Exponentials of derivations in prime Gradings characteristic - PowerPoint PPT Presentation

Exponentials of derivations S. Mattarei Ordinary exponentials Truncated exponentials Exponentials of derivations in prime Gradings characteristic Artin-Hasse exponentials Laguerre polynomials Sandro Mattarei University of Lincoln Bath, February 2016 1 / 33

Exponentials of derivations S. Mattarei Taking a break from maths: Ordinary exponentials Truncated exponentials Gradings Artin-Hasse exponentials Laguerre polynomials 2 / 33

Exponentials Summary of derivations S. Mattarei Ordinary exponentials Truncated Traditional exponentials in characteristic zero 1 exponentials Gradings Artin-Hasse 2 Truncated exponentials exponentials Laguerre polynomials Application to gradings of algebras 3 4 Artin-Hasse exponentials Laguerre polynomials 5 3 / 33

Exponentials Derivations and automorphisms of derivations S. Mattarei Let A be a non-associative algebra over a field F . Ordinary • A derivation of A is a linear map D : A → A such that exponentials Truncated D ( a · b ) = ( Da ) · b + a · ( Db ) , for a , b ∈ A . exponentials Gradings Artin-Hasse Lemma exponentials Laguerre Assume char ( F ) = 0 . If D is a nilpotent derivation of A, then polynomials exp D = � ∞ k = 0 D k / k ! is an automorphism of A. • D being a derivation is equivalent to D ◦ m = m ◦ ( D ⊗ id + id ⊗ D ) , where m : A ⊗ A → A is the multiplication map. • The Lemma follows from exp ( X + Y ) = exp ( X ) · exp ( Y ) after setting X = D ⊗ id and Y = id ⊗ D . 5 / 33

Exponentials Proof of the Lemma of derivations S. Mattarei Ordinary Proof. exponentials Truncated Because exponentials D k ◦ m = m ◦ ( D ⊗ id + id ⊗ D ) k Gradings Artin-Hasse for k ≥ 0, we have exponentials Laguerre polynomials ( exp D ) ◦ m = m ◦ exp ( D ⊗ id + id ⊗ D ) = m ◦ exp ( D ⊗ id ) ◦ exp ( id ⊗ D ) � � � � = m ◦ ( exp D ) ⊗ id ◦ id ⊗ ( exp D ) Evaluating on x ⊗ y , for x , y ∈ A , we get ( exp D )( x · y ) = ( exp D )( x ) · ( exp D )( y ) and hence exp D is an automorphism of A . 6 / 33

Exponentials Example: A polynomial algebra of derivations S. Mattarei Ordinary Example exponentials Truncated Let A = F [ X ] , D = d / dX , α, β ∈ F . Then exponentials Gradings • exp ( β D ) f ( X ) = f ( X + β ) (Taylor’s formula); Artin-Hasse (if e α makes sense). • exp ( α XD ) f ( X ) = f ( e α X ) exponentials Laguerre polynomials • In fact, all automorphisms of F [ X ] as an F -algebra are given by substitutions X �→ aX + b , for a ∈ F ∗ , b ∈ F . • The derivation algebra is much larger, � � F · X k + 1 D , W 1 = Der ( F [ X ]) = Der ( F [ X ]) k = k ≥− 1 k ≥− 1 but exp does not apply to derivations of positive degree. 7 / 33

Exponentials Example: The Lie algebra W 1 of derivations S. Mattarei • W 1 = Der ( F [ X ]) is the Lie algebra of polynomial vector Ordinary exponentials fields on the line (usually with F = R or C ). Truncated • W 1 has a Z -graded basis given by the X i + 1 D , where exponentials D = d / dX , this element having degree i , for i ≥ − 1. Gradings Artin-Hasse • Lie bracket: exponentials Laguerre [ X i + 1 D , X j + 1 D ] = ( j − i ) X i + j + 1 D . polynomials In particular, consider the inner derivation ad D = [ D , · ] . Example Lie algebra W 1 = Der ( F [ X ]) . Then exp ( ad D ) is an automorphism of W 1 . Explicitly: exp ( ad D ) X i + 1 D = ( X + 1 ) i + 1 D 8 / 33

Exponentials Exponentials in positive characteristic of derivations S. Mattarei From now on assume char ( F ) = p > 0. Ordinary exponentials • For exp ( D ) to make sense we need at least D p = 0, but Truncated exponentials then what we really apply is the truncated exponential Gradings Artin-Hasse p − 1 exponentials � D k / k ! E ( D ) = Laguerre polynomials k = 0 • This is defined for any derivation D but it need not be an automorphism, even when D p = 0. • In the theory of modular Lie algebras, this is good : certain E ( D ) can be used to pass from some torus to another torus with more desirable properties ( toral switching: [Winter 1969], [Block-Wilson 1982], [Premet 1986/89]). 10 / 33

Exponentials What fails with the truncated exponential of derivations S. Mattarei We compute E ( X ) · E ( Y ) , Ordinary exponentials Y 2 Y 3 Y p − 1 1 Y . . . . . . Truncated 2 ! 3 ! ( p − 1 )! exponentials XY 2 XY p − 2 XY p − 1 X XY 2 ! ( p − 2 )! ( p − 1 )! Gradings X 2 X 2 Y X 2 Y p − 3 X 2 Y p − 2 X 2 Y p − 1 Artin-Hasse 2 ! 2 ! 2 !( p − 3 )! 2 !( p − 2 )! 2 !( p − 1 )! exponentials . . X 3 . Laguerre 3 ! polynomials . . X p − 3 Y 2 . ( p − 3 )! 2 ! . . . . X p − 2 Y X p − 2 Y 2 . . ( p − 2 )! ( p − 2 )! 2 ! X p − 1 X p − 1 Y X p − 1 Y 2 X p − 1 Y p − 1 . . . . . . ( p − 1 )! ( p − 1 )! ( p − 1 )! 2 ! ( p − 1 )!( p − 1 )! and find 2 p − 2 p − 1 X i Y k − i � � E ( X ) · E ( Y ) − E ( X + Y ) = i !( k − i )! . k = p i = k + 1 − p 11 / 33

Exponentials A closer look at the term of degree p of derivations S. Mattarei Ordinary exponentials Truncated • The term with k = p in E ( X ) · E ( Y ) − E ( X + Y ) is exponentials Gradings p − 1 X i Y p − i = ( X + Y ) p − X p − Y p Artin-Hasse � p � 1 � exponentials . p ! i p ! Laguerre i = 1 polynomials • Modulo p it can also be written as p − 1 ( − 1 ) i � X i Y p − i . i i = 1 12 / 33

Exponentials The obstruction formula of derivations S. Mattarei Ordinary exponentials Truncated exponentials • Setting X = D ⊗ id and Y = id ⊗ D yields the Gradings obstruction formula Artin-Hasse exponentials 2 p − 2 p − 1 ( D i x )( D k − i y ) Laguerre � � polynomials E ( D ) x · E ( D ) y − E ( D )( xy ) = , i !( k − i )! k = p i = k + 1 − p for D any derivation of A , and x , y ∈ A . • In particular, if p is odd and D ( p + 1 ) / 2 = 0, then E ( D ) is an automorphism of A . 13 / 33

Exponentials Example: A truncated polynomial ring of derivations S. Mattarei Example Ordinary exponentials If A = F [ X ] / ( X p ) and D = d / dX , then D p = 0, and Truncated exponentials E ( D ) X k = ( X + 1 ) k Gradings for 0 ≤ k < p . Artin-Hasse exponentials Here X p = 0, but ( X + 1 ) p = 1, and hence Laguerre polynomials E ( D ) is not an automorphism of A . • However, A = F 1 ⊕ FX ⊕ · · · ⊕ FX p − 1 is a Z -grading of A , and E ( D ) maps it to A = F 1 ⊕ F ( X + 1 ) ⊕ · · · ⊕ F ( X + 1 ) p − 1 , which is a ( genuine ) Z / p Z -grading of A . 14 / 33

Exponentials Why did E ( D ) turn a grading into another? of derivations S. Mattarei Lemma Ordinary exponentials If D is a derivation of A with D p = 0 , for x , y ∈ A we have Truncated exponentials p − 1 Gradings ( − 1 ) i � ( D i x )( D p − i y ) . E ( D ) x · E ( D ) y − E ( D )( xy ) = E ( D ) Artin-Hasse i exponentials i = 1 Laguerre polynomials • The sum at the RHS equals the term with k = p of the obstruction formula. That is the primary obstruction cocycle p − 1 D i D p − i � ( p − i )! ∈ Z 2 ( A , A ) Sq p ( D ) = i ! ⌣ i = 1 which arises in Gerstenhaber’s deformation theory . 16 / 33

Exponentials Truncated exponentials and gradings of derivations S. Mattarei Theorem (grading switching with D p = 0) Ordinary exponentials Truncated • Let A = � k A k be a Z / m Z -grading of A; exponentials • let D be a derivation of A, homogeneous of degree d, Gradings with m | pd, such that D p = 0 . Artin-Hasse exponentials Laguerre Then polynomials � A = E ( D ) A k k is a Z / m Z -grading of A. • In our example with A = F [ X ] / ( X p ) , its derivation D = d / dX had degree − 1, and A was graded over Z , but then also over Z / m Z with m = p . • Less trivial application: construction of gradings over a group having elements of order p 2 . 17 / 33

Exponentials Two basic methods to produce gradings of derivations S. Mattarei Ordinary � ( D − α · id ) i � � exponentials • If D ∈ Der ( A ) and A α = ker , Truncated i > 0 exponentials then A = � α ∈ F A α is a grading over the additive group Gradings of F (or a subgroup). Artin-Hasse exponentials • With ψ ∈ Aut ( A ) in place of D we get a grading Laguerre A = � α ∈ F ∗ A α over the multiplicative group of F . polynomials • Combining the two methods one can get gradings over any f.g. abelian group with no elements of order p 2 . • These methods alone are unable to produce genuine Z / p s Z -gradings with s > 1, which do occur in practice. • ‘genuine’ means that the grading does not simply come from a Z / m Z -grading with m = 0 or a larger power of p by viewing the degrees modulo p s . 18 / 33

Weakening the condition D p = 0 Exponentials of derivations S. Mattarei • The Artin-Hasse exponential series Ordinary ∞ exponentials p + X p 2 � X p i X + X p � � � � E p ( X ) = exp p 2 + · · · = exp Truncated p i exponentials i = 0 Gradings has coefficients in the (rational) p -adic integers. Artin-Hasse exponentials • For example, the term of degree p is ( p − 1 )!+ 1 X p . p ! Laguerre polynomials Lemma There exist integers a ij , with a ij = 0 if p ∤ i + j, such that for D a nilpotent derivation of A, and for x , y ∈ A, we have � a ij D i x · D j y . E p ( D ) x · E p ( D ) y − E p ( D )( xy ) = E p ( D ) i , j > 0 � a ij X i Y j � � • Proof: E p ( X ) · E p ( Y ) = E p ( X + Y ) · 1 + . 20 / 33