Nonassociative Lie Theory Ivan P . Shestakov The International - PowerPoint PPT Presentation

Nonassociative Lie Theory Ivan P . Shestakov The International Conference on Group Theory in Honor of the 70th Birthday of Professor Victor D. Mazurov Novosibirsk, July 16-20, 2013 Sobolev Institute of Mathematics Siberian Branch of the Russian Academy of Sciences Ivan Shestakov Nonassociative Lie Theory

� Lie Groups Lie � Associative Algebras � Hopf Algebras Algebras � Formal Groups � Analytic Loops Sabinin � Nonassociative Algebras � Nonassociative Hopf Algebras Algebras � Formal Loops

Local Loops and Tangent Algebras. Definition Quasigroup (nonassociative group) is a set with a binary operation � Q , ·� where equations a · x = b , y · a = b have unique solutions in Q for any a , b ∈ Q . Loop is a quasigroup with the unit element e . The solutions of the equations above define the operations of left and right division a \ b = x and b / a = y . In terms of these operations, we can give Equivalent definition: Loop is an algebraic system � M , · , \ , /, e � such that a \ ( a · b ) = b = a · ( a \ b ) ( a · b ) / b = a = ( a / b ) · b e · a = a = a · e Ivan Shestakov Nonassociative Lie Theory

Local loops. Let M be smooth finite-dimensional manifold. A local multiplication on open U ⊂ M is a smooth map F : U × U → M . If there exists e ∈ U with the property that F | e × U = Id ( U ) = F | U × e , the local multiplication F is called unital, or a local loop . The point e is referred to as the unit . Notation: F ( x , y ) , x · y , xy . Ivan Shestakov Nonassociative Lie Theory

Left and right divisions For any local loop there exist two local multiplications V × V → M with V ⊂ U , denoted by x / y and y \ x . As above, they are defined by ( x / y ) · y = x and y · ( y \ x ) = x . and are called the right and the left divison , respectively. The existence of both divisions follows from the fact that the right and left multiplication maps R y = F | U × y : U → M , L y = F | y × U : U → M are close to the inclusion map U ֒ → M when y is close to e . In particular, if y is sufficiently close to e , both maps R y and L y are one-to-one and their images contain a neighbourhood of e . We take V to be the largest neighbourhood on which both divisions are defined. Ivan Shestakov Nonassociative Lie Theory

Example 1: Invertible elements in algebras. Call an element a of a unital algebra invertible if both equations ax = 1 and xa = 1 have a unique solution. Let A be a finite-dimensional unital algebra over R . Then the invertible elements of A form a local loop. This local loop is not necessarily a loop. Consider, for instance, the generalized Cayley-Dickson algebras C n on R 2 n . When n > 3 there exist pairs of invertible elements in C n whose product is zero. Ivan Shestakov Nonassociative Lie Theory

Example 2: Homogeneous spaces. Let M be a homogeneous space for a Lie group G and U ⊂ M a neighbourhood of a point e ∈ M . Consider the mapping p : G → M , g �→ g ( e ) . Assume that we are given a section of p over U , that is, a smooth map i : U → G such that i ( e ) is the unit in G and p ◦ i = Id U . Then M is a local loop, with the multiplication U × U → M defined as ( x , y ) �→ p ( i ( x ) i ( y )) . When p is actually a homomorphism of Lie groups, that is, when the stabilizer G e of the element e is a normal subgroup in G , this local loop structure is the same thing as the product on M restricted to U × U . There are many important examples of homogeneous spaces, among them spheres, hyperbolic spaces and Grassmannians. Ivan Shestakov Nonassociative Lie Theory

Example 3: Analytic local loops. Consider an n -tuple of power series F ( x , y ) = ( F 1 ( x , y ) , . . . , F n ( x , y )) where x , y ∈ R n , and assume that all of them converge in some neighbourhood of the origin in R 2 n . Then the map ( x , y ) �→ F ( x , y ) defines a local loop on R n , with the origin as the unit, if and only if F ( 0 , y ) = y and F ( x , 0 ) = x for all x , y ∈ R n . A local loop on an analytic manifold whose multiplication can be written in this form in some coordinate chart is called analytic . Ivan Shestakov Nonassociative Lie Theory

Tangent algebras A.I.Malcev (1955): Analytic loop L ⇒ the tangent algebra T ( L ) , [ x , y ] = − [ y , x ] Malcev algebra T ( L ) , Moufang loop L , ⇒ [ a , a ] = 0 , a ( b ( ac )) = (( ab ) a ) c [ J ( a , b , c ) , a ] = J ( a , b , [ a , c ]) Alternative algebra A , ⇒ Malcev algebra A ( − ) , a ( bb ) = ( ab ) b , [ a , b ] = ab − ba a ( ab ) = ( aa ) b Alternative algebra A ⇒ Moufang loop U ( A ) Ivan Shestakov Nonassociative Lie Theory

Malcev algebras, alternative algebras, and Moufang loops E.N.Kuzmin (1971): Malcev algebras ⇒ Moufang loops I.P.Shestakov (2004): Moufang loops �⇒ Alternative algebras Malcev Problem: ??? Malcev algebras ⇒ Alternative algebras Ivan Shestakov Nonassociative Lie Theory

Akivis algebras M.Akivis (1976): Analytic loop L ⇒ Akivis algebra Ak ( L ) Akivis algebra ( A , + , [ · , · ] , �· , · , ·� ) : [ x , x ] = 0 � ( − 1 ) σ � x σ ( 1 ) , x σ ( 2 ) , x σ ( 3 ) � = J ( x 1 , x 2 , x 3 ) σ ∈ S 3 Algebra B → Ak B = � B , [ x , y ] , ( x , y , z ) � , where [ x , y ] = xy − yx , ( x , y , z ) = xy ) z − x ( yz ) . I.Shestakov (1999): Every Akivis algebra A can be embedded into the algebra Ak B for a suitable algebra B. Ivan Shestakov Nonassociative Lie Theory

Local loops and Sabinin algebras L 1 , L 2 local analytic loops. L 1 → Ak ( L 1 ) ∼ = Ak ( L 2 ) ← L 2 L 1 �∼ = L 2 L.Sabinin, P.Mikheev (1987): Local analytic ⇔ Hyperalgebras loops (Sabinin algebras) L → Sab ( L ) → L Ivan Shestakov Nonassociative Lie Theory

Primitive elements of bialgebras Bialgebra B = � B , + , m , ∆ � : � B , + , m � an algebra: m : B ⊗ B → B � B , + , ∆ � a coalgebra: ∆ : B → B ⊗ B ∆ is a homomorphism of algebras. Prim ( B , ∆) = { w ∈ B | ∆( w ) = w ⊗ 1 + 1 ⊗ w } . 1. B = F � X � , free associative algebra (char =0). ∆( x i ) = x i ⊗ 1 + 1 ⊗ x i . Prim ( F � X � , ∆) = Lie � X � . Ivan Shestakov Nonassociative Lie Theory

Primitive elements of bialgebras 1. B = F { X } , free nonassociative algebra (char =0). ∆( x i ) = x i ⊗ 1 + 1 ⊗ x i . K.Strambach: Prim ( F { X } , ∆) = Ak � X � ? I.Sh.+ U.Umirbaev (2001): p = ( x 2 , x , x ) − x ( x , x , x ) − ( x , x , x ) x ∈ Prim ( F { X } , ∆) , p �∈ Ak � X � Problem: To describe Prim ( F { X } , ∆) . Ivan Shestakov Nonassociative Lie Theory

Primitive elements of bialgebras I.Sh.+ U.Umirbaev: Prim ( F { X } , ∆) is generated (starting with X) by [ x , y ] , ( x , y , z ) and p ( x 1 , . . . , x n ; y 1 , . . . , y m ; z ) . Let u = x 1 x 2 · · · x n , v = y 1 y 2 · · · y m ; denote p ( x 1 , . . . , x n ; y 1 , . . . , y m ; z ) as p ( u , v , z ) . Then the equality � ( u , v , z ) = u ( 1 ) v ( 1 ) p ( u ( 2 ) , v ( 2 ) , z ) ( u ) , ( v ) defines the primitive elements p ( u , v , z ) inductively. p ( x 1 , y 1 , z ) = ( x , y , z ) p ( x 1 x 2 , y , z ) = ( x 1 x 2 , y , z ) − x 1 ( x 2 , y , z ) − x 2 ( x 1 , y , z ) . Ivan Shestakov Nonassociative Lie Theory

Primitive elements of bialgebras Theorem I.Sh.+ U.U.: Let � C , · , δ � be a unital bialgebra over a field F of characteristic 0. Then the space Prim ( C , δ ) is closed relatively the operations p ( u , v , z ) . If C is generated as an algebra by Prim ( C , δ ) then C has a PBW-base over Prim ( C , δ ) . Ivan Shestakov Nonassociative Lie Theory

Sabinin algebras V a vector space, T ( V ) the tensor algebra over V , ∆ : T ( V ) → T ( V ) ⊗ T ( V ) , v �→ 1 ⊗ v + v ⊗ 1 , v ∈ V . �− ; − , −� : T ( V ) ⊗ V ⊗ V → V , w ⊗ y ⊗ z �→ � w ; y , z � . � w ; y , y � = 0 , � w ⊗ u ⊗ v ⊗ w ′ ; y , z � − � w ⊗ v ⊗ u ⊗ w ′ ; y , z � � � w ( 1 ) ⊗ � w ( 2 ) ; u , v � ⊗ w ′ ; y , z � = 0 , + ( w ) � � ( � w ⊗ x ; y , z � + � w ( 1 ) ; � w ( 2 ) ; y , z � , x � ) = 0 . x , y , z ( w ) x , y , z , u , v ∈ V ; w , w ′ ∈ T ( V ) . Ivan Shestakov Nonassociative Lie Theory

Sabinin algebras Examples: - Lie algebras: � 1 ; a , b � = [ a , b ] , � x ; a , b � = 0 , x ∈ VT ( V ) . - Lie triple systems: � 1 ; a , b � = 0 , � u ; a , b � = [ a , b , u ] , u ∈ V ; � x ; a , b � = 0 , x ∈ V ⊗ i , i > 1. - Malcev algebras: � 1 ; a , b � = [ a , b ] , . . . Ivan Shestakov Nonassociative Lie Theory

Shestakov-Umirbaev functor I.Sh.+ U.U.: Let A be an arbitrary algebra. Define � 1 ; a , b � = [ a , b ] � a ; b , c � = ( a , b , c ) − ( a , c , b ) � a 1 , . . . , a n ; b , c � = p ( a 1 · · · a n ; b , c ) − p ( a 1 · · · a n ; c , b ) , where a , b , c , a 1 , . . . , a n ∈ A . Then A ( ∼ ) = � A , �· · · �� is a Sabinin algebra. If A is a bialgebra then Prim A is a subalgebra of the Sabinin algebra A ( ∼ ) . Ivan Shestakov Nonassociative Lie Theory