Principal subspaces of twisted modules for certain lattice vertex - PowerPoint PPT Presentation

Principal subspaces of twisted modules for certain lattice vertex operator algebras Christopher Sadowski Joint work with Michael Penn and Gautam Webb Department of Mathematics and Computer Science Ursinus College Collegeville, PA, USA Representation Theory XVI Dubrovnik, Croatia, June 24 - 29, 2019 Christopher Sadowski Lattice Principal Subspace

Outline Preliminaries and motivation Christopher Sadowski Lattice Principal Subspace

Outline Preliminaries and motivation Lattice constructions, twisted modules, and the principal subspace Christopher Sadowski Lattice Principal Subspace

Outline Preliminaries and motivation Lattice constructions, twisted modules, and the principal subspace Presentations, recursions, and characters Christopher Sadowski Lattice Principal Subspace

Outline Preliminaries and motivation Lattice constructions, twisted modules, and the principal subspace Presentations, recursions, and characters Some interesting examples Christopher Sadowski Lattice Principal Subspace

Motivating Preliminaries Let g be a finite dimensional simple Lie algebra of type A , D , E and of rank n . Fix: Christopher Sadowski Lattice Principal Subspace

Motivating Preliminaries Let g be a finite dimensional simple Lie algebra of type A , D , E and of rank n . Fix: a Cartan subalgebra h ⊂ g Christopher Sadowski Lattice Principal Subspace

Motivating Preliminaries Let g be a finite dimensional simple Lie algebra of type A , D , E and of rank n . Fix: a Cartan subalgebra h ⊂ g the Killing form �· , ·� Christopher Sadowski Lattice Principal Subspace

Motivating Preliminaries Let g be a finite dimensional simple Lie algebra of type A , D , E and of rank n . Fix: a Cartan subalgebra h ⊂ g the Killing form �· , ·� a set of roots ∆ and a set of simple roots { α 1 , . . . , α n } Christopher Sadowski Lattice Principal Subspace

Motivating Preliminaries Let g be a finite dimensional simple Lie algebra of type A , D , E and of rank n . Fix: a Cartan subalgebra h ⊂ g the Killing form �· , ·� a set of roots ∆ and a set of simple roots { α 1 , . . . , α n } let ∆ + denote the set of positive roots Christopher Sadowski Lattice Principal Subspace

Motivating Preliminaries Let g be a finite dimensional simple Lie algebra of type A , D , E and of rank n . Fix: a Cartan subalgebra h ⊂ g the Killing form �· , ·� a set of roots ∆ and a set of simple roots { α 1 , . . . , α n } let ∆ + denote the set of positive roots let x α denote a nonzero root vector for the root α . Christopher Sadowski Lattice Principal Subspace

Motivating Preliminaries Let g be a finite dimensional simple Lie algebra of type A , D , E and of rank n . Fix: a Cartan subalgebra h ⊂ g the Killing form �· , ·� a set of roots ∆ and a set of simple roots { α 1 , . . . , α n } let ∆ + denote the set of positive roots let x α denote a nonzero root vector for the root α . denote by { λ 1 , . . . , λ n } the simple weights, dual to the simple roots: � λ i , α j � = δ i , j Christopher Sadowski Lattice Principal Subspace

Motivating Preliminaries Let g be a finite dimensional simple Lie algebra of type A , D , E and of rank n . Fix: a Cartan subalgebra h ⊂ g the Killing form �· , ·� a set of roots ∆ and a set of simple roots { α 1 , . . . , α n } let ∆ + denote the set of positive roots let x α denote a nonzero root vector for the root α . denote by { λ 1 , . . . , λ n } the simple weights, dual to the simple roots: � λ i , α j � = δ i , j Let L = ⊕ n i =1 Z α i be the root lattice Christopher Sadowski Lattice Principal Subspace

Motivating Preliminaries Let g be a finite dimensional simple Lie algebra of type A , D , E and of rank n . Fix: a Cartan subalgebra h ⊂ g the Killing form �· , ·� a set of roots ∆ and a set of simple roots { α 1 , . . . , α n } let ∆ + denote the set of positive roots let x α denote a nonzero root vector for the root α . denote by { λ 1 , . . . , λ n } the simple weights, dual to the simple roots: � λ i , α j � = δ i , j Let L = ⊕ n i =1 Z α i be the root lattice Let P = ⊕ n i =1 Z λ i be the weight lattice. Christopher Sadowski Lattice Principal Subspace

Motivating Preliminaries Let g be a finite dimensional simple Lie algebra of type A , D , E and of rank n . Fix: a Cartan subalgebra h ⊂ g the Killing form �· , ·� a set of roots ∆ and a set of simple roots { α 1 , . . . , α n } let ∆ + denote the set of positive roots let x α denote a nonzero root vector for the root α . denote by { λ 1 , . . . , λ n } the simple weights, dual to the simple roots: � λ i , α j � = δ i , j Let L = ⊕ n i =1 Z α i be the root lattice Let P = ⊕ n i =1 Z λ i be the weight lattice. Let ∆ + denote the set of positive roots, and let x α denote a nonzero root vector for the root α . Define also the subalgebra � n = C x α α ∈ ∆ + Christopher Sadowski Lattice Principal Subspace

Motivating Preliminaries We consider the affine Lie algebra ˆ g , and denote by V L the vertex operator algebra constructed from L (cf. [LL]). Christopher Sadowski Lattice Principal Subspace

Motivating Preliminaries We consider the affine Lie algebra ˆ g , and denote by V L the vertex operator algebra constructed from L (cf. [LL]). V L gives a realization of the level 1 basic ˆ g -module L (Λ 0 ), and V L e λ i gives a realization of the basic ˆ g -module L (Λ i ), in both cases with the action of x α ⊗ t n given by the n -th mode of the vertex operator � x α ( n ) x − n − 1 . Y ( ι ( e α ) , x ) = n ∈ Z Christopher Sadowski Lattice Principal Subspace

Principal subspaces Consider the subalgebra of � g : n = n ⊗ C [ t , t − 1 ] ¯ Christopher Sadowski Lattice Principal Subspace

Principal subspaces Consider the subalgebra of � g : n = n ⊗ C [ t , t − 1 ] ¯ Let v Λ be the highest weight vector of L (Λ). The principal subspace W (Λ) of L (Λ) is defined by W (Λ) = U (¯ n ) · v Λ . Principal subspaces were originally defined and studied by Feigin and Stoyanovsky. Christopher Sadowski Lattice Principal Subspace

Principal subspaces The principal subspace inherits certain compatible gradings from L (Λ). First, we have the conformal weight grading : � W (Λ) = W (Λ) s + h Λ , s ∈ Z Given a monomial x β 1 ( m 1 ) . . . x β r ( m r ) v Λ ∈ W (Λ) , its conformal weight is − m 1 − · · · − m r + h Λ , where h Λ ∈ Q is determined by Λ. This grading is given by the Virasoro L (0) operator’s eigenvalues when acting on W (Λ). Christopher Sadowski Lattice Principal Subspace

Principal subspaces Second, W (Λ) has λ i -charge gradings : � W (Λ) = W (Λ) r i + � λ i , Λ � r i ∈ Z for each i = 1 , . . . , n . Given a monomial x β 1 ( m 1 ) . . . x β r ( m r ) v Λ ∈ W (Λ) , it’s λ i -charge is � r � λ i , β j � + � λ i , Λ � j =1 Christopher Sadowski Lattice Principal Subspace

Principal subspaces Second, W (Λ) has λ i -charge gradings : � W (Λ) = W (Λ) r i + � λ i , Λ � r i ∈ Z for each i = 1 , . . . , n . Given a monomial x β 1 ( m 1 ) . . . x β r ( m r ) v Λ ∈ W (Λ) , it’s λ i -charge is � r � λ i , β j � + � λ i , Λ � j =1 These gradings are given by the eigenvalues of each λ i (0), i = 1 , . . . , n , acting on W (Λ) and “count” the number of α i ’s appearing as subscripts in each monomial. Christopher Sadowski Lattice Principal Subspace

Principal subspaces These gradings are compatible, and we have that: � W (Λ) = W (Λ) r 1 + � λ 1 , Λ � ,..., r n + � λ n , Λ � ; s + h Λ . r 1 ,..., r n , s ∈ Z Christopher Sadowski Lattice Principal Subspace

Principal subspaces These gradings are compatible, and we have that: � W (Λ) = W (Λ) r 1 + � λ 1 , Λ � ,..., r n + � λ n , Λ � ; s + h Λ . r 1 ,..., r n , s ∈ Z We define the multigraded dimensions of W (Λ) by: χ W (Λ) ( x 1 , . . . , x n , q ) = tr W (Λ) x λ 1 1 · · · x λ n n q L (0) . and a modified version W (Λ) ( x 1 , . . . , x n , q ) = x −� Λ ,λ 1 � . . . x −� Λ ,λ n � q −� Λ , Λ � / 2 tr W (Λ) x λ 1 χ ′ 1 · · · x λ n n q L (0) n 1 in order to have series with integer powers. Christopher Sadowski Lattice Principal Subspace

Principal subspaces In a series of papers, Capparelli, Calinescu, Lepowsky, and Milas studied the principal subspaces of basic modules for all the cases mentioned above, and also studied the principal subspaces of the higher level � sl (2)-modules. They constructed exact sequences among principal subspaces: 0 → W (Λ i ) → W (Λ 0 ) → W (Λ i ) → 0 , where the maps used arise naturally from the lattice construction of L (Λ) and intertwining operators among standard modules. Christopher Sadowski Lattice Principal Subspace