Generalized Weyl algebras and their global dimension V. V. Bavula 1 - PDF document

Generalized Weyl algebras and their global dimension V. V. Bavula 1 Generalized Weyl algebras Definition of the generalized Weyl algebras . Let D be a ring, σ = ( σ 1 , ..., σ n ) a set of commuting automorphisms of D , ( σ i σ j = σ j σ i ), a = ( a 1 , ..., a n ) a set of elements of the centre Z ( D ) of D , such that σ i ( a j ) = a j for all i ̸ = j . The generalized Weyl algebra A = D ( σ, a ) (briefly GWA) of degree n with a base ring D is the ring generated by D and the 2 n indeterminates X + 1 , ..., X + n ,X − 1 , ..., X − n subject to the defining relations: i X + X + X − i X − i = a i , i = σ i ( a i ) , X ± i α = σ ± 1 i ( α ) X ± i , ∀ α ∈ D, [ X − i , X − j ] = [ X + i , X + j ] = [ X + i , X − j ] = 0 , ∀ i ̸ = j, where [ x, y ] = xy − yx . We say that a and σ are the sets of defining elements and automorphisms of A respectively. We use also the following notation: X i := X + and Y i := X − i . i Z -grading on GWA . For an vector k = ( k 1 , . . . , k n ) ∈ Z n we put v k = v k 1 (1) · · · v k n ( n ) where for 1 ≤ i ≤ n and m ≥ 0: v ± m ( i ) = ( X ± i ) m , v 0 ( i ) = 1 . In the case n = 1, we write v m for v m (1). It follows from the definition of the GWA that A = ⊕ k ∈ Z n A k is a Z n -graded algebra ( A k A e ⊆ A k + e , for all k, e ∈ Z n ), where A k = Dv k = v k D . 1

The category of generalized Weyl algebras is closed under the tensor product (over a base field) of algebras: A ⊗ A ′ = D ⊗ D ′ ( σ ∪ σ ′ , a ∪ a ′ ) . Noetherian property . Proposition 1.1 Let A be a generalized Weyl algebra with base ring D . Then 1. if D is left (right) Noetherian, then A is left (right) Noetherian; 2. if D is a domain and a i ̸ = 0 , for all i = 1 , ..., n , then A is a domain. The Weyl algebra A n . Define the n -th Weyl algebra , A n = A n ( K ), over a field (a ring) K to be the associative K -algebra with identity generated by the 2 n indeterminates X 1 , ..., X n , ∂ 1 , ..., ∂ n , subject to the relations: [ X i , X j ] = [ ∂ i , ∂ j ] = [ ∂ i , X j ] = 0 for i ̸ = j, [ ∂ i , X i ] = 1 for all i. The Weyl algebra A n is the generalized Weyl algebra A = D ( σ ; a ) of degree n where D = K [ H 1 , ..., H n ] is a polynomial ring in n variables with coefficients in K . The sets of defining elements and automorphisms of A are { a i = H i | 1 ≤ i ≤ n } and { σ i | σ i ( H j ) = H j − δ ij } , respectively, where δ ij is the Kronecker delta. Moreover, the map A n → A, X i �→ X + i , ∂ i �→ X − i , ∂ i X i �→ H i , i = 1 , . . . , n, is an algebra isomorphism. Examples of generalized Weyl algebras of degree 1. Let A = D ( σ, a ) be a generalized Weyl algebra of degree 1, a ∈ Z ( D ), σ ∈ Aut( D ). The ring A is generated by D , X = X + 1 and Y = X − 1 subject to the defining relations: Xα = σ ( α ) X and Y α = σ − 1 ( α ) Y, ∀ α ∈ D, Y X = a and XY = σ ( a ) . The algebra A = ⊕ n ∈ Z A n 2

is Z -graded, where A n = Dv n , v n = X n ( n > 0) , v n = Y − n ( n < 0) , v 0 = 1 . It follows from the above relations that v n v m = ( n, m ) v n + m = v n + m < n, m > for some ( n, m ) ∈ D . If n > 0 and m > 0 then n ≥ m : ( n, − m ) = σ n ( a ) · · · σ n − m +1 ( a ) , ( − n, m ) = σ − n +1 ( a ) · · · σ − n + m ( a ) , ( n, − m ) = σ n ( a ) · · · σ ( a ) , ( − n, m ) = σ − n +1 ( a ) · · · a, n ≤ m : in other cases ( n, m ) = 1. Let A ( i ) = D i ( σ i , a i ) ( i = 1 , ..., n ) be a GWA of degree 1 over a field K and assume that each σ i is a K -automorphism, then their tensor product ⊗ n 1 A ( i ) = ( ⊗ n 1 D i )(( σ i ) , ( a i )) is a GWA of degree n over K . This construction allows us to build a great deal of examples of generalized Weyl alge- bras of degree n . For example, the n -th Weyl algebra A n ( K ) can be written in this way as A n ( K ) = A 1 ⊗ · · · ⊗ A 1 , n times. Example 1. A = K [ H ]( σ, a ) , where σ is an arbitrary automorphism of K [ H ] , i.e. σ ( H ) = λH + µ, λ ̸ = 0 , µ ∈ K, a ∈ K [ H ] . When σ ( H ) = H − 1 and a = H we get the 1st Weyl algebra: A 1 ≃ K [ H ]( σ, a = H ) . Let F m = A G 1 be the fixed ring where G is the cyclic group of order m , acting on the Weyl algebra A 1 as follows: ∂ �→ ω∂, X �→ ω − 1 X, ω : A 1 → A 1 , where ω is a primitive m ’th root of unity. Then F m = K < ∂ m , ∂X, X m > ≃ K [ H ]( σ, a = m m H ( H − 1 /m ) . . . ( H − ( m − 1) /m ) , 3

∂ m ↔ Y, X m ↔ X, ∂X/m ↔ H, is a GWA of degree 1, where σ ( H ) = H − 1 , char K = 0. Case, µ = 0 . Let Λ = K < X, Y | XY = λY X >, the quantum plane , then Λ ≃ K [ H ]( σ, a = H ) , σ ( H ) = λH. Let A ( S 2 K < X, Y, H | XH = λHX, Y H = λ − 1 HY, λ ) = λ 1 / 2 Y X = − ( c − H )( d + H ) , λ − 1 / 2 XY = − ( c − λH )( d + λH ) > be the algebra of functions on the quantum 2-dimensional sphere , then A ( S 2 λ ) ≃ K [ H ]( σ, a = − λ − 1 / 2 ( c − H )( d + H )) , σ ( H ) = λH. The quantum Weyl algebra A 1 ( q ) = < x, ∂ | ∂x − qx∂ = 1 > of degree 1 over K ( q ̸ = 0 ∈ K ) is the GWA A 1 ( q ) = K [ H ]( σ, a = H ) of degree 1 where σ ( H ) = q − 1 ( H − 1). Example 2. A = D ( σ, a ) , where D = K [ H, ( H − µ/ (1 − λ )) − 1 ] , σ ( H ) = λH + µ, λ ̸ = 0 , 1 , µ ∈ K, a ̸ = 0 ∈ D . In particular, when µ = 0 we have A = K [ H, H − 1 ]( σ, a ) , σ ( H ) = λH. Example 3. Consider the K -algebra Λ( b ), deformation of Usl (2) which is generated by X, Y, Z subject to the relations: [ H, X ] = X, [ H, Y ] = − Y, [ X, Y ] = b ̸ = 0 ∈ K [ H ] . Then Λ( b ) ∼ = K [ H, C ]( σ, a = C − α ) 4

where σ : K [ H, C ] → K [ H, C ] , H → H − 1 , C → C, and α ∈ K [ H ] is a solution of the equation α − σ ( α ) = b. For b = 2 H , Λ( b ) = Usl (2) . If K is a field of characteristic zero, then the center of Λ( b ) is K [ C ] . For any λ ∈ K the factor algebra Λ( b, λ ) := Λ( b ) / Λ( b )( C − λ ) is isomorphic to the GWA from Example 1 with the defining element λ − α . Let U ( λ ) := Usl (2) /Usl (2)( C − λ ) ≃ K [ H ]( σ, λ − H ( H + 1)) be the infinite dimensional primitive factor of Usl (2). Example 4. The quantum Heisenberg algebra: H q = K < X, Y, H | XH = q 2 HX, Y H = q − 2 HY, XY − q − 2 Y X = q − 1 H >, q ∈ K, q 4 ̸ = 1 is isomorphic to the GWA of degree 1: H q ≃ k [ H, C ]( σ ; a = ρ − 1 C − µH = q 2 ( C − H/q (1 − q 4 )) , σ ( H ) = q 2 H, σ ( C ) = q − 2 C. The element Ω = HC belongs to the centre of H q . For each λ ̸ = 0 ∈ K the factor algebra H q ( λ ) := H q / (Ω − λ ) is the GWA of degree 1: H q ( λ ) ≃ K [ H, H − 1 ]( σ ; a = q 2 ( λH − 1 − H/q (1 − q 4 )) , σ ( H ) = q 2 H. The ambiskew polynomial rings E are GWAs . Let D be an ring, σ ∈ Aut( D ). Suppose that elements b and ρ belong to the centre of D , moreover, ρ is invertible and σ -stable, i.e. σ ( ρ ) = ρ. Then the ambiskew polynomial ring E = D < σ ; b, ρ > is obtained by adjoining to D two symbols X and Y subject to the relations: Xα = σ ( α ) X, Y α = σ − 1 ( α ) Y, ∀ α ∈ D ; XY − ρY X = b. 5

If D = K [ H ] is the polynomial ring, ρ = 1, b = 2 H , and σ ( H ) = H − 1 , we get the universal enveloping algebra Usl (2)). The ring E is the iterated skew polynomial ring E = D [ Y ; σ − 1 ][ X ; σ, ∂ ] where ∂ is the σ − derivation of D [ Y ; σ − 1 ] such that ∂D = 0 and ∂Y = b (here σ is extended from D to D [ Y ; σ − 1 ] by the rule: σ ( Y ) = ρY ). Lemma 1.2 shows that the rings E = D < σ ; b, ρ > are generalized Weyl algebras of degree 1. Lemma 1.2 Each iterated skew polynomial ring E is the generalized Weyl algebra of degree 1 with base polynomial ring D [ H ] and defining automorphism σ : σ ( H ) = ρ ( H ) + b ( σ acts on D as before): D < σ ; b, ρ > ≃ D [ H ]( σ ; a = H ) , X ↔ X, Y ↔ Y, d ↔ d ( ∀ d ∈ D ) , Y X ↔ H. An element d of a ring D is normal if dD = Dd . Lemma 1.3 The following are equivalent. 1. C = ρ ( Y X + α ) = XY + σ ( α ) is normal in D < σ ; b, ρ > ; 2. ρα − σ ( α ) = b for some α ∈ D ; 3. D [ H ] = D [ C 1 ] for some C 1 ∈ D [ H ] such that σ ( C 1 ) = γC 1 where γ ∈ D is invertible and σ ( γ ) = γ. Corollary 1.4 Let E be as in Lemma 1.3. Then D < σ ; b, ρ > ≃ D [ C ]( σ, a = ρ − 1 C − α ) , σ ( C ) = ρC. Putting ρ = 1 we obtain the following Lemma and Corollary. Lemma 1.5 The following are equivalent. 1. C = Y X + α = XY + σ ( α ) is central in D < σ ; b, ρ = 1 > ; 2. α − σ ( α ) = b for some α ∈ D ; 3. D [ H ] = D [ C 1 ] for some C 1 ∈ D [ H ] such that σ ( C 1 ) = C 1 (then C 1 = γC for some central invertible σ -stable element γ ). Corollary 1.6 Suppose that Lemma 1.5 holds. Then D < σ ; b, ρ = 1 > ≃ D [ C ]( σ, a = C − α ) , σ ( C ) = C. 6