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Nonabelian Multiplicative Integration on Surfaces Amnon Yekutieli Department of Mathematics Ben Gurion University Notes available at http://www.math.bgu.ac.il/~amyekut/lectures Updated 29 Sep 2015 Amnon Yekutieli (BGU) Multiplicative

  1. 1. Some Preliminaries For n ≥ 0 we let ∆ n be the n -dimensional real simplex. This is a polyhedron embedded in R n + 1 . If we use the barycentric coordinates t 0 , . . . , t n on R n + 1 , then ∆ n is the compact subset defined by n ∑ t i ≥ 0 and t i = 1. i = 0 The vertices of ∆ n are v 0 , . . . , v n , where v i : = ( 0, . . . , 1, . . . , 0 ) with 1 in the i -th position. For n = 1 we can identify ∆ 1 with the unit line segment I 1 . But then we use the coordinate t : = t 1 . Amnon Yekutieli (BGU) Multiplicative Integration 4 / 45

  2. 1. Some Preliminaries Figure : The simplices ∆ n for n = 1, 2, 3. Amnon Yekutieli (BGU) Multiplicative Integration 5 / 45

  3. 1. Some Preliminaries Let X be an n -dimensional manifold (differentiable of type C ∞ ) or a convex polyhedron (such as ∆ n ). We denote by n Ω p ( X ) � Ω ( X ) = p = 0 the de Rham algebra of smooth differential forms on X . In degree 0 we have Ω 0 ( X ) = O ( X ) , the ring of smooth R -valued functions on X . The de Rham algebra comes with the exterior derivative d : Ω p ( X ) → Ω p + 1 ( X ) . If Y is a manifold, and f : X → Y is a smooth map, then there is a pullback operation f ∗ : Ω p ( Y ) → Ω p ( X ) . Amnon Yekutieli (BGU) Multiplicative Integration 6 / 45

  4. 1. Some Preliminaries Let X be an n -dimensional manifold (differentiable of type C ∞ ) or a convex polyhedron (such as ∆ n ). We denote by n Ω p ( X ) � Ω ( X ) = p = 0 the de Rham algebra of smooth differential forms on X . In degree 0 we have Ω 0 ( X ) = O ( X ) , the ring of smooth R -valued functions on X . The de Rham algebra comes with the exterior derivative d : Ω p ( X ) → Ω p + 1 ( X ) . If Y is a manifold, and f : X → Y is a smooth map, then there is a pullback operation f ∗ : Ω p ( Y ) → Ω p ( X ) . Amnon Yekutieli (BGU) Multiplicative Integration 6 / 45

  5. 1. Some Preliminaries Let X be an n -dimensional manifold (differentiable of type C ∞ ) or a convex polyhedron (such as ∆ n ). We denote by n Ω p ( X ) � Ω ( X ) = p = 0 the de Rham algebra of smooth differential forms on X . In degree 0 we have Ω 0 ( X ) = O ( X ) , the ring of smooth R -valued functions on X . The de Rham algebra comes with the exterior derivative d : Ω p ( X ) → Ω p + 1 ( X ) . If Y is a manifold, and f : X → Y is a smooth map, then there is a pullback operation f ∗ : Ω p ( Y ) → Ω p ( X ) . Amnon Yekutieli (BGU) Multiplicative Integration 6 / 45

  6. 1. Some Preliminaries Let X be an n -dimensional manifold (differentiable of type C ∞ ) or a convex polyhedron (such as ∆ n ). We denote by n Ω p ( X ) � Ω ( X ) = p = 0 the de Rham algebra of smooth differential forms on X . In degree 0 we have Ω 0 ( X ) = O ( X ) , the ring of smooth R -valued functions on X . The de Rham algebra comes with the exterior derivative d : Ω p ( X ) → Ω p + 1 ( X ) . If Y is a manifold, and f : X → Y is a smooth map, then there is a pullback operation f ∗ : Ω p ( Y ) → Ω p ( X ) . Amnon Yekutieli (BGU) Multiplicative Integration 6 / 45

  7. 1. Some Preliminaries Let X be an n -dimensional manifold (differentiable of type C ∞ ) or a convex polyhedron (such as ∆ n ). We denote by n Ω p ( X ) � Ω ( X ) = p = 0 the de Rham algebra of smooth differential forms on X . In degree 0 we have Ω 0 ( X ) = O ( X ) , the ring of smooth R -valued functions on X . The de Rham algebra comes with the exterior derivative d : Ω p ( X ) → Ω p + 1 ( X ) . If Y is a manifold, and f : X → Y is a smooth map, then there is a pullback operation f ∗ : Ω p ( Y ) → Ω p ( X ) . Amnon Yekutieli (BGU) Multiplicative Integration 6 / 45

  8. 2. MI on Curves 2. MI on Curves Let X be a manifold. By a path (or string) in X we mean a smooth map σ : ∆ 1 → X . Let G be a Lie group with Lie algebra g . Suppose σ is a path in X , and α is a g -valued 1-form on X , i.e. α ∈ Ω 1 ( X ) ⊗ g . We wish to define the nonabelian multiplicative integral of α on σ , which is an element of the group G . Amnon Yekutieli (BGU) Multiplicative Integration 7 / 45

  9. 2. MI on Curves 2. MI on Curves Let X be a manifold. By a path (or string) in X we mean a smooth map σ : ∆ 1 → X . Let G be a Lie group with Lie algebra g . Suppose σ is a path in X , and α is a g -valued 1-form on X , i.e. α ∈ Ω 1 ( X ) ⊗ g . We wish to define the nonabelian multiplicative integral of α on σ , which is an element of the group G . Amnon Yekutieli (BGU) Multiplicative Integration 7 / 45

  10. 2. MI on Curves 2. MI on Curves Let X be a manifold. By a path (or string) in X we mean a smooth map σ : ∆ 1 → X . Let G be a Lie group with Lie algebra g . Suppose σ is a path in X , and α is a g -valued 1-form on X , i.e. α ∈ Ω 1 ( X ) ⊗ g . We wish to define the nonabelian multiplicative integral of α on σ , which is an element of the group G . Amnon Yekutieli (BGU) Multiplicative Integration 7 / 45

  11. 2. MI on Curves 2. MI on Curves Let X be a manifold. By a path (or string) in X we mean a smooth map σ : ∆ 1 → X . Let G be a Lie group with Lie algebra g . Suppose σ is a path in X , and α is a g -valued 1-form on X , i.e. α ∈ Ω 1 ( X ) ⊗ g . We wish to define the nonabelian multiplicative integral of α on σ , which is an element of the group G . Amnon Yekutieli (BGU) Multiplicative Integration 7 / 45

  12. 2. MI on Curves 2. MI on Curves Let X be a manifold. By a path (or string) in X we mean a smooth map σ : ∆ 1 → X . Let G be a Lie group with Lie algebra g . Suppose σ is a path in X , and α is a g -valued 1-form on X , i.e. α ∈ Ω 1 ( X ) ⊗ g . We wish to define the nonabelian multiplicative integral of α on σ , which is an element of the group G . Amnon Yekutieli (BGU) Multiplicative Integration 7 / 45

  13. 2. MI on Curves Take k ≥ 0. We partition ∆ 1 into 2 k equal line segments, starting from v 0 . Composing with σ we get paths σ 1 , . . . , σ 2 k : ∆ 1 → X , that we call the k -th binary subdivision of σ . The case k = 2 is depicted below. Amnon Yekutieli (BGU) Multiplicative Integration 8 / 45

  14. 2. MI on Curves Take k ≥ 0. We partition ∆ 1 into 2 k equal line segments, starting from v 0 . Composing with σ we get paths σ 1 , . . . , σ 2 k : ∆ 1 → X , that we call the k -th binary subdivision of σ . The case k = 2 is depicted below. Amnon Yekutieli (BGU) Multiplicative Integration 8 / 45

  15. 2. MI on Curves Take k ≥ 0. We partition ∆ 1 into 2 k equal line segments, starting from v 0 . Composing with σ we get paths σ 1 , . . . , σ 2 k : ∆ 1 → X , that we call the k -th binary subdivision of σ . The case k = 2 is depicted below. Amnon Yekutieli (BGU) Multiplicative Integration 8 / 45

  16. 2. MI on Curves For each i there is the usual integral � � ∆ 1 σ ∗ α = i ( α ) ∈ g . σ i The k -th Riemann product is 2 k � ∈ G , � � ∏ RP k ( α | σ ) : = exp G α (2.1) σ i i = 1 where the product goes from left to right. It is not hard to prove that the limit MI ( α | σ ) : = lim k → ∞ RP k ( α | σ ) ∈ G . (2.2) exists. This is the multiplicative integral of α on σ . Amnon Yekutieli (BGU) Multiplicative Integration 9 / 45

  17. 2. MI on Curves For each i there is the usual integral � � ∆ 1 σ ∗ α = i ( α ) ∈ g . σ i The k -th Riemann product is 2 k � ∈ G , � � ∏ RP k ( α | σ ) : = exp G α (2.1) σ i i = 1 where the product goes from left to right. It is not hard to prove that the limit MI ( α | σ ) : = lim k → ∞ RP k ( α | σ ) ∈ G . (2.2) exists. This is the multiplicative integral of α on σ . Amnon Yekutieli (BGU) Multiplicative Integration 9 / 45

  18. 2. MI on Curves For each i there is the usual integral � � ∆ 1 σ ∗ α = i ( α ) ∈ g . σ i The k -th Riemann product is 2 k � ∈ G , � � ∏ RP k ( α | σ ) : = exp G α (2.1) σ i i = 1 where the product goes from left to right. It is not hard to prove that the limit MI ( α | σ ) : = lim k → ∞ RP k ( α | σ ) ∈ G . (2.2) exists. This is the multiplicative integral of α on σ . Amnon Yekutieli (BGU) Multiplicative Integration 9 / 45

  19. 2. MI on Curves For each i there is the usual integral � � ∆ 1 σ ∗ α = i ( α ) ∈ g . σ i The k -th Riemann product is 2 k � ∈ G , � � ∏ RP k ( α | σ ) : = exp G α (2.1) σ i i = 1 where the product goes from left to right. It is not hard to prove that the limit MI ( α | σ ) : = lim k → ∞ RP k ( α | σ ) ∈ G . (2.2) exists. This is the multiplicative integral of α on σ . Amnon Yekutieli (BGU) Multiplicative Integration 9 / 45

  20. 2. MI on Curves The operation MI ( α | σ ) has several nice properties. If G is abelian then � � � MI ( α | σ ) = exp G σ α . Another nice property is the geometric multiplicativity, which I shall now explain. Suppose we partition ∆ 1 into two segments or arbitrary length, starting from v 0 . This gives rise to paths σ 1 , σ 2 : ∆ 1 → X as shown on the next slide. Amnon Yekutieli (BGU) Multiplicative Integration 10 / 45

  21. 2. MI on Curves The operation MI ( α | σ ) has several nice properties. If G is abelian then � � � MI ( α | σ ) = exp G σ α . Another nice property is the geometric multiplicativity, which I shall now explain. Suppose we partition ∆ 1 into two segments or arbitrary length, starting from v 0 . This gives rise to paths σ 1 , σ 2 : ∆ 1 → X as shown on the next slide. Amnon Yekutieli (BGU) Multiplicative Integration 10 / 45

  22. 2. MI on Curves The operation MI ( α | σ ) has several nice properties. If G is abelian then � � � MI ( α | σ ) = exp G σ α . Another nice property is the geometric multiplicativity, which I shall now explain. Suppose we partition ∆ 1 into two segments or arbitrary length, starting from v 0 . This gives rise to paths σ 1 , σ 2 : ∆ 1 → X as shown on the next slide. Amnon Yekutieli (BGU) Multiplicative Integration 10 / 45

  23. 2. MI on Curves The operation MI ( α | σ ) has several nice properties. If G is abelian then � � � MI ( α | σ ) = exp G σ α . Another nice property is the geometric multiplicativity, which I shall now explain. Suppose we partition ∆ 1 into two segments or arbitrary length, starting from v 0 . This gives rise to paths σ 1 , σ 2 : ∆ 1 → X as shown on the next slide. Amnon Yekutieli (BGU) Multiplicative Integration 10 / 45

  24. 2. MI on Curves The operation MI ( α | σ ) has several nice properties. If G is abelian then � � � MI ( α | σ ) = exp G σ α . Another nice property is the geometric multiplicativity, which I shall now explain. Suppose we partition ∆ 1 into two segments or arbitrary length, starting from v 0 . This gives rise to paths σ 1 , σ 2 : ∆ 1 → X as shown on the next slide. Amnon Yekutieli (BGU) Multiplicative Integration 10 / 45

  25. 2. MI on Curves Then MI ( α | σ ) = MI ( α | σ 1 ) · MI ( α | σ 2 ) (2.3) in the group G . Amnon Yekutieli (BGU) Multiplicative Integration 11 / 45

  26. 2. MI on Curves Then MI ( α | σ ) = MI ( α | σ 1 ) · MI ( α | σ 2 ) (2.3) in the group G . Amnon Yekutieli (BGU) Multiplicative Integration 11 / 45

  27. 2. MI on Curves For G = GL n ( R ) there is an interpretation of the 1-dimensional MI in terms of ordinary differential equations. Consider a smooth function f : I 1 → M n ( R ) . � � In other words f = f i , j ( t ) , an n × n matrix of smooth functions of the real variable t . Let g : I 1 → M n ( R ) be the unique smooth solution of the matrix ODE d d t g ( t ) = g ( t ) · f ( t ) with initial condition g ( 0 ) = 1. On the other hand, f defines a matrix 1-form α : = f ( t ) · d t ∈ Ω 1 ( I 1 ) ⊗ M n ( R ) . It is not hard to show that MI ( α | I 1 ) = g ( 1 ) . Amnon Yekutieli (BGU) Multiplicative Integration 12 / 45

  28. 2. MI on Curves For G = GL n ( R ) there is an interpretation of the 1-dimensional MI in terms of ordinary differential equations. Consider a smooth function f : I 1 → M n ( R ) . � � In other words f = f i , j ( t ) , an n × n matrix of smooth functions of the real variable t . Let g : I 1 → M n ( R ) be the unique smooth solution of the matrix ODE d d t g ( t ) = g ( t ) · f ( t ) with initial condition g ( 0 ) = 1. On the other hand, f defines a matrix 1-form α : = f ( t ) · d t ∈ Ω 1 ( I 1 ) ⊗ M n ( R ) . It is not hard to show that MI ( α | I 1 ) = g ( 1 ) . Amnon Yekutieli (BGU) Multiplicative Integration 12 / 45

  29. 2. MI on Curves For G = GL n ( R ) there is an interpretation of the 1-dimensional MI in terms of ordinary differential equations. Consider a smooth function f : I 1 → M n ( R ) . � � In other words f = f i , j ( t ) , an n × n matrix of smooth functions of the real variable t . Let g : I 1 → M n ( R ) be the unique smooth solution of the matrix ODE d d t g ( t ) = g ( t ) · f ( t ) with initial condition g ( 0 ) = 1. On the other hand, f defines a matrix 1-form α : = f ( t ) · d t ∈ Ω 1 ( I 1 ) ⊗ M n ( R ) . It is not hard to show that MI ( α | I 1 ) = g ( 1 ) . Amnon Yekutieli (BGU) Multiplicative Integration 12 / 45

  30. 2. MI on Curves For G = GL n ( R ) there is an interpretation of the 1-dimensional MI in terms of ordinary differential equations. Consider a smooth function f : I 1 → M n ( R ) . � � In other words f = f i , j ( t ) , an n × n matrix of smooth functions of the real variable t . Let g : I 1 → M n ( R ) be the unique smooth solution of the matrix ODE d d t g ( t ) = g ( t ) · f ( t ) with initial condition g ( 0 ) = 1. On the other hand, f defines a matrix 1-form α : = f ( t ) · d t ∈ Ω 1 ( I 1 ) ⊗ M n ( R ) . It is not hard to show that MI ( α | I 1 ) = g ( 1 ) . Amnon Yekutieli (BGU) Multiplicative Integration 12 / 45

  31. 2. MI on Curves For G = GL n ( R ) there is an interpretation of the 1-dimensional MI in terms of ordinary differential equations. Consider a smooth function f : I 1 → M n ( R ) . � � In other words f = f i , j ( t ) , an n × n matrix of smooth functions of the real variable t . Let g : I 1 → M n ( R ) be the unique smooth solution of the matrix ODE d d t g ( t ) = g ( t ) · f ( t ) with initial condition g ( 0 ) = 1. On the other hand, f defines a matrix 1-form α : = f ( t ) · d t ∈ Ω 1 ( I 1 ) ⊗ M n ( R ) . It is not hard to show that MI ( α | I 1 ) = g ( 1 ) . Amnon Yekutieli (BGU) Multiplicative Integration 12 / 45

  32. 2. MI on Curves For G = GL n ( R ) there is an interpretation of the 1-dimensional MI in terms of ordinary differential equations. Consider a smooth function f : I 1 → M n ( R ) . � � In other words f = f i , j ( t ) , an n × n matrix of smooth functions of the real variable t . Let g : I 1 → M n ( R ) be the unique smooth solution of the matrix ODE d d t g ( t ) = g ( t ) · f ( t ) with initial condition g ( 0 ) = 1. On the other hand, f defines a matrix 1-form α : = f ( t ) · d t ∈ Ω 1 ( I 1 ) ⊗ M n ( R ) . It is not hard to show that MI ( α | I 1 ) = g ( 1 ) . Amnon Yekutieli (BGU) Multiplicative Integration 12 / 45

  33. 2. MI on Curves 1-dimensional MI is used in various areas, such as mathematical physics and probability. There are various names and notations for this operation. One name is path ordered exponential integral, with corresponding notation � Pexp σ α . In probability this operation is called a time dependent continuous Markov process. Indeed, the geometric multiplicativity (2.3) is a manifestation of the Markov property. Amnon Yekutieli (BGU) Multiplicative Integration 13 / 45

  34. 2. MI on Curves 1-dimensional MI is used in various areas, such as mathematical physics and probability. There are various names and notations for this operation. One name is path ordered exponential integral, with corresponding notation � Pexp σ α . In probability this operation is called a time dependent continuous Markov process. Indeed, the geometric multiplicativity (2.3) is a manifestation of the Markov property. Amnon Yekutieli (BGU) Multiplicative Integration 13 / 45

  35. 2. MI on Curves 1-dimensional MI is used in various areas, such as mathematical physics and probability. There are various names and notations for this operation. One name is path ordered exponential integral, with corresponding notation � Pexp σ α . In probability this operation is called a time dependent continuous Markov process. Indeed, the geometric multiplicativity (2.3) is a manifestation of the Markov property. Amnon Yekutieli (BGU) Multiplicative Integration 13 / 45

  36. 2. MI on Curves 1-dimensional MI is used in various areas, such as mathematical physics and probability. There are various names and notations for this operation. One name is path ordered exponential integral, with corresponding notation � Pexp σ α . In probability this operation is called a time dependent continuous Markov process. Indeed, the geometric multiplicativity (2.3) is a manifestation of the Markov property. Amnon Yekutieli (BGU) Multiplicative Integration 13 / 45

  37. 2. MI on Curves In differential geometry the 1-dimensional MI has the following interpretation. Suppose E is a vector bundle of rank n over X , with a connection ∇ . Assume E is trivial; so for a choice of basis the connection ∇ has a matrix α ∈ Ω 1 ( X ) ⊗ M n ( R ) . Let σ be a path in X . Then the element MI ( α | σ ) ∈ GL n ( R ) is the holonomy of ∇ along σ . Amnon Yekutieli (BGU) Multiplicative Integration 14 / 45

  38. 2. MI on Curves In differential geometry the 1-dimensional MI has the following interpretation. Suppose E is a vector bundle of rank n over X , with a connection ∇ . Assume E is trivial; so for a choice of basis the connection ∇ has a matrix α ∈ Ω 1 ( X ) ⊗ M n ( R ) . Let σ be a path in X . Then the element MI ( α | σ ) ∈ GL n ( R ) is the holonomy of ∇ along σ . Amnon Yekutieli (BGU) Multiplicative Integration 14 / 45

  39. 2. MI on Curves In differential geometry the 1-dimensional MI has the following interpretation. Suppose E is a vector bundle of rank n over X , with a connection ∇ . Assume E is trivial; so for a choice of basis the connection ∇ has a matrix α ∈ Ω 1 ( X ) ⊗ M n ( R ) . Let σ be a path in X . Then the element MI ( α | σ ) ∈ GL n ( R ) is the holonomy of ∇ along σ . Amnon Yekutieli (BGU) Multiplicative Integration 14 / 45

  40. 2. MI on Curves In differential geometry the 1-dimensional MI has the following interpretation. Suppose E is a vector bundle of rank n over X , with a connection ∇ . Assume E is trivial; so for a choice of basis the connection ∇ has a matrix α ∈ Ω 1 ( X ) ⊗ M n ( R ) . Let σ be a path in X . Then the element MI ( α | σ ) ∈ GL n ( R ) is the holonomy of ∇ along σ . Amnon Yekutieli (BGU) Multiplicative Integration 14 / 45

  41. 3. MI on Surfaces – a Naive Attempt 3. MI on Surfaces – a Naive Attempt Consider another Lie group H , with Lie algebra h . As before X is a manifold. Let β be an h -valued 2-form on X , i.e. β ∈ Ω 2 ( X ) ⊗ h . Let τ : ∆ 2 → X be a smooth map. So τ is a triangle in X : Amnon Yekutieli (BGU) Multiplicative Integration 15 / 45

  42. 3. MI on Surfaces – a Naive Attempt 3. MI on Surfaces – a Naive Attempt Consider another Lie group H , with Lie algebra h . As before X is a manifold. Let β be an h -valued 2-form on X , i.e. β ∈ Ω 2 ( X ) ⊗ h . Let τ : ∆ 2 → X be a smooth map. So τ is a triangle in X : Amnon Yekutieli (BGU) Multiplicative Integration 15 / 45

  43. 3. MI on Surfaces – a Naive Attempt 3. MI on Surfaces – a Naive Attempt Consider another Lie group H , with Lie algebra h . As before X is a manifold. Let β be an h -valued 2-form on X , i.e. β ∈ Ω 2 ( X ) ⊗ h . Let τ : ∆ 2 → X be a smooth map. So τ is a triangle in X : Amnon Yekutieli (BGU) Multiplicative Integration 15 / 45

  44. 3. MI on Surfaces – a Naive Attempt 3. MI on Surfaces – a Naive Attempt Consider another Lie group H , with Lie algebra h . As before X is a manifold. Let β be an h -valued 2-form on X , i.e. β ∈ Ω 2 ( X ) ⊗ h . Let τ : ∆ 2 → X be a smooth map. So τ is a triangle in X : Amnon Yekutieli (BGU) Multiplicative Integration 15 / 45

  45. 3. MI on Surfaces – a Naive Attempt 3. MI on Surfaces – a Naive Attempt Consider another Lie group H , with Lie algebra h . As before X is a manifold. Let β be an h -valued 2-form on X , i.e. β ∈ Ω 2 ( X ) ⊗ h . Let τ : ∆ 2 → X be a smooth map. So τ is a triangle in X : Amnon Yekutieli (BGU) Multiplicative Integration 15 / 45

  46. 3. MI on Surfaces – a Naive Attempt We would like to construct a multiplicative integral MI ( β | τ ) ∈ H . For any k ≥ 0 we partition the simplex ∆ 2 into 4 k triangles labeled 1, . . . , 4 k , by the recursive rule shown below. Composing with τ : ∆ 2 → X we obtain, for each k , a sequence of maps τ 1 , . . . , τ 4 k : ∆ 2 → X . Amnon Yekutieli (BGU) Multiplicative Integration 16 / 45

  47. 3. MI on Surfaces – a Naive Attempt We would like to construct a multiplicative integral MI ( β | τ ) ∈ H . For any k ≥ 0 we partition the simplex ∆ 2 into 4 k triangles labeled 1, . . . , 4 k , by the recursive rule shown below. Composing with τ : ∆ 2 → X we obtain, for each k , a sequence of maps τ 1 , . . . , τ 4 k : ∆ 2 → X . Amnon Yekutieli (BGU) Multiplicative Integration 16 / 45

  48. 3. MI on Surfaces – a Naive Attempt We would like to construct a multiplicative integral MI ( β | τ ) ∈ H . For any k ≥ 0 we partition the simplex ∆ 2 into 4 k triangles labeled 1, . . . , 4 k , by the recursive rule shown below. Composing with τ : ∆ 2 → X we obtain, for each k , a sequence of maps τ 1 , . . . , τ 4 k : ∆ 2 → X . Amnon Yekutieli (BGU) Multiplicative Integration 16 / 45

  49. 3. MI on Surfaces – a Naive Attempt We would like to construct a multiplicative integral MI ( β | τ ) ∈ H . For any k ≥ 0 we partition the simplex ∆ 2 into 4 k triangles labeled 1, . . . , 4 k , by the recursive rule shown below. Composing with τ : ∆ 2 → X we obtain, for each k , a sequence of maps τ 1 , . . . , τ 4 k : ∆ 2 → X . Amnon Yekutieli (BGU) Multiplicative Integration 16 / 45

  50. 3. MI on Surfaces – a Naive Attempt We would like to construct a multiplicative integral MI ( β | τ ) ∈ H . For any k ≥ 0 we partition the simplex ∆ 2 into 4 k triangles labeled 1, . . . , 4 k , by the recursive rule shown below. Composing with τ : ∆ 2 → X we obtain, for each k , a sequence of maps τ 1 , . . . , τ 4 k : ∆ 2 → X . Amnon Yekutieli (BGU) Multiplicative Integration 16 / 45

  51. 3. MI on Surfaces – a Naive Attempt We would like to construct a multiplicative integral MI ( β | τ ) ∈ H . For any k ≥ 0 we partition the simplex ∆ 2 into 4 k triangles labeled 1, . . . , 4 k , by the recursive rule shown below. Composing with τ : ∆ 2 → X we obtain, for each k , a sequence of maps τ 1 , . . . , τ 4 k : ∆ 2 → X . Amnon Yekutieli (BGU) Multiplicative Integration 16 / 45

  52. 3. MI on Surfaces – a Naive Attempt We would like to construct a multiplicative integral MI ( β | τ ) ∈ H . For any k ≥ 0 we partition the simplex ∆ 2 into 4 k triangles labeled 1, . . . , 4 k , by the recursive rule shown below. Composing with τ : ∆ 2 → X we obtain, for each k , a sequence of maps τ 1 , . . . , τ 4 k : ∆ 2 → X . Amnon Yekutieli (BGU) Multiplicative Integration 16 / 45

  53. 3. MI on Surfaces – a Naive Attempt We then define the k -th Riemann Product 4 k � ∈ H . � � ∏ RP k ( β | τ ) : = exp H β (3.1) τ i i = 1 The geometry involved in these Riemann products is thus of a fractal nature. The limit MI ( β | τ ) : = lim k → ∞ RP k ( β | τ ) ∈ H (3.2) exists. We know that when H is abelian there is equality � � � MI ( β | τ ) = exp H τ β . Amnon Yekutieli (BGU) Multiplicative Integration 17 / 45

  54. 3. MI on Surfaces – a Naive Attempt We then define the k -th Riemann Product 4 k � ∈ H . � � ∏ RP k ( β | τ ) : = exp H β (3.1) τ i i = 1 The geometry involved in these Riemann products is thus of a fractal nature. The limit MI ( β | τ ) : = lim k → ∞ RP k ( β | τ ) ∈ H (3.2) exists. We know that when H is abelian there is equality � � � MI ( β | τ ) = exp H τ β . Amnon Yekutieli (BGU) Multiplicative Integration 17 / 45

  55. 3. MI on Surfaces – a Naive Attempt We then define the k -th Riemann Product 4 k � ∈ H . � � ∏ RP k ( β | τ ) : = exp H β (3.1) τ i i = 1 The geometry involved in these Riemann products is thus of a fractal nature. The limit MI ( β | τ ) : = lim k → ∞ RP k ( β | τ ) ∈ H (3.2) exists. We know that when H is abelian there is equality � � � MI ( β | τ ) = exp H τ β . Amnon Yekutieli (BGU) Multiplicative Integration 17 / 45

  56. 3. MI on Surfaces – a Naive Attempt We then define the k -th Riemann Product 4 k � ∈ H . � � ∏ RP k ( β | τ ) : = exp H β (3.1) τ i i = 1 The geometry involved in these Riemann products is thus of a fractal nature. The limit MI ( β | τ ) : = lim k → ∞ RP k ( β | τ ) ∈ H (3.2) exists. We know that when H is abelian there is equality � � � MI ( β | τ ) = exp H τ β . Amnon Yekutieli (BGU) Multiplicative Integration 17 / 45

  57. 3. MI on Surfaces – a Naive Attempt What about “geometric multiplicativity” ? Suppose we partition ∆ 2 into two triangles, by passing a straight line from v 2 to an arbitrary point on the opposite edge. We get two smooth maps τ 1 , τ 2 : ∆ 2 → X as shown below. Amnon Yekutieli (BGU) Multiplicative Integration 18 / 45

  58. 3. MI on Surfaces – a Naive Attempt What about “geometric multiplicativity” ? Suppose we partition ∆ 2 into two triangles, by passing a straight line from v 2 to an arbitrary point on the opposite edge. We get two smooth maps τ 1 , τ 2 : ∆ 2 → X as shown below. Amnon Yekutieli (BGU) Multiplicative Integration 18 / 45

  59. 3. MI on Surfaces – a Naive Attempt What about “geometric multiplicativity” ? Suppose we partition ∆ 2 into two triangles, by passing a straight line from v 2 to an arbitrary point on the opposite edge. We get two smooth maps τ 1 , τ 2 : ∆ 2 → X as shown below. Amnon Yekutieli (BGU) Multiplicative Integration 18 / 45

  60. 3. MI on Surfaces – a Naive Attempt What about “geometric multiplicativity” ? Suppose we partition ∆ 2 into two triangles, by passing a straight line from v 2 to an arbitrary point on the opposite edge. We get two smooth maps τ 1 , τ 2 : ∆ 2 → X as shown below. Amnon Yekutieli (BGU) Multiplicative Integration 18 / 45

  61. 3. MI on Surfaces – a Naive Attempt We would like MI ( β | τ ) to be the product of MI ( β | τ 1 ) and MI ( β | τ 2 ) . But the product in which order? Remember that the group H is not abelian. The answer: in general, neither order works! In the next section we are going to produce a more refined MI, both in terms of the fractal geometry and in terms of the Lie theory, in an attempt to solve this problem. Amnon Yekutieli (BGU) Multiplicative Integration 19 / 45

  62. 3. MI on Surfaces – a Naive Attempt We would like MI ( β | τ ) to be the product of MI ( β | τ 1 ) and MI ( β | τ 2 ) . But the product in which order? Remember that the group H is not abelian. The answer: in general, neither order works! In the next section we are going to produce a more refined MI, both in terms of the fractal geometry and in terms of the Lie theory, in an attempt to solve this problem. Amnon Yekutieli (BGU) Multiplicative Integration 19 / 45

  63. 3. MI on Surfaces – a Naive Attempt We would like MI ( β | τ ) to be the product of MI ( β | τ 1 ) and MI ( β | τ 2 ) . But the product in which order? Remember that the group H is not abelian. The answer: in general, neither order works! In the next section we are going to produce a more refined MI, both in terms of the fractal geometry and in terms of the Lie theory, in an attempt to solve this problem. Amnon Yekutieli (BGU) Multiplicative Integration 19 / 45

  64. 3. MI on Surfaces – a Naive Attempt We would like MI ( β | τ ) to be the product of MI ( β | τ 1 ) and MI ( β | τ 2 ) . But the product in which order? Remember that the group H is not abelian. The answer: in general, neither order works! In the next section we are going to produce a more refined MI, both in terms of the fractal geometry and in terms of the Lie theory, in an attempt to solve this problem. Amnon Yekutieli (BGU) Multiplicative Integration 19 / 45

  65. 4. Twisting the 2-Dimensional MI 4. Twisting the 2 -Dimensional MI Definition 4.1. A Lie crossed module is data ( G , H , Ψ , Φ ) consisting of: ◮ Lie groups G and H . ◮ An analytic action Ψ of G on H by automorphisms of Lie groups, called the twisting. ◮ A map of Lie groups Φ : H → G , called the feedback. The conditions are: (i) The feedback Φ is G -equivariant, with respect to the twisting Ψ , and the conjugation action Ad G of G on itself. (ii) Ψ ◦ Φ = Ad H , as actions of H on itself. Amnon Yekutieli (BGU) Multiplicative Integration 20 / 45

  66. 4. Twisting the 2-Dimensional MI 4. Twisting the 2 -Dimensional MI Definition 4.1. A Lie crossed module is data ( G , H , Ψ , Φ ) consisting of: ◮ Lie groups G and H . ◮ An analytic action Ψ of G on H by automorphisms of Lie groups, called the twisting. ◮ A map of Lie groups Φ : H → G , called the feedback. The conditions are: (i) The feedback Φ is G -equivariant, with respect to the twisting Ψ , and the conjugation action Ad G of G on itself. (ii) Ψ ◦ Φ = Ad H , as actions of H on itself. Amnon Yekutieli (BGU) Multiplicative Integration 20 / 45

  67. 4. Twisting the 2-Dimensional MI 4. Twisting the 2 -Dimensional MI Definition 4.1. A Lie crossed module is data ( G , H , Ψ , Φ ) consisting of: ◮ Lie groups G and H . ◮ An analytic action Ψ of G on H by automorphisms of Lie groups, called the twisting. ◮ A map of Lie groups Φ : H → G , called the feedback. The conditions are: (i) The feedback Φ is G -equivariant, with respect to the twisting Ψ , and the conjugation action Ad G of G on itself. (ii) Ψ ◦ Φ = Ad H , as actions of H on itself. Amnon Yekutieli (BGU) Multiplicative Integration 20 / 45

  68. 4. Twisting the 2-Dimensional MI 4. Twisting the 2 -Dimensional MI Definition 4.1. A Lie crossed module is data ( G , H , Ψ , Φ ) consisting of: ◮ Lie groups G and H . ◮ An analytic action Ψ of G on H by automorphisms of Lie groups, called the twisting. ◮ A map of Lie groups Φ : H → G , called the feedback. The conditions are: (i) The feedback Φ is G -equivariant, with respect to the twisting Ψ , and the conjugation action Ad G of G on itself. (ii) Ψ ◦ Φ = Ad H , as actions of H on itself. Amnon Yekutieli (BGU) Multiplicative Integration 20 / 45

  69. 4. Twisting the 2-Dimensional MI 4. Twisting the 2 -Dimensional MI Definition 4.1. A Lie crossed module is data ( G , H , Ψ , Φ ) consisting of: ◮ Lie groups G and H . ◮ An analytic action Ψ of G on H by automorphisms of Lie groups, called the twisting. ◮ A map of Lie groups Φ : H → G , called the feedback. The conditions are: (i) The feedback Φ is G -equivariant, with respect to the twisting Ψ , and the conjugation action Ad G of G on itself. (ii) Ψ ◦ Φ = Ad H , as actions of H on itself. Amnon Yekutieli (BGU) Multiplicative Integration 20 / 45

  70. 4. Twisting the 2-Dimensional MI 4. Twisting the 2 -Dimensional MI Definition 4.1. A Lie crossed module is data ( G , H , Ψ , Φ ) consisting of: ◮ Lie groups G and H . ◮ An analytic action Ψ of G on H by automorphisms of Lie groups, called the twisting. ◮ A map of Lie groups Φ : H → G , called the feedback. The conditions are: (i) The feedback Φ is G -equivariant, with respect to the twisting Ψ , and the conjugation action Ad G of G on itself. (ii) Ψ ◦ Φ = Ad H , as actions of H on itself. Amnon Yekutieli (BGU) Multiplicative Integration 20 / 45

  71. 4. Twisting the 2-Dimensional MI 4. Twisting the 2 -Dimensional MI Definition 4.1. A Lie crossed module is data ( G , H , Ψ , Φ ) consisting of: ◮ Lie groups G and H . ◮ An analytic action Ψ of G on H by automorphisms of Lie groups, called the twisting. ◮ A map of Lie groups Φ : H → G , called the feedback. The conditions are: (i) The feedback Φ is G -equivariant, with respect to the twisting Ψ , and the conjugation action Ad G of G on itself. (ii) Ψ ◦ Φ = Ad H , as actions of H on itself. Amnon Yekutieli (BGU) Multiplicative Integration 20 / 45

  72. 4. Twisting the 2-Dimensional MI 4. Twisting the 2 -Dimensional MI Definition 4.1. A Lie crossed module is data ( G , H , Ψ , Φ ) consisting of: ◮ Lie groups G and H . ◮ An analytic action Ψ of G on H by automorphisms of Lie groups, called the twisting. ◮ A map of Lie groups Φ : H → G , called the feedback. The conditions are: (i) The feedback Φ is G -equivariant, with respect to the twisting Ψ , and the conjugation action Ad G of G on itself. (ii) Ψ ◦ Φ = Ad H , as actions of H on itself. Amnon Yekutieli (BGU) Multiplicative Integration 20 / 45

  73. � � 4. Twisting the 2-Dimensional MI Here is a commutative diagram of groups depicting the situation: Φ Ψ � Aut ( H ) H G (4.2) Ad H The subgroup H 0 : = Ker ( Φ ) ⊂ H is called the inertia group. Note that H 0 ⊂ Ker ( Ad H ) = Z ( H ) , where Z ( H ) is the center of the group H . We shall denote the Lie algebras of G and H by g and h respectively. Amnon Yekutieli (BGU) Multiplicative Integration 21 / 45

  74. � � 4. Twisting the 2-Dimensional MI Here is a commutative diagram of groups depicting the situation: Φ Ψ � Aut ( H ) H G (4.2) Ad H The subgroup H 0 : = Ker ( Φ ) ⊂ H is called the inertia group. Note that H 0 ⊂ Ker ( Ad H ) = Z ( H ) , where Z ( H ) is the center of the group H . We shall denote the Lie algebras of G and H by g and h respectively. Amnon Yekutieli (BGU) Multiplicative Integration 21 / 45

  75. � � 4. Twisting the 2-Dimensional MI Here is a commutative diagram of groups depicting the situation: Φ Ψ � Aut ( H ) H G (4.2) Ad H The subgroup H 0 : = Ker ( Φ ) ⊂ H is called the inertia group. Note that H 0 ⊂ Ker ( Ad H ) = Z ( H ) , where Z ( H ) is the center of the group H . We shall denote the Lie algebras of G and H by g and h respectively. Amnon Yekutieli (BGU) Multiplicative Integration 21 / 45

  76. � � 4. Twisting the 2-Dimensional MI Here is a commutative diagram of groups depicting the situation: Φ Ψ � Aut ( H ) H G (4.2) Ad H The subgroup H 0 : = Ker ( Φ ) ⊂ H is called the inertia group. Note that H 0 ⊂ Ker ( Ad H ) = Z ( H ) , where Z ( H ) is the center of the group H . We shall denote the Lie algebras of G and H by g and h respectively. Amnon Yekutieli (BGU) Multiplicative Integration 21 / 45

  77. 4. Twisting the 2-Dimensional MI Here are a few examples of Lie crossed modules ( G , H , Ψ , Φ ) . Example 4.3. H is any Lie group, G = H , Ψ = Ad G and Φ = id. Here H 0 is the trivial group. Example 4.4. H is an abelian Lie group, and G is the trivial group. Here H 0 = H . Example 4.5. Suppose 1 → N → H Φ − → G → 1 is a central extension of Lie groups. There is an induced action Ψ of G on H (this is an easy exercise in group theory). Here H 0 = N of course. This example contains both previous examples. Amnon Yekutieli (BGU) Multiplicative Integration 22 / 45

  78. 4. Twisting the 2-Dimensional MI Here are a few examples of Lie crossed modules ( G , H , Ψ , Φ ) . Example 4.3. H is any Lie group, G = H , Ψ = Ad G and Φ = id. Here H 0 is the trivial group. Example 4.4. H is an abelian Lie group, and G is the trivial group. Here H 0 = H . Example 4.5. Suppose 1 → N → H Φ − → G → 1 is a central extension of Lie groups. There is an induced action Ψ of G on H (this is an easy exercise in group theory). Here H 0 = N of course. This example contains both previous examples. Amnon Yekutieli (BGU) Multiplicative Integration 22 / 45

  79. 4. Twisting the 2-Dimensional MI Here are a few examples of Lie crossed modules ( G , H , Ψ , Φ ) . Example 4.3. H is any Lie group, G = H , Ψ = Ad G and Φ = id. Here H 0 is the trivial group. Example 4.4. H is an abelian Lie group, and G is the trivial group. Here H 0 = H . Example 4.5. Suppose 1 → N → H Φ − → G → 1 is a central extension of Lie groups. There is an induced action Ψ of G on H (this is an easy exercise in group theory). Here H 0 = N of course. This example contains both previous examples. Amnon Yekutieli (BGU) Multiplicative Integration 22 / 45

  80. 4. Twisting the 2-Dimensional MI Here are a few examples of Lie crossed modules ( G , H , Ψ , Φ ) . Example 4.3. H is any Lie group, G = H , Ψ = Ad G and Φ = id. Here H 0 is the trivial group. Example 4.4. H is an abelian Lie group, and G is the trivial group. Here H 0 = H . Example 4.5. Suppose 1 → N → H Φ − → G → 1 is a central extension of Lie groups. There is an induced action Ψ of G on H (this is an easy exercise in group theory). Here H 0 = N of course. This example contains both previous examples. Amnon Yekutieli (BGU) Multiplicative Integration 22 / 45

  81. 4. Twisting the 2-Dimensional MI Here are a few examples of Lie crossed modules ( G , H , Ψ , Φ ) . Example 4.3. H is any Lie group, G = H , Ψ = Ad G and Φ = id. Here H 0 is the trivial group. Example 4.4. H is an abelian Lie group, and G is the trivial group. Here H 0 = H . Example 4.5. Suppose 1 → N → H Φ − → G → 1 is a central extension of Lie groups. There is an induced action Ψ of G on H (this is an easy exercise in group theory). Here H 0 = N of course. This example contains both previous examples. Amnon Yekutieli (BGU) Multiplicative Integration 22 / 45

  82. 4. Twisting the 2-Dimensional MI Here are a few examples of Lie crossed modules ( G , H , Ψ , Φ ) . Example 4.3. H is any Lie group, G = H , Ψ = Ad G and Φ = id. Here H 0 is the trivial group. Example 4.4. H is an abelian Lie group, and G is the trivial group. Here H 0 = H . Example 4.5. Suppose 1 → N → H Φ − → G → 1 is a central extension of Lie groups. There is an induced action Ψ of G on H (this is an easy exercise in group theory). Here H 0 = N of course. This example contains both previous examples. Amnon Yekutieli (BGU) Multiplicative Integration 22 / 45

  83. 4. Twisting the 2-Dimensional MI Example 4.6. Consider a nonabelian unipotent group H , e.g. � 1 ∗ ∗ � H = ⊂ GL 3 ( R ) . 0 1 ∗ 0 0 1 Here exp H : h → H is a diffeomorphism. This implies that the group G : = Aut ( H ) is a Lie group (isomorphic to a closed subgroup of GL ( h ) ). We get a Lie crossed module ( G , H , Ψ , Φ ) with Φ : = Ad H . The inertia group here is of intermediate size: 1 � H 0 � H . This is the sort of thing that comes up in twisted deformation quantization (the Deligne crossed groupoid). Amnon Yekutieli (BGU) Multiplicative Integration 23 / 45

  84. 4. Twisting the 2-Dimensional MI Example 4.6. Consider a nonabelian unipotent group H , e.g. � 1 ∗ ∗ � H = ⊂ GL 3 ( R ) . 0 1 ∗ 0 0 1 Here exp H : h → H is a diffeomorphism. This implies that the group G : = Aut ( H ) is a Lie group (isomorphic to a closed subgroup of GL ( h ) ). We get a Lie crossed module ( G , H , Ψ , Φ ) with Φ : = Ad H . The inertia group here is of intermediate size: 1 � H 0 � H . This is the sort of thing that comes up in twisted deformation quantization (the Deligne crossed groupoid). Amnon Yekutieli (BGU) Multiplicative Integration 23 / 45

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