The algebra of polynomial integro-differential operators and its - - PDF document

The algebra of polynomial integro-differential

• perators and its

group of automorphisms

• V. V. Bavula (University of Sheffield)

∗

∗1. V. V. Bavula, The algebra of integro-differential op-

erators on a polynomial algebra, Journal of the London

• Math. Soc., 83 (2011) no. 2, 517-543.

2.

• V. V. Bavula, The group of automorphisms of the

algebra of polynomial integro-differential operators, J.

• f Algebra, 348 (2011) 233–263.

Talks/talkalgintdif-09.tex

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K is a field of characteristic zero Pn = K[x1, . . . , xn] = ⊕α∈NnKxα is a polyno- mial algebra ∂1 :=

∂ ∂x1, . . . , ∂n := ∂ ∂xn ∈ DerK(Pn)

EndK(Pn), the endomorphism algebra the Weyl algebra An := K⟨x1, . . . , xn, ∂1, . . . , ∂n⟩ ⊆ EndK(Pn) In := An⟨

∫

1, . . . ,

∫

n⟩ ⊆ EndK(Pn) is the alge-

bra of polynomial integro-differential op- erators

∫

i : Pn → Pn, xα → (αi + 1)−1xixα

In is a prime, central, catenary, self-dual, non- Noetherian algebra

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cl.Kdim(In) = n, the classical Krull dimension GK (In) = 2n, the Gelfand-Kirillov dimension wdim(In) = n, the weak dimension wdim(In/p) = n, for all p ∈ Spec(In) n ≤ gl.dim(In) ≤ 2n, n ≤ gl.dim(In/p) ≤ 2n

• Conjecture. gl.dim(In) = n

The set J (In) of ideals of the algebra In is a finite distributive lattice |J (In)| = dn where dn is the Dedekind number The ideals of J (In) commute, ab = ba, and are idempotent ideals, a2 = a, ab = a ∩ b

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All the prime ideals of J (In) are classified All the ideals ideals of J (In) are classified GK (In/p) = 2n for all p ∈ Spec(In) ∃ exactly n height one primes of In, say p1, . . . , pn ∃! maximal ideal an and an = p1 + · · · + pn In = K⟨∂1, . . . , ∂n, H1, . . . , Hn,

∫

1, . . . ,

∫

n⟩ where

H1 := ∂1x1, . . . , Hn := ∂nxn ∈ AutK(Pn) An = K⟨∂1, . . . , ∂n, H±1

1 , . . . , H±1 n ,

∫

1, . . . ,

∫

n⟩ ⊆

EndK(Pn) is the Jacobian algebra. Clearly, In ⊂ An The map J (In) → J (An), a → ae := AnaAn is an isomorphism, i.e. (ab)e = aebe, (a+b)e = ae+be, (a∩b)e = ae∩be

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The involution ∗ on In: ∂∗

i =

∫

i,

∫ ∗

i

= ∂i, H∗

i = Hi

a∗ = a for all ideals a of In Pn is the only (up to iso) faithful and simple In-module Each ideal of In is an essential left and right In-submodule of In The group I∗

n of units of In:

I∗

n = K∗ ⋉ (1 + an)∗ ⊇ GL∞(K) ⋉ · · · ⋉ GL∞(K)

• 2n−1 times

Let A be an algebra. Then A ⊗ In is prime iff A is prime Hilbert’s Syzygy Thm: d(Pn ⊗ B) = d(Pn) + d(B) = n + d(B), d = wdim, gldim

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An analogue of Hilbert’s Syzygy Thm for In: wdim(In ⊗ B) = wdim(In) + wdim(B) = n + wdim(B) ∀ f.g. Noetherian K-algebras B, K is a.c. and uncountable The factor algebra In/a is Noetherian iff a = an, the maximal ideal of In Let Gn := AutK−alg(In) Gn = Sn ⋉ Tn ⋉ Inn(In) where Sn is the sym- metric group, Tn is the n-dimensional torus, Inn(In) is the group of inner automorphisms The map (1 + an)∗ → Inn(In), u → ωu, is a group isomorphism (ωu(a) := uau−1, a ∈ In) The centre Z(Gn) = K∗ IGn

n

= K and IInn(In)

n

= K, the algebras of in- variants

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(Rigidity of the group Gn) Let σ, τ ∈ Gn. Then the following statements are equivalent.

• 1. σ = τ.
• 2. σ(

∫

1) = τ(

∫

1), . . . , σ(

∫

n) = τ(

∫

n).

• 3. σ(∂1) = τ(∂1), . . . , σ(∂n) = τ(∂n).
• 4. σ(x1) = τ(x1), . . . , σ(xn) = τ(xn).

The algebras Pn, K[∂1, . . . , ∂n], K[

∫

1, . . . ,

∫

n] are

maximal commutative subalgebras of In There is an explicit inversion formula for σ ∈ Gn (too technical to explain) (A criterion of being an inner automor- phism) Let σ ∈ Gn. TFAE

• 1. σ ∈ Inn(In).
• 2. σ(∂i) ≡ ∂i mod an.
• 3. σ(

∫

i) ≡

∫

i

mod an.

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an is the only non-zero prime Gn-invariant ideal

• f In

[Gn : StGn(a)] < ∞ ∀ ideals a of In where StGn(a) := {σ ∈ Gn | σ(a) = a}, the stabilizer

• f a in Gn

Let p ∈ Spec(In). Then StGn(p) is a maximal subgroup of Gn iff n > 1 and ht(p) = 1; in this case [Gn : StGn(p)] = n There are exactly n + 2 Gn-invariant ideals of In (they are found explicitly)

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For each ideal a of In, Min(a) denotes the set

• f minimal primes over a. Two distinct prime

ideals p and q are called incomparable if nei- ther p ⊂ q nor p ⊃ q. The algebras In have beautiful ideal theory as the following unique factorization properties demonstrate.

• Theorem. 1. Each ideal a of In such that a ̸=

In is a unique product of incomparable primes, i.e. if a = q1 · · · qs = r1 · · · rt are two such prod- ucts then s = t and q1 = rσ(1), . . . , qs = rσ(s) for a permutation σ of {1, . . . , n}. 2. Each ideal a of In such that a ̸= In is a unique intersection of incomparable primes, i.e. if a = q1 ∩ · · · ∩ qs = r1 ∩ · · · ∩ rt are two such intersections then s = t and q1 = rσ(1), . . . , qs = rσ(s) for a permutation σ of {1, . . . , n}.

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3. For each ideal a of In such that a ̸= In, the sets of incomparable primes in statements 1 and 2 are the same, and so a = q1 · · · qs = q1 ∩ · · · ∩ qs.

• 4. The ideals q1, . . . , qs in statement 3 are the

minimal primes of a, and so a = ∏

p∈Min(a) p =

∩p∈Min(a)p.

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The next theorem gives all decompositions of an ideal as a product or intersection of ideals.

• Theorem. Let a be an ideal of In, and M

be the minimal elements with respect to inclusion of the set of minimal primes of a set of ideals a1, . . . , ak of In. Then

• 1. a = a1 · · · ak iff Min(a) = M.
• 2. a = a1 ∩ · · · ∩ ak iff Min(a) = M.

This is a rare example of a non-commutative algebra of Krull dimension > 1 where one has a complete picture of decompositions of ideals.

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