Marina Avitabile

Laguerre Polynomials of Derivations

Marina Avitabile

Universita di Milano - Bicocca

Trento, June 2013

Joint work with Sandro Mattarei

Marina Avitabile (MiB) Laguerre Polynomials of Derivations Trento, June 2013 1 / 25

1 Gradings of non-associative algebras

2 The exponential of a derivationThe Artin-Hasse exponential of a derivation

3 Laguerre polynomialsLaguerre polynomials modulo p

4 A model special case

5 General Case

6 Toral switching


Gradings

Definition

Let A be a (finite-dimensional) non-associative algebra over a field F. Agrading of A over an abelian group G is a direct sum decomposition:

A =⊕g∈G

Ag

such that AgAh ⊆ Ag+h.

A derivation D of A is a linear map D : A→ A such that

D(a · b) = D(a) · b + a · D(b)

for all a, b ∈ A.


Gradings

Let D be a derivation of A, with all its characteristic roots in F. Thedirect sum decomposition of A into (generalized) eigenspaces for D

A =⊕α∈F

Aα

(where Aα = {x ∈ A : (D − α id)i (x) = 0, for some i > 0}) is a grading ofA over (F,+), i.e. AαAβ ⊆ Aα+β.Let σ be an automorphism of A, with all its characteristic roots in F. Thedirect sum decomposition of A into generalized eigenspaces for σ

A =⊕α∈F

Aα

is a grading of A over (F∗, ·), i.e. AαAβ ⊆ Aαβ.


The exponential of a derivation

Let char(F) = 0 and D be a nilpotent derivation of A, with Dn = 0.The exponential map

exp(D) =n−1∑i=0

D i

i !

defines an automorphism of A.

D ◦m = m ◦ (D ⊗ id + id⊗D), where m : A⊗ A→ A is themultiplication map.

Set X = D ⊗ id and Y = id⊗D, then exp(X + Y ) = exp(X ) · exp(Y )(if X and Y commute).



From now on assume char(F) = p > 0

S. MattareiArtin-Hasse exponentials of derivationsJ. Algebra 294 (2005), 1–18

Let D be a nilpotent derivation of A with Dp = 0, then

exp(D) =

p−1∑i=0

D i

i !

and it defines a bijective linear map on A.



Define the truncated exponential of D as

E (D) =

p−1∑i=0

D i

i !.

Direct computation shows that

E (D)x · E (D)y − E (D)(xy) =

2p−2∑t=p

p−1∑i=t+1−p

(D ix)(Dt−iy)

i !(t − i)!.

If p is odd and Dp+1

2 = 0 then each term in the sum vanishes and exp(D)is an automorphism of A, but in general E (D) it is not an automorphismof A even if Dp = 0.



However, under certain hypotheses, the (truncated) exponential of aderivation has the property of sending a grading of A into another gradingof A.

Lemma (S. Mattarei)

Let A be a non-associative algebra over a field of positive characteristic p,with derivation D such that Dp = 0. Then

exp(D)x · exp(D)y − exp(D)(xy) = exp(D)

(p−1∑i=1

(−1)i

iD ix · Dp−iy

)

for all x , y ∈ A.

Let A = ⊕Ai be a grading of A over the integers modulo m. A derivationD of A is graded of degree d if D(Ai ) ⊆ Ai+d for every i .



Theorem (S. Mattarei)

Let A = ⊕Ai be a non-associative algebra over a field of positivecharacteristic p, graded over the integers modulo m. Suppose that A has agraded derivation D of degree d, with m | pd, such that Dp = 0. Thenthe direct sum decomposition A = ⊕ exp(D)Ai is a grading of A over theintegers modulo m.

Proof.

Verify that exp(D)As · exp(D)At ⊆ exp(D)As+t .Let x ∈ As and y ∈ At , then D i (x) ∈ As+di and Dp−i (y) ∈ At+d(p−i) and

D i (x) · Dp−i (y) ∈ As+t+pd = As+t . The previous lemma yields

exp(D)x · exp(D)y = exp(D)(xy)︸︷︷︸exp(D)(As+t)

+ exp(D)

(p−1∑i=0

(−1)i

iD ix · Dp−iy

)︸︷︷︸

exp(D)(As+t)


The Artin-Hasse exponential of a derivation

The assumption that Dp = 0 can be relaxed by considering theArtin-Hasse exponential series which is defined as

Ep(X ) = exp

( ∞∑i=0

X pi

pi

)=∞∏i=0

exp

(X pi

pi

)∈ Zp[[X ]].

Theorem (S. Mattarei)

Let A = ⊕Ai be a non-associative algebra over a field of positivecharacteristic p, graded over the integers modulo m. Suppose that A has anilpotent graded derivation D of degree d, with m | pd. Then the directsum decomposition A = ⊕Ep(D)Ai is a grading of A over the integersmodulo m.


The Artin-Hasse exponential of a derivation

Ideas from the proof.

Let D be a nilpotent derivation of A. Then there exist integers ai ,j , withai ,j = 0 when p 6 | i + j such that

Ep(D)x · Ep(D)y − Ep(D)(xy) = Ep(D)

∞∑i ,j=1

ai ,jDix · D jy

for all x , y ∈ A.Set X = D ⊗ id and Y = id⊗D, then the identity above is equivalent to

(Ep(X + Y ))−1Ep(X ) · Ep(Y ) = 1 +∞∑

i ,j=1

ai ,jXiY j

in Q[[X ,Y ]], where ai ,j = 0 when where p 6 | i + j .


Laguerre polynomials

The classical (generalized) Laguerre polynomial of degree n ≥ 0 is definedas

L(α)n (X ) =

n∑k=0

(α + n

n − k

)(−X )k

k!

where α is a parameter, classically in the complex numbers. We may also

view L(α)n (x) as a polynomial in two indeterminates X and α.

Recurrence relations

1 L(γ)n (X ) = L

(γ+1)n (X )− L

(γ+1)n−1 (X )

2 nL(γ+1)n (X ) = (n − X )L

(γ+1)n−1 (X ) + (n + γ)L

(γ)n−1(X )

3ddX L

(γ)n (X ) = −L

(γ+1)n−1 (X ) = L

(γ)n (X )− L

(γ+1)n (X )


Laguerre polynomials modulo p

Let p be a fixed prime, we consider Laguerre polynomials of degree p − 1

L(α)p−1(X ) =

p−1∑k=0

(α + p − 1

p − 1− k

)(−X )k

k!

≡(α + p − 1

p − 1

) p−1∑k=0

X k

(α + k)(α + k − 1) · · · (α + 1)mod p

because(p−1

k

)≡ (−1)k mod p.

Special case α = 0,

L(0)p−1(X ) ≡ E (X ) =

p−1∑k=0

X k

k!mod p.



We have

pL(γ)p (X ) = p

p∑k=0

(γ + 1− p

p − k

)(−X )k

k!

≡ X p − (γp − γ) mod p

Crucial congruence

Xd

dXL

(γ)p−1(X ) ≡ (X − γ)L

(γ)p−1(X ) + X p − (γp − γ) mod p

Special case γ = 0,

XE ′(X ) ≡ XE (X ) + X p mod p



Lemma

We have L(Zp)p−1 (Zp − Z ) =

p−1∏i=1

(1 + Z/i)i in Fp[Z ].

Sketch of the proof.

Laguerre polynomials satisfy

(Zp − Z )L(Zp+1)p−1 (Zp − Z ) = ZpL

(Zp)p−1 (Zp − Z ) (1)

Since L(0)(0) = 1 we can write L(Zp)p−1 (Zp − Z ) =

s∏i=1

(1− Z/αi ) in Fp[Z ].

Thenp−1∏j=1

(Z − j)s∏

i=1

(Z − (αi − 1)) = Zp−1s∏

i=1

(Z − αi )



Let α ∈ Fp be a root of L(Zp)p−1 (Zp − Z ) with multiplicity m, then

1 if α = 0 then α + 1 is a root with multiplicity m + p − 1

2 if α ∈ F∗p then α + 1 is a root with multiplicity m − 1

3 if α 6∈ Fp then α + 1 is a root with multiplicity m.

Since 0 is not a root, it follows that the elements of F∗p are roots of

L(Zp)p−1 (Zp − Z ) with the claimed multiplicities.

There are no further roots.

1 ∂L(Zp)p−1 (Zp − Z ) ≤ p(p − 1) and ∂

∏p−1i=1 (1 + Z/i)i = p(p−1)

2

2 ZpL(Zp)p−1 (Zp − Z )L

(−Zp)p−1 (−Zp + Z ) is invariant under the substitution

Z → Z + 1

3 ZpL(Zp)p−1 (Zp − Z )L

(−Zp)p−1 (−Zp + Z ) has zero derivative



Therefore ZpL(Zp)p−1 (Zp − Z )L

(−Zp)p−1 (−Zp + Z ) is a polynomial in Zp2 − Zp.

Since its degree cannot exceed 2p2 − p then it cannot exceed p2. This

proves that ∂L(Zp)p−1 (Zp − Z ) ≤ p(p−1)

2 and completes the proof.


An exponential-like property of Laguerre polynomials

We now turn the modular differential equation into an analougue of thefunctional equation exp(X ) exp(Y ) = exp(X + Y ) for the classicalexponential

Proposition

Consider the subring R = Fp[α, β, ((α + β)p−1 − 1)−1] of the ringFp(α, β) of rational expressions in the indeterminates α and β, and let Xand Y be further indeterminates. Then there exists rational expressionsci (α, β) ∈ R such that

L(α)p−1(X )L

(β)p−1(Y ) ≡ L

(α+β)p−1 (X + Y )

(c0(α, β) +

p−1∑i=1

ci (α, β)X iY p−i

)

in R[X ,Y ], modulo the ideal generated by X p − (αp − α) andY p − (βp − β).


An exponential-like property of Laguerre polynomials

Special case: α = 0 = β, L(0)p−1(X ) = E (X )

E (X ) · E (Y ) ≡ E (X + Y )

(1 +

p−1∑i=1

(−1)iX iY p−i/i

)

in Fp[X ,Y ] modulo the ideal generated by X p and Y p.


A model special case

Theorem

Let A =⊕

k Ak be a Z/mZ-grading of A;

let D ∈ Der(A), graded of degree d, with m | pd, such thatDp2

= Dp;

let A =⊕

a∈FpA(a) be the decomposition of A into generalized

eigenspaces for D;

assuming Fpp ⊆ F, fix γ ∈ F with γp − γ = 1;

let LD : A→ A be the linear map on A whose restriction to A(a)

coincides with L(aγ)p−1(D).

Then A =⊕

k LD(Ak) is a Z/mZ-grading of A.


The linear map LD is bijective: (LD)p acts on A(a) as multiplication bythe nontrivial scalar


A model special case


L((aγ)p)p−1 (Dp) = L

((aγ)p)p−1 (a) = L

((aγ)p)p−1 ((aγ)p − (aγ)).

The direct sum decomposition A =⊕

k LD(Ak) is a grading of A over theintegers modulo m.

Dp(Ak) ⊆ Ak , Ak = ⊕a∈FpAk ∩ A(a);

let x ∈ Ak ∩ A(a) and y ∈ Al ∩ A(b);

for any θ ∈ F, L(θ)p−1(D) ◦m = m ◦ L

(θ)p−1(D ⊗ id + id⊗D).

The proposition yields

LDx · LDy = LD

(c0(aγ, bγ)xy +

p−1∑i=1

ci (aγ, bγ)D ix · Dp−iy

)

thus LDAk · LDAl ⊆ LDAk+l .


General Case

Theorem

Let A =⊕

k Ak be a Z/mZ-grading of A;

let D ∈ Der(A), graded of degree d, with m | pd, such that Dpr isdiagonalizable over F ;

let A =⊕

ρ∈F A(ρ) be the decomposition of A into generalizedeigenspaces for D;

assuming F large enough, there is a p-polynomial g(T ) ∈ F [T ], such

that g(D)p − g(D) = Dpr , set h(T ) =∑r−1

i=1 T pi ;

let LD : A→ A be the linear map on A whose restriction to A(ρ)

coincides with L((g(ρ)−h(D))p−1 (D).

Then A =⊕

k LD(Ak) is a Z/mZ-grading of A.

On the subalgebra ker(Dpr ) the map LD coincides with the Artin-Hasseexponential series.


General Case

Set

α = α− h(X ) = α− (X p + X p2+ · · ·X pr−1

)

β = β − h(Y ) = β − (Y p + Y p2+ · · ·Y pr−1

)

thus α + β = α + β − h(X + Y ).We have

L(α)p−1(X )L

(β)p−1(Y ) ≡ L

(α+β)p−1 (X + Y ) ·

(c0(α, β) +

p−1∑i=1

ci (α, β)X iY p−i

)

modulo the ideal (X p − (αp − α),Y p − (βp − β)), that is the ideal

generated by (X pr − (αp − α),Y pr − (βp − β)).


Toral switching

Replace a torus T of a restricted Lie algebra L with another torus Tx , byapplying to T a sort of exponential in the inner derivation ad x , for acertain root vector x ∈ L. This techinique goes back to Winter and it hasbeen generalized by Block, Wilson and Premet. A crucial step in the toralswitching process is the keep control on the root space decomposition withrespect to the new torus, by constructing a linear map E (x , λ) mappingthe root spaces with respect to T bijectively onto the root spaces withrespect to Tx .The linear map E (x , λ) coincides with our map LD and the toral switchingprocess, a part for the strictly Lie-theoretic aspects, can be viewed as aspecial instance of our theorem in the general case.


Grading switching

Grading switching

applies to nonassociative algebras;

is not restricted to gradings over groups of exponent p.

M. Avitabile and S. MattareiLaguerre polynomials of derivationssubmitted (arXiv:1211.4432)

It finds one application (to thin Lie algebras) in

M. Avitabile and S. MattareiNottingham Lie algebras with diamonds of finite and infinite typesubmitted (arXiv:1211.4436)


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