 1 ζ(2)ζ(2) = 2ζ(2, 2) + ζ(4) = 4ζ(3, 1) + 2ζ(2, 2) K. Ebrahimi-Fard 9 February

4 Shuffle and uasi-shuffle products Recall Two alphabets: X := {x 0, x 1 } Y := {y 1, y 2, y 3,...} words and phrases: X, Q X Y, Q Y 1 v = v 1 = v for v X x k1 x k p x k p+1 x kp+ = x k1 ( xk2 x k p x k p+1 x kp+ ) + x kp+1 ( xk1 x k p x k p+2 x kp+ ) 1 - u = u - 1 = u for u Y ( (y k1 y k p) - (y kp+1 y kp+ ) = y k1 yk2 y k p - y ) k p+1 y kp+ ( + y kp+1 yk1 y k p - y ) k p+2 y kp+ ( + y k1 +k yk2 p+1 y k p - y ) k p+2 y kp+ K. Ebrahimi-Fard 9 February

5 Recall Algebra morphisms Y conv := Y /y 1 Y, i.e., words y n1 y n r with n 1 > 1 ζ - (y n1 y n r) := ζ(n 1,..., n r ) ζ - (v - v ) = ζ - (v)ζ - (v ) X conv := x 0X x 1, i.e., words x n x 1 x n r 1 0 x 1 ζ (x n x 1 x n r 1 0 x 1 ) := ζ(n 1,..., n r ) ζ (u u ) = ζ (u)ζ (u ) K. Ebrahimi-Fard 9 February

6 Recall Renormalisation of MZVs Provide an extension procedure for MZVs to arbitrary integer arguments such that: (A) the meromorphic continuation is verified whenever it is defined and (B) the uasi-shuffle relation is satisfied. Connes Kreimer Birkhoff decomposition ψ : H A ψ = ψ ( 1) ψ + Regularisation parameter λ ζ λ (k 1,..., k n ) := f λ,k1 (m 1 ) f λ,k n(m n ) A f λ,l (x) := exp( lxλ) x l f λ,l (x) := 1 x l λ f λ,l (x) := l x (1 x ) l, λ = log() λ 0, f λ,l (x) f l (x) K. Ebrahimi-Fard 9 February

7 -analogues (regularised) of Multiple Zeta Values Schlesinger Zudilin model ζ SZ (k 1,..., k n ) := k 1 m 1 + +k nmn m 1 ] k 1 m n ] k n Q]] -number m] := 1 m 1 = m 1 modified Schlesinger Zudilin model SZ (k 1,..., k n ) := k 1 m 1 + +k nmn (1 m 1) k 1 (1 m n) k n Regularised mod. Schlesinger Zudilin model SZ,reg (k 1,..., k n ) := k 1 m k n mn (1 m 1) k 1 (1 m n) k n K. Ebrahimi-Fard 9 February

8 -analogues of Multiple Zeta Values Jackson integral Jf](t) := t 0 f(x)d x = (1 ) n 0 f( n t) n t E f](t) := f(t), M f g](t) := (fg)(t) = f(t)g(t) (1 ) n 0 f( n t) n t = (1 ) n 0 E n Mid f] ] (t) = (1 )P M id f](t) P f](t) := n 0 E n f] = f(t) + f(t) + f(2 t) + f( 3 t) + Rota Baxter map of weight 1 P f]p g] = P P f]g ] + P fp g] ] P fg ] K. Ebrahimi-Fard 9 February

9 -analogues of Multiple Zeta Values Replace Riemann by Jackson integral and... ζ(k 1,..., k n ) = k := k kn, ω i (t) := 1>t 1 > >t k >0 ω 1 (t 1 ) ω k (t k ) 1 t dt, i {k 1, k 1 + k 2,..., k}, and ω i (t) = dt t otherwise ζ (k 1,..., k n ) = >t 1 > >t k >0 k := k kn, ω i (t) := d t 1 t, i {k 1, k 1 + k 2,..., k}, and ω i (t) = d t t ζ (2) = (1 ) 2 (2) = 0 d t 1 t 1 t1 0 ω 1 (t 1) ω k (t k) otherwise d t 2 t ] = (1 ) 2 P P () = (1 ) 2 1 t 2 1 t m>0 m (1 m ) 2 K. Ebrahimi-Fard 9 February

10 Ohno Okuda Zudilin () model -analogues of Multiple Zeta Values ζ (k 1,..., k n ) = (1 ) k P P P }{{} = (1 ) k k 1 = (1 ) k (k 1,..., k n ) Recall: P is Rota Baxter map of weight 1 Note: operator P is invertible t 1 t P P P }{{} kn m 1 (1 m 1) k 1 (1 m n) k n P f]p g] = P P f]g ] + P fp g] ] P fg ] P 1 f](t) = Df](t) := f(t) f(t) = (id E )f](t) t 1 t ]]]] ]] () K. Ebrahimi-Fard 9 February

11 -shuffle relations I -shuffle relation for model The -difference operator D satisfies a modified Leibniz rule D fg] = D f]g + fd g] D f]d g] P f]p g] = P P f]g + fp g] fg] D f]d g] = D f]g + fdg] D fg] D f]p g] = D fp g]] + D f]g fg The iterated sum defining ζ (k 1,..., k n ) makes sense in Q]] for any k 1,..., k n Z: ζ (k 1,..., k n ) = (1 ) k (k 1,..., k n ) = (1 ) k P k 1 t 1 t P k n t 1 t ] ]() K. Ebrahimi-Fard 9 February

12 Examples (0) = m = m>0 1 (0, 0) = m 1 >m 2 >0 m1 = m>0 (m 1) m = ( 1 ) 2 (a, 0) = m 1 >m 2 >0 m 1 (1 m 1) a = m>0 (m 1) m (1 m 1) a (1) = m>0 m 1 m ( a) = D a t 1 t ]() = m (1 m ) a m>0 K. Ebrahimi-Fard 9 February

13 -shuffle relations II Alphabet X := {d, y, p} X words and good words: X good := X y, dp = pd = 1, and k 1,..., k n Z w = p k 1 y p kn y, (pn 1y p n ky) := (n 1,..., n k ) The -shuffle product is given on Q X recursively by 1 v = v 1 = v: (yv) u = v (yu) = y(v u) pv pu = p(v pu) + p(pv u) p(v u) dv du = v du + dv u d(v u) dv pu = pu dv = d(v pu) + dv u v u K. Ebrahimi-Fard 9 February

14 -shuffle and Euler-type identity Theorem The -shuffle product is commutative and associative. For v, u W we have: Euler-type identity:, (v), (u) =, (v u). (2) (2) = 2 (2, 2) + 4 (3, 1) 4 (2, 1) (3, 0) + (2, 0) 1 < a b ( a) ( b) = a j=0 b j i=1 ( 1) j( a + b 1 i j a 1 )( a j ) ( j, i) + + b j=0 j=1 max(1,a j) i=1 ( 1) j( a + b 1 i j b 1 )( b j a ( 1) j( a + b 1 j ) ( j, 0) j 1, a j, b j ) ( j, i) K. Ebrahimi-Fard 9 February

15 -uasi-shuffle relations (a) (b) = (a, b) + (b, a) + (a + b) (a, b 1) (b, a 1) (a + b 1) Alphabet Ỹ := {z i : i Z} with product z i z j := z i+j, Ỹ words, (z n 1 z nk ) := (n 1,..., n k ) Define linear map T on Q Ỹ : T (z nv) := z n v z n 1 v The -uasi-shuffle product is given on Q Ỹ through the classical uasi-shuffle product: z m u z n v := z m ( u - T (z n v) ) + z n ( T (zm u) - v ) + (z m+n z m+n 1 )(u - v) In particular we have: z a z b = z a (T z b ) + z b (T z a ) + T z a+b = z a z b + z a z b + z a+b z a z b 1 z b z a 1 z a+b 1 K. Ebrahimi-Fard 9 February

16 Weight drop terms and regularised relations Theorem The -uasi-shuffle product is commutative and associative. For u, v Ỹ we have:, (u v) =, (u), (v). In terms of -MZVs, the previous example becomes: ζ (a)ζ (b) = ζ (a, b) + ζ ( + (1 ) (b, a) + ζ (a + b) ζ (a, b 1) ζ (b, a 1) ζ (a + b 1) In the limit 1, the weight drop terms disappear classical uasi-shuffle relation ) (1) (2) = = (1, 2) + (2, 1) + (1, 2) + 2 (2, 1) (3) (2, 0) (1, 1) (1, 1) (2, 0) (2) (3) (2) = (2, 1) ζ (3) (1 )ζ (2) = ζ (2, 1) K. Ebrahimi-Fard 9 February

17 Renormalisation of MZVs Remarks: i) The -uasi-shuffle is not a uasi-shuffle in the sense of Hoffman, i.e., no Hopf algebra structure known. This implies that this model is not suited for renormalisation of -regularised MZVs using Connes Kreimer factorisation. ii) On the other hand, a Hopf algebra can be defined with respect to the -shuffle product. See Singer s 2nd talk. iii) Other MZVs with double -shuffle (for admissible arguments): SZ (k 1,..., k n ) := BZ (k 1,..., k n ) := k 1 m 1 + +k nmn (1 m 1) k 1 (1 m n) k n (k 1 1)m 1 + +(k n 1)mn (1 m 1) k 1 (1 m n) k n K. Ebrahimi-Fard 9 February

18 Another Rota Baxter representation for MZVs weight 1 RB map P f](t) := n 0 En f] = f(t) + f(t) + f(2 t) + f( 3 t) + P f]p g] = P P f]g ] + P fp g] ] P fg ] weight 1 RB map P f](t) := n 1 En f] = f(t) + f(2 t) + f( 3 t) + P f] P g] = P P f]g ] + P f P g] ] + P fg ] P f]p g] = P fp g]] + P P f]g] K. Ebrahimi-Fard 9 February

19 modified Schlesinger Zudilin model Double shuffle for other -MZVs SZ (k 1,..., k n ) := SZ (a) SZ modified Bradley Zhao model BZ BZ (k 1,..., k n ) := (a) BZ (b) = BZ = P k t 1 1 t P k n (b) = SZ (a, b) + SZ k 1 m 1 + +k nmn (1 m 1) k 1 (1 m n) k n t 1 t ] ](1) SZ (b, a) + (a + b) (k 1 1)m 1 + +(k n 1)mn (1 m 1) k 1 (1 m n) k n = P k 1 1 t P 1 t P k n 1 t P 1 t ] ](1) (a, b) + BZ (b, a) + BZ BZ (a + b) + (a + b 1). K. Ebrahimi-Fard 9 February

20 Regularised mod. Schlesinger Zudilin model SZ,reg (k 1,..., k n ) := k 1 m k n mn (1 m 1) k 1 (1 m n) k n defined for k 1,..., k n strictly negative. 1) (-)uasi-shuffle: for a, b N. SZ,reg ( a) SZ,reg ( b) = ( m (1 m )) a ( n (1 n )) b m>0 n>0 = ( m (1 m )) a ( n (1 n )) b + ( n (1 n )) b ( m (1 m )) a m>n>0 n>m>0 + ( m (1 m )) a+b m>0 = SZ,reg ( a, b) + SZ,reg ( b, a) + SZ,reg ( a b) 2) no -shuffle representation K. Ebrahimi-Fard 9 February