Hirota equation and the quantum plane

Transcription

1 Hirota equation and the quantum plane dam oliwa University of Warmia and Mazury (Olsztyn, Poland) Mathematisches Forschungsinstitut Oberwolfach 3 July 202 dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane / 29

2 Outline esargues maps and Hirota s discrete KP equation 2 The normalization map Pentagonal property of the normalization map Ultralocality and quantum the normalization map case The bialgebra structure of the quantum plane 3 The Veblen flip map dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 2 / 29

3 . oliwa, esargues maps and the Hirota-Miwa equation, Proc. R. Soc. 466 (200) oliwa, S. M. Sergeev, The pentagon relation and incidence geometry, arxiv: oliwa, Hirota equation and the quantum plane dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 3 / 29

4 Outline esargues maps and Hirota s discrete KP equation esargues maps and Hirota s discrete KP equation 2 The normalization map Pentagonal property of the normalization map Ultralocality and quantum the normalization map case The bialgebra structure of the quantum plane 3 The Veblen flip map dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 4 / 29

5 esargues maps and Hirota s discrete KP equation esargues maps Maps φ : Z K P M (), K, M 2, such that the points φ(n), φ (i) (n) and φ (j) (n) are collinear, for all n Z K, i j Notation: φ (i) (n,..., n i,..., n K ) = φ(n,..., n i +,..., n K ) φ (i) φ (j) φ (ij) φ (k) φ φ (ik) φ (jk) Remark y the Menelaus theorem this configuration gives geometric interpretation of the discrete S-KP equation [Konopelchenko & Schief 02] Remark 2 Four dimensional consistency of discrete S-KP has combinatorics of the esargues theorem [Schief 09] dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 5 / 29

6 esargues maps and Hirota s discrete KP equation The esargues map system In homogeneous coordinates φ : Z K M+ φ + φ (i) ij + φ (j) ji = 0, i j, where ij : Z K. The compatibility condition of the above linear system reads where i, j, k are distinct ij ik + kj ki =, ik(j) jk = jk(i) ik, [ 0] Gauge transformations: φ = φf, where F : Z K - gauge function results in Ãij = F (i) ij F dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 6 / 29

7 esargues maps and Hirota s discrete KP equation The Hirota system gauge One can find homogeneous coordinates such that ji = ij = U ij φ (i) φ (j) = φu ij, i j K, When = F is comutative then the functions U ij can be parametrized in terms of a single potential τ : Z K F U ij = ττ (ij) τ (i) τ (j), i < j K The nonlinear system reads [Hirota 8], [Miwa 82] τ (i) τ (jk) τ (j) τ (ik) + τ (k) τ (ij) = 0, i < j < k K dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 7 / 29

8 Outline The normalization map esargues maps and Hirota s discrete KP equation 2 The normalization map Pentagonal property of the normalization map Ultralocality and quantum the normalization map case The bialgebra structure of the quantum plane 3 The Veblen flip map dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 8 / 29

9 The normalization map The first part of the esargues map equations φ (i) φ (j) φ (ij) φ (k) φ φ (ik) φ (jk) ij ik + kj ki =, i, j, k distinct dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 9 / 29

10 The normalization map Pentagonal property of the normalization map The normalization map in the simplex representation C y C 2 x W ~ ~ C 2 φ C =φ y + φ x, φ =φ ỹ + φ x, φ =φ C y 2 + φ x 2, φ =φ C ỹ 2 + φ x 2, x = x 2 + x y 2, ỹ = y y 2 x 2 = y x x 2, ỹ 2 = y 2 + x x 2 dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 0 / 29

11 The normalization map Pentagonal property of the normalization map The pentagon property of the normalization map What happens when we add a fifth point? C 2 W 23 3 W 2 C 2 3 E C 3 2 E E W 2 W 23 C 2 3 C 2 3 W 3 E E W 23 W 3 W 2 = W 2 W 23 dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane / 29

12 The normalization map Pentagonal property of the normalization map = x, y W 2 W 23 x 2, y 2 = x 3, y 3 x 3 + x 2 y 3 + x y 2 y 3, y y 2 y 3 y x (x 3 + x 2 y 3 ), y 2 y 3 + x y 2 x 2 x 3, y 3 + x (x 3 + x 2 y 3 ) 2 x 3 x, y = W 23 W 3 W 2 x 2, y 2 x 3, y 3 = dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 2 / 29

13 The normalization map Ultralocality and quantum the normalization map case Ultra-locality and quantum k Z() fixed subfield of the division ring i division k-subalgebra generated by x i, y i, i =, 2 General position assumptions x, y, x 2, y 2 do not belong to k and are linearly independent 2 = k Ultra-locality "variables with different indices commute" Proposition ssume that the map W preserves the ultra-locality condition x i x j = x j x i, y i y j = y j y i, x i y j = y j x i, i j, then under general position assumptions there exists q k such that x i y i = qy i x i, x i ỹ i = qỹ i x i, i =, 2 dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 3 / 29

14 The normalization map Ultralocality and quantum the normalization map case The Poisson property of the normalization map Corollary The normalization map provides automorphism of the division ring k q (x, y, x 2, y 2 ) of quantum rational functions The classical limit q Observation The normalization map provides a Poisson automorphism of the commutative field of rational functions k(x, y, x 2, y 2 ) equipped with the Poisson structure {x i, y i } = x i y i, {x i, x j } = {y i, y j } = {x i, y j } = 0, i j dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 4 / 29

15 The normalization map Cartier Foata property Ultralocality and quantum the normalization map case Cartier Foata matrix "elements in different rows commute" Proposition Given 2 by 2 Cartier Foata matrix ( ) y x M =, x 2 y 2 with elements satisfying the generic position conditions. If its inverse ( ) ( ) M y x y 2 = x 2 x 2 y 2 x y ( ) ( ) x y x 2 y 2 y 2 x 2 y x is Cartier Foata matrix as well, then there exists q k such that x i y i = qy i x i, i =, 2. dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 5 / 29

16 The normalization map The bialgebra structure of the quantum plane Quantum plane k q [x, y] = k{x, y xy qyx = 0} The normalization map W defines unital morfism of algebras (coproduct) : k q [x, y] k q [x, y] k q [x, y] = k q [x, y, x 2, y 2 ] given on generators by (x) = x + x y = x, (y) = y y = ỹ which due to pentagonal property of W satisfies the coassociativity condition (id ) = ( id) 2 (x) = x + x y + x y y, 2 (y) = y y y dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 6 / 29

17 The normalization map The bialgebra structure of the quantum plane Quantum plane: counit and antipode The unital morphism ɛ : k q [x, y] k (counit) satisfying (id ɛ) = id = (ɛ id) is given by ɛ(x) = 0, ɛ(y) = To construct the coresponding Hopf algebra enlarge the quantum plane by the inverse of y and find on k q [x, y, y ] a linear map (antipode) by requiring S(a () )a (2) = a () S(a (2) ) = ɛ(a), (a) = a () a (2) (a) (a) (a) We have S(x) = xy, S(y) = y dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 7 / 29

18 The normalization map Quantum pentagon equation The bialgebra structure of the quantum plane Using quantum dilogarithm function (Faddeev, Woronowicz) we can find W k q (x, y, x 2, y 2 ) such that for any rational function F k q (x, y, x 2, y 2 ) in particular WF(x, y, x 2, y 2 )W = F( x, ỹ, x 2, ỹ 2 ) (a) = W(a )W, a k q [x, y, y ] fter eventual rescaling such a W satisfies the quantum pentagon equation W 2 W 3 W 23 = W 23 W 2 dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 8 / 29

19 The normalization map Quantization as reduction The bialgebra structure of the quantum plane Non commutative integrable map Ultra local integrable reduction Quantum integrable map Classical limit Integrable (commutative) Poisson map dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 9 / 29

20 Outline The Veblen flip map esargues maps and Hirota s discrete KP equation 2 The normalization map Pentagonal property of the normalization map Ultralocality and quantum the normalization map case The bialgebra structure of the quantum plane 3 The Veblen flip map dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 20 / 29

21 The Veblen flip map The Veblen configuration and the second part of the esargues map equations φ (i) φ (j) φ (ij) φ (k) φ φ (ik) φ (jk) ik(j) jk = jk(i) ik, i, j, k distinct dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 2 / 29

22 The Veblen flip map The Veblen flip in P M (), division ring C 2 u C w C C C 2 2 The Veblen flip and its simplex representation φ M+, and u, w φ =φ C w + φ C u φ C =φ w 2 + φ C u 2 2 φ =φ w + φ u φ C =φ w 2 + φ Cu 2 dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 22 / 29

23 The Veblen flip map The Veblen flip map W G : G free gauge parameter φ = u = Gw, w = w 2w u 2 = u 2w u, w 2 = Gw u ( φ C u w + φ C u 2 ) G = ( φ w φ w 2 ) G dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 23 / 29

24 The Veblen flip map The esargues configuration C C E E E C E CE C The esargues configuration and its 4-simplex combinatorics dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 24 / 29

25 The Veblen flip map The pentagon property of the Veblen flip C 3 2 E C W E C W 2 E 2 3 W 23 C E 2 3 W 3 E C 2 3 U V W 2 X Y Z The pentagon relation for Veblen flips in the 4-simplex representation dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 25 / 29

26 The Veblen flip map The pentagon property of the Veblen map Proposition The Veblen flip map W G : u = Gw, w = w 2w u 2 = u 2w u, w 2 = Gw u satisfies the functional dynamical pentagon relation W Z 2 W Y 3 W X 23 = W V 23 W U 2 provided the parameters/functions U, V on the right hand side are related to the left hand functions X, Y, Z by the conditions V = Yw 2, ZX = Uw 2 dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 26 / 29

27 The Veblen flip map Ultra-local/quantum reduction of the Veblen flip map "Fix" the gauge parameter G of the Veblen flip map G(u, w, u 2, w 2 ) = (αu 2 + βw 2 u )w, α, β k, Under the ultra-locality condition the map W (α,β) u = αu 2 + βu w 2, w = w w 2 u 2 = w u u 2, w 2 = αu u 2 + βu selects the Weyl (q-plane) commutation relations satisfies the functional pentagon equation provided W (α V,β V ) 2 W (α U,β U ) 23 = W (α Z,β Z ) 23 W (α Y,β Y ) 3 W (α X,β X ) 2, α X = α V β Z, α Y = α U α V, α Z = α U β X, β U = β Y β Z, β V = β X β Y dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 27 / 29

28 The Veblen flip map Girard esargues (59 66) The relevance of a geometric theorem is determined by what the theorem tells us about space, and not by the eventual difficulty of the proof. The esargues theorem of projective geometry comes as close as a proof can to the Zen ideal. It can be summarized in two words: "I see!" Nevertheless, esargues theorem, far from trivial despite the simplicity of its proof, has many more applications both in geometry and beyond than any theorem in number theory, maybe even more then all the theorems in analytic number theory combined. [Gian Carlo Rota, The Phenomenology of Mathematical Proof] dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 28 / 29

29 Conclusion The Veblen flip map the pentagonal property of the Veblen and normalization maps are equivalent to four dimensional consistency of the esargues map equation from the ultra-locality assumption of the normalization map (and the Veblen map as well) we obtain for free the quantum plane (Weyl) commutation relations and Poisson property of the corresponding classical map integrability (multidimensional consistency) of the ultra-local normalization map provides with the bi-algebra structure maps of the quantum plane the pentagonal property of the q-commutative maps can be then reformulated in the operator form relation to quantum integrable systems? relation to Topological Quantum Field Theory (via Pachner moves)? dam oliwa (Univ. Warmia & Mazury) Hirota equation and the quantum plane 29 / 29