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

The algebra of polynomial integro-differential operators 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. of Algebra , 348 (2011) 233–263. Talks/talkalgintdif-09.tex 1

K is a field of characteristic zero P n = K [ x 1 , . . . , x n ] = ⊕ α ∈ N n Kx α is a polyno- mial algebra ∂ ∂ ∂ 1 := ∂x 1 , . . . , ∂ n := ∂x n ∈ Der K ( P n ) End K ( P n ), the endomorphism algebra the Weyl algebra A n := K ⟨ x 1 , . . . , x n , ∂ 1 , . . . , ∂ n ⟩ ⊆ End K ( P n ) ∫ ∫ I n := A n ⟨ n ⟩ ⊆ End K ( P n ) is the alge- 1 , . . . , bra of polynomial integro-differential op- erators ∫ i : P n → P n , x α �→ ( α i + 1) − 1 x i x α I n is a prime, central, catenary, self-dual, non- Noetherian algebra 2

cl . Kdim( I n ) = n , the classical Krull dimension GK ( I n ) = 2 n , the Gelfand-Kirillov dimension wdim( I n ) = n , the weak dimension wdim( I n / p ) = n , for all p ∈ Spec( I n ) n ≤ gl . dim( I n ) ≤ 2 n , n ≤ gl . dim( I n / p ) ≤ 2 n Conjecture . gl . dim( I n ) = n The set J ( I n ) of ideals of the algebra I n is a finite distributive lattice |J ( I n ) | = d n where d n is the Dedekind number The ideals of J ( I n ) commute, ab = ba , and are idempotent ideals, a 2 = a , ab = a ∩ b 3

All the prime ideals of J ( I n ) are classified All the ideals ideals of J ( I n ) are classified GK ( I n / p ) = 2 n for all p ∈ Spec( I n ) ∃ exactly n height one primes of I n , say p 1 , . . . , p n ∃ ! maximal ideal a n and a n = p 1 + · · · + p n ∫ ∫ I n = K ⟨ ∂ 1 , . . . , ∂ n , H 1 , . . . , H n , n ⟩ where 1 , . . . , H 1 := ∂ 1 x 1 , . . . , H n := ∂ n x n ∈ Aut K ( P n ) A n = K ⟨ ∂ 1 , . . . , ∂ n , H ± 1 ∫ ∫ 1 , . . . , H ± 1 n , 1 , . . . , n ⟩ ⊆ End K ( P n ) is the Jacobian algebra . Clearly, I n ⊂ A n The map J ( I n ) → J ( A n ), a �→ a e := A n a A n is an isomorphism, i.e. ( ab ) e = a e b e , ( a + b ) e = a e + b e , ( a ∩ b ) e = a e ∩ b e 4

The involution ∗ on I n : ∫ ∗ ∫ ∂ ∗ H ∗ i = i , = ∂ i , i = H i i a ∗ = a for all ideals a of I n P n is the only (up to iso) faithful and simple I n -module Each ideal of I n is an essential left and right I n -submodule of I n The group I ∗ n of units of I n : n = K ∗ ⋉ (1 + a n ) ∗ ⊇ GL ∞ (K) ⋉ · · · ⋉ GL ∞ (K) I ∗ � �� � 2 n − 1 times Let A be an algebra. Then A ⊗ I n is prime iff A is prime Hilbert’s Syzygy Thm : d ( P n ⊗ B ) = d ( P n ) + d ( B ) = n + d ( B ), d = wdim , gldim 5

An analogue of Hilbert’s Syzygy Thm for I n : wdim( I n ⊗ B ) = wdim( I n ) + wdim( B ) = n + wdim( B ) ∀ f.g. Noetherian K -algebras B , K is a.c. and uncountable The factor algebra I n / a is Noetherian iff a = a n , the maximal ideal of I n Let G n := Aut K − alg ( I n ) G n = S n ⋉ T n ⋉ Inn( I n ) where S n is the sym- metric group, T n is the n -dimensional torus, Inn( I n ) is the group of inner automorphisms The map (1 + a n ) ∗ → Inn( I n ), u �→ ω u , is a group isomorphism ( ω u ( a ) := uau − 1 , a ∈ I n ) The centre Z (G n ) = K ∗ = K and I Inn( I n ) I G n = K , the algebras of in- n n variants 6

(Rigidity of the group G n ) Let σ, τ ∈ G n . 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 ( x 1 ) = τ ( x 1 ) , . . . , σ ( x n ) = τ ( x n ). ∫ ∫ The algebras P n , K [ ∂ 1 , . . . , ∂ n ], K [ 1 , . . . , n ] are maximal commutative subalgebras of I n There is an explicit inversion formula for σ ∈ G n (too technical to explain) (A criterion of being an inner automor- phism) Let σ ∈ G n . TFAE 1. σ ∈ Inn( I n ). 2. σ ( ∂ i ) ≡ ∂ i mod a n . ∫ ∫ 3. σ ( i ) ≡ mod a n . i 7

a n is the only non-zero prime G n -invariant ideal of I n [G n : St G n ( a )] < ∞ ∀ ideals a of I n where St G n ( a ) := { σ ∈ G n | σ ( a ) = a } , the stabilizer of a in G n Let p ∈ Spec( I n ). Then St G n ( p ) is a maximal subgroup of G n iff n > 1 and ht( p ) = 1; in this case [G n : St G n ( p )] = n There are exactly n + 2 G n -invariant ideals of I n (they are found explicitly) 8

For each ideal a of I n , Min( a ) denotes the set of minimal primes over a . Two distinct prime ideals p and q are called incomparable if nei- ther p ⊂ q nor p ⊃ q . The algebras I n have beautiful ideal theory as the following unique factorization properties demonstrate. Theorem . 1. Each ideal a of I n such that a ̸ = I n is a unique product of incomparable primes, i.e. if a = q 1 · · · q s = r 1 · · · r t are two such prod- ucts then s = t and q 1 = r σ (1) , . . . , q s = r σ ( s ) for a permutation σ of { 1 , . . . , n } . 2. Each ideal a of I n such that a ̸ = I n is a unique intersection of incomparable primes, i.e. if a = q 1 ∩ · · · ∩ q s = r 1 ∩ · · · ∩ r t are two such intersections then s = t and q 1 = r σ (1) , . . . , q s = r σ ( s ) for a permutation σ of { 1 , . . . , n } . 9

3. For each ideal a of I n such that a ̸ = I n , the sets of incomparable primes in statements 1 and 2 are the same, and so a = q 1 · · · q s = q 1 ∩ · · · ∩ q s . 4. The ideals q 1 , . . . , q s in statement 3 are the minimal primes of a , and so a = ∏ p ∈ Min( a ) p = ∩ p ∈ Min( a ) p . 10

The next theorem gives all decompositions of an ideal as a product or intersection of ideals. • Theorem . Let a be an ideal of I n , and M be the minimal elements with respect to inclusion of the set of minimal primes of a set of ideals a 1 , . . . , a k of I n . Then 1. a = a 1 · · · a k iff Min( a ) = M . 2. a = a 1 ∩ · · · ∩ a k 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. 11