Symmetric polynomials and modules over affine sl ( 2 ) at admissible - PowerPoint PPT Presentation

Symmetric polynomials and modules over affine sl ( 2 ) at admissible levels Simon Wood The Australian National University Joint work with David Ridout Conference on Lie algebras, vertex operator algebras, and related topics A conference in honor of J. Lepowsky and R. Wilson (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 1 / 13

Affine vertex operator algebras g = g ⊗ C [ t , t − 1 ] ⊕ C K . Let g be a simple Lie algebra with affinisation � Then, for k � = − h ∨ , � � V k ( g ) = Ind � � � g g ⊗ C [ t ] ⊕ C K C k , C k = k · id , g ⊗ C [ t ] C k = 0 , K is a universal affine vertex operator algebra . For certain levels k , there exist proper ideals. V k ( g ) L k ( g ) = � max ideal � . Idea and goal Determine module theory of L k ( g ) from that of V k ( g ) . (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 2 / 13

Example g = sl ( 2 ) = span { E , H , F } For g = sl ( 2 ) , there exists a (unique) proper ideal I if and only if k + 2 = u u ≥ 2 , v ≥ 1 , gcd ( u , v ) = 1 , v , L k ( sl ( 2 )) = V k ( sl ( 2 )) . I Such levels are called admissible. The ideal is generated by a singular vector χ of sl ( 2 ) -weight 2 ( u − 1 ) and conformal weight ( u − 1 ) v . Integral levels For v = 1 , χ = ( E − 1 ) u − 1 1 u − 2 . (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 3 / 13

The Zhu algebra Moral “definition”: Zhu algebra of vertex operator algebra V A ( V ) ≃ { 0-modes of V acting on vectors annihilated by pos. modes. } There is a 1-1 correspondence between simple N -gradable modules over a vertex operator algebra V and simple modules over the Zhu algebra A ( V ) . A ( V ) -module M Top grade N -grading V -module M (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 4 / 13

Classification strategy Let π : V ։ A ( V ) . Theorem [Frenkel,Zhu] For V k ( g ) , Zhu’s algebra is A ( V k ( g )) ≃ U ( g ) . For any ideal I ⊂ V k ( g ) , the image π ( I ) is an ideal of A ( V k ( g )) and � � V k ( g ) = A ( V k ( g )) A I π ( I ) For χ ∈ V k ( g ) singular, such that � χ � = I ⇒ � π ( χ ) � = π ( I ) . Classifying N -gradable weight L k ( g ) -modules A V k ( g ) -module M is a L k ( g ) -module. ⇐ ⇒ I annihilates M . 1 - 1 → simple U ( g ) Simple N -gradable L k ( g ) -modules ← π ( I ) -weight modules. U ( g ) -weight modules ⇒ U ( g ) π ( I ) -weight modules ⇒ N -gradable L k ( g ) -modules. (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 5 / 13

Example g = sl ( 2 ) = span { E , F , H } Theorem [Gabriel] Any simple sl ( 2 ) weight module with finite dimensional weight spaces is isomorphic to one of the following: Finite-dimensional modules F λ , λ ∈ Z ≥ 0 . Highest and lowest weight. Weights: λ , λ − 2 ,..., 2 − λ , − λ Infinite-dimensional highest weight modules H λ , λ ∈ C \ Z ≥ 0 . Weights: λ , λ − 2 , λ − 4 ,... Infinite-dimensional lowest weight modules L λ , λ ∈ C \ Z ≤ 0 . Weights: ..., λ + 4 , λ + 2 , λ Infinite-dimensional weight modules W λ ; ∆ , λ , ∆ ∈ C and 2 ∆ � = µ ( µ + 2 ) for any µ ∈ λ + 2 Z , where ∆ is the eigenvalue of the quadratic Casimir and W λ ; ∆ ∼ = W λ + 2; ∆ . Neither highest nor lowest weight. Weights: ..., 2 + λ , λ , λ − 2 ,... All weight spaces are 1 dimensional. (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 6 / 13

Example g = sl ( 2 ) = span { E , F , H } For k ∈ Z ≥ 0 , the ideal of V k ( sl ( 2 )) is generated by the singular vector χ = ( E − 1 ) k + 1 1 k . In A ( V k ( sl ( 2 ))) ≃ U ( sl ( 2 )) , we have π (( E − 1 ) k + 1 1 k ) = E k + 1 . � � V k ( sl ( 2 )) ≃ U ( sl ( 2 )) The generator E is nilpotent in A � E k + 1 � . � E k + 1 1 k � The simple N -gradable U ( sl ( 2 )) � E k + 1 � -weight modules are the simple V k ( sl ( 2 )) -weight modules with top grade F λ , λ = 0 ,..., k . Upshot Easy if the singular vector is easy, very hard if not. (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 7 / 13

General admissible levels Let ∆ r , s = r 2 − 1 + s 2 u 2 k + 2 = u λ r , s = r − 1 − su v 2 − rsu v , v , v . 2 2 Theorem [Adamovi´ c, Milas] [Ridout,SW] Any simple N -gradable L k ( sl ( 2 )) -module is isomorphic to one of the following: The simple quotients induced from the finite-dimensional modules F r − 1 , where 1 ≤ r ≤ u − 1 . The simple quotients induced from the infinite-dimensional highest weight modules H λ r , s , where 1 ≤ r ≤ u − 1 and 1 ≤ s ≤ v − 1 . The simple quotients induced from the infinite-dimensional lowest weight modules L − λ r , s , where 1 ≤ r ≤ u − 1 and 1 ≤ s ≤ v − 1 . The simple quotients induced from the infinite-dimensional weight modules W λ , ∆ r , s , where 1 ≤ r ≤ u − 1 and 1 ≤ s ≤ v − 1 , 2 ∆ r , s � = µ ( µ + 2 ) for all µ ∈ λ + 2 Z and W λ , ∆ r , s ∼ = W λ , ∆ u − r , v − s . (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 8 / 13

Proof idea: Wakimoto free field realisation V k ( sl ( 2 )) is a vertex operator subalgebra of { rank 1 Heisenberg }⊗{ βγ -ghosts } . 1 1 a ( z ) a ( w ) ∼ γ ( z ) β ( w ) ∼ β ( z ) β ( w ) ∼ 0 ∼ γ ( z ) γ ( w ) . ( z − w ) 2 , z − w , E ( z ) = β ( z ) , √ H ( z ) = 2 : β ( z ) γ ( z ) : + 2 k + 4 a ( z ) , √ F ( z ) = : β ( z ) γ ( z ) γ ( z ) : + 2 k + 4: a ( z ) γ ( z ) : + k ∂γ ( z ) . Screening operator � � � 2 S ( z ) = : β ( z ) exp − k + 2 φ ( z ) ∂φ ( z ) = a ( z ) . : , (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 9 / 13

Proof idea: Wakimoto free field realisation The singular vector of V u v − 2 ( sl ( 2 )) , in the free field realisation, can be realised by the screening operator. � S [ u − 1 ] | q � = S ( z 1 ) ··· S ( z u − 1 ) | q � d z � = β ( z 1 ) ··· β ( z u − 1 ) � � � � � � � v u u − 1 p m z a − m | q � d z 1 ··· d z u − 1 1 − z i 2 v z − v − 1 ∏ ∏ ∏ × − exp i , z 1 ··· z u − 1 z j u m 1 ≤ i � = j ≤ u − 1 i = 1 m ≥ 1 where � u − 1 � � 2 v ∑ z m q = ( u − 1 ) = p m z i . u , i = 1 (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 10 / 13

Proof idea: Wakimoto free field realisation The singular vector of V u v − 2 ( sl ( 2 )) , in the free field realisation, can be realised by the screening operator. � S [ u − 1 ] | q � = S ( z 1 ) ··· S ( z u − 1 ) | q � d z � = β ( z 1 ) ··· β ( z u − 1 ) � � � � � � � v u − 1 p m z a − m | q � d z 1 ··· d z u − 1 1 − z i u 2 v z − v − 1 ∏ ∏ ∏ × − exp i , z 1 ··· z u − 1 z j u m 1 ≤ i � = j ≤ u − 1 i = 1 m ≥ 1 � �� � Inner prod. of Jack symm. poly. (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 10 / 13

Proof idea: Wakimoto free field realisation The singular vector of V u v − 2 ( sl ( 2 )) , in the free field realisation, can be realised by the screening operator. � S [ u − 1 ] | q � = S ( z 1 ) ··· S ( z u − 1 ) | q � d z � = β ( z 1 ) ··· β ( z u − 1 ) � � � � � � � v u u − 1 p m z a − m | q � d z 1 ··· d z u − 1 1 − z i 2 v z − v − 1 ∏ ∏ ∏ × − exp i , z 1 ··· z u − 1 z j u m 1 ≤ i � = j ≤ u − 1 i = 1 m ≥ 1 � �� � Jack poly. (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 10 / 13

Proof idea: Wakimoto free field realisation The singular vector of V u v − 2 ( sl ( 2 )) , in the free field realisation, can be realised by the screening operator. � S [ u − 1 ] | q � = S ( z 1 ) ··· S ( z u − 1 ) | q � d z � = β ( z 1 ) ··· β ( z u − 1 ) � � � � � � � v u u − 1 p m 1 − z i 2 v z a − m | q � d z 1 ··· d z u − 1 z − v − 1 ∏ ∏ ∏ × − exp i z 1 ··· z u − 1 , z j u m 1 ≤ i � = j ≤ u − 1 i = 1 m ≥ 1 � �� � easy expansion in Jack poly. (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 10 / 13

Proof idea: The generator of the ideal Choose a generator of sl ( 2 ) -weight 0. S [ u − 1 ] | q � = const · S [ u − 1 ] γ u − 1 χ = F u − 1 | q � 0 0 Compute eigenvalue of zero-mode χ 0 on a general top grade vector to determine image in A ( V u v − 2 ( sl ( 2 ))) . χ 0 | p , τ � = f ( p , τ ) | p , τ � , p = Heisenberg weight , τ = βγ -weight . Theorem [Ridout, SW] The polynomial f ( p , τ ) , in free field data, is also a polynomial in 1 sl ( 2 ) -data. f ( λ , ∆ ) = g u ( λ , ∆ ) ∏ ( ∆ − ∆ r , s ) , 2 r , s g u + 2 ( λ , ∆ ) = ( 2 u + 1 ) λ ( u + 1 ) 2 g u + 1 ( λ , ∆ ) − 2 ∆ − ( u − 1 )( u + 1 ) g u ( λ , ∆ ) ( u + 1 ) 2 g 1 ( λ , ∆ ) = 1 , g 2 ( λ , ∆ ) = λ . (Simon Wood, ANU) Sym polys and admissible levels Lie algebras and VOAs 11 / 13