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Representations of Weyl Algebras Representations of Weyl algebras Vyacheslav Futorny University of S˜aoPaulo, Brazil Quantum 2013 V.Futorny Representations of Weyl algebras Weyl algebras The n-th Weyl algebra An = An(|): generators: xi ;@i , i = 1;:::; n relations xi xj = xj xi ;@i @j = @j @i ; @i xj − xj @i = δij ; i; j = 1;:::; n: An is a simple Noetherian domain; V.Futorny Representations of Weyl algebras Weyl algebras The n-th Weyl algebra An = An(|): generators: xi ;@i , i = 1;:::; n relations xi xj = xj xi ;@i @j = @j @i ; @i xj − xj @i = δij ; i; j = 1;:::; n: An is a simple Noetherian domain; V.Futorny Representations of Weyl algebras Dixmier problem: End(An) ' Aut(An)? e.g. 8x; y 2 A1, s.t. xy − yx = 1 ) < x; y >= A1? V.Futorny Representations of Weyl algebras Dixmier problem: End(An) ' Aut(An)? e.g. 8x; y 2 A1, s.t. xy − yx = 1 ) < x; y >= A1? V.Futorny Representations of Weyl algebras Let Pn = C[x1;:::; xn]. 8φ 2 End(Pn) define φi = φ(xi ). Then φ defines a polynomial function n n F : C ! C ; F (z1;:::; zn) = (φ1(z1); : : : ; φn(zn)). Let JF (¯x) = (@fi =@xj ); ∆(F ) = det(JF ): ∗ If F is invertible then ∆(F ) 2 C . V.Futorny Representations of Weyl algebras Let Pn = C[x1;:::; xn]. 8φ 2 End(Pn) define φi = φ(xi ). Then φ defines a polynomial function n n F : C ! C ; F (z1;:::; zn) = (φ1(z1); : : : ; φn(zn)). Let JF (¯x) = (@fi =@xj ); ∆(F ) = det(JF ): ∗ If F is invertible then ∆(F ) 2 C . V.Futorny Representations of Weyl algebras Let Pn = C[x1;:::; xn]. 8φ 2 End(Pn) define φi = φ(xi ). Then φ defines a polynomial function n n F : C ! C ; F (z1;:::; zn) = (φ1(z1); : : : ; φn(zn)). Let JF (¯x) = (@fi =@xj ); ∆(F ) = det(JF ): ∗ If F is invertible then ∆(F ) 2 C . V.Futorny Representations of Weyl algebras Let Pn = C[x1;:::; xn]. 8φ 2 End(Pn) define φi = φ(xi ). Then φ defines a polynomial function n n F : C ! C ; F (z1;:::; zn) = (φ1(z1); : : : ; φn(zn)). Let JF (¯x) = (@fi =@xj ); ∆(F ) = det(JF ): ∗ If F is invertible then ∆(F ) 2 C . V.Futorny Representations of Weyl algebras n n Jacobian Conjecture: Let F : C ! C be a polynomial ∗ function such that ∆(F ) 2 C . Then F is invertible (Keller, 1939). Dixmier problem ) Jacobian Conjecture (Bass, McConnel, Wright, 1982; Bavula, 2001) Van den Essen: Jac(2n) ) Dix(n) ) Jac(n) V.Futorny Representations of Weyl algebras n n Jacobian Conjecture: Let F : C ! C be a polynomial ∗ function such that ∆(F ) 2 C . Then F is invertible (Keller, 1939). Dixmier problem ) Jacobian Conjecture (Bass, McConnel, Wright, 1982; Bavula, 2001) Van den Essen: Jac(2n) ) Dix(n) ) Jac(n) V.Futorny Representations of Weyl algebras n n Jacobian Conjecture: Let F : C ! C be a polynomial ∗ function such that ∆(F ) 2 C . Then F is invertible (Keller, 1939). Dixmier problem ) Jacobian Conjecture (Bass, McConnel, Wright, 1982; Bavula, 2001) Van den Essen: Jac(2n) ) Dix(n) ) Jac(n) V.Futorny Representations of Weyl algebras In An, the elements ti = @i xi , i = 1;:::; n, generate the polynomial algebra D = C[ti j i = 1;:::; n], which is a maximal commutative subalgebra of An. Denote by G the group generated by the automorphisms σi , i = 1;:::; n, of D, where σi (tj ) = tj − δi;j a1. Then G acts on the set maxD of maximal ideals of D. A module VV for An is said to be a weight module if L VV = m2maxD VVm, where VVm = fv 2 VV j mv = 0g. If VV is a weight module and VVm 6= 0 for m 2 maxD, then m is said to be a weight of VV , and the set fm 2 maxD j VVm 6= 0g of weights of VV is the support of VV . We have xi VVm ⊆ VVσ (m) and @i VVm ⊆ VV −1 . i σi (m) V.Futorny Representations of Weyl algebras In An, the elements ti = @i xi , i = 1;:::; n, generate the polynomial algebra D = C[ti j i = 1;:::; n], which is a maximal commutative subalgebra of An. Denote by G the group generated by the automorphisms σi , i = 1;:::; n, of D, where σi (tj ) = tj − δi;j a1. Then G acts on the set maxD of maximal ideals of D. A module VV for An is said to be a weight module if L VV = m2maxD VVm, where VVm = fv 2 VV j mv = 0g. If VV is a weight module and VVm 6= 0 for m 2 maxD, then m is said to be a weight of VV , and the set fm 2 maxD j VVm 6= 0g of weights of VV is the support of VV . We have xi VVm ⊆ VVσ (m) and @i VVm ⊆ VV −1 . i σi (m) V.Futorny Representations of Weyl algebras 1. All indecomposable weight modules for An, n < 1 were classified by Bavula, Bekkert (1999). (i) Every irreducible weight module of A1 is isomorphic to one of λ1 ±1 ±1 the following x1 C[x1 ], λ1 2= Z, C[x1], C[x1 ]=C[x1]. Nn (ii) Every irreducible weight module of An = i=1 Afig is isomorphic to the outer tensor product of n simple modules of Afig, i = 1; :::; n. 2. All irreducible weight modules for A1 (with some restrictions) were classified by Benkart, Bekkert and V.F. (2000) 3. All irreducible weight modules for A1 were classified by V.F., Grantcharov, Mazorchuk (2012). V.Futorny Representations of Weyl algebras 1. All indecomposable weight modules for An, n < 1 were classified by Bavula, Bekkert (1999). (i) Every irreducible weight module of A1 is isomorphic to one of λ1 ±1 ±1 the following x1 C[x1 ], λ1 2= Z, C[x1], C[x1 ]=C[x1]. Nn (ii) Every irreducible weight module of An = i=1 Afig is isomorphic to the outer tensor product of n simple modules of Afig, i = 1; :::; n. 2. All irreducible weight modules for A1 (with some restrictions) were classified by Benkart, Bekkert and V.F. (2000) 3. All irreducible weight modules for A1 were classified by V.F., Grantcharov, Mazorchuk (2012). V.Futorny Representations of Weyl algebras 1. All indecomposable weight modules for An, n < 1 were classified by Bavula, Bekkert (1999). (i) Every irreducible weight module of A1 is isomorphic to one of λ1 ±1 ±1 the following x1 C[x1 ], λ1 2= Z, C[x1], C[x1 ]=C[x1]. Nn (ii) Every irreducible weight module of An = i=1 Afig is isomorphic to the outer tensor product of n simple modules of Afig, i = 1; :::; n. 2. All irreducible weight modules for A1 (with some restrictions) were classified by Benkart, Bekkert and V.F. (2000) 3. All irreducible weight modules for A1 were classified by V.F., Grantcharov, Mazorchuk (2012). V.Futorny Representations of Weyl algebras 1. All indecomposable weight modules for An, n < 1 were classified by Bavula, Bekkert (1999). (i) Every irreducible weight module of A1 is isomorphic to one of λ1 ±1 ±1 the following x1 C[x1 ], λ1 2= Z, C[x1], C[x1 ]=C[x1]. Nn (ii) Every irreducible weight module of An = i=1 Afig is isomorphic to the outer tensor product of n simple modules of Afig, i = 1; :::; n. 2. All irreducible weight modules for A1 (with some restrictions) were classified by Benkart, Bekkert and V.F. (2000) 3. All irreducible weight modules for A1 were classified by V.F., Grantcharov, Mazorchuk (2012). V.Futorny Representations of Weyl algebras N λ λ1 λ2 For λ 2 C we will consider formal expressions x = x1 x2 ::: and assume that xλxµ = xλ+µ. In particular, the standard basis of ±1 n N N C[x ] can be written as fx j n 2 Zfing. For λ 2 C define λ ±1 λ ±1 x C[x ] := fx f j f 2 C[x ]g: V.Futorny Representations of Weyl algebras λ ±1 (i) The module x C[x ] is a weight module with basis consisting λ+n N of the monomials x , n 2 Zfin. In particular, λ ±1 N Supp(x C[x ]) = λ + Zfin. λ ±1 (ii) x C[x ] is isomorphic to the A1-module with underlying ±1 vector space C[x ] and action −1 xi 7! xi ; @i 7! @i + λi xi : λ ±1 µ ±1 (iii) The A1-modules x C[x ] and x C[x ] are isomorphic if N and only if λ − µ 2 Zfin. V.Futorny Representations of Weyl algebras Let λ 2 CN be such that for every i 2 N, if λi 2 Z then λi ≥ 0. λ ±1 Then the A1-module x C[x ] has a unique irreducible submodule λ ±1 n M = fm 2 x C[x ] j for every i 2 I(λ);@i m = 0; for some n > 0g; where I(λ) stands for the set of all i 2 N for which λi is integer. λ ±1 The unique irreducible submodule M of x C[x ] will be denoted λ ±1 by x C[x ]@. V.Futorny Representations of Weyl algebras Let λ 2 CN be such that for every i 2 N, if λi 2 Z then λi ≥ 0. λ ±1 Then the A1-module x C[x ] has a unique irreducible submodule λ ±1 n M = fm 2 x C[x ] j for every i 2 I(λ);@i m = 0; for some n > 0g; where I(λ) stands for the set of all i 2 N for which λi is integer.
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