Finite-dimensional irreducible modules for an even subalgebra of U q - PowerPoint PPT Presentation

Finite-dimensional irreducible modules for an even subalgebra of U q ( sl 2 ) Alison Gordon Lynch University of Wisconsin-Madison gordon@math.wisc.edu June 5, 2014 Alison Gordon Lynch (Wisconsin) June 5, 2014

Introduction Fix a field F and fix 0 � = q ∈ F not a root of unity. In this talk, we consider a subalgebra of the F -algebra U q ( sl 2 ). Alison Gordon Lynch (Wisconsin) June 5, 2014

The Lie algebra sl 2 The Lie algebra sl 2 consists of the 2 x 2 matrices over F with trace 0. For x , y ∈ sl 2 , [ x , y ] = xy − yx . Alison Gordon Lynch (Wisconsin) June 5, 2014

The Lie algebra sl 2 The Lie algebra sl 2 consists of the 2 x 2 matrices over F with trace 0. For x , y ∈ sl 2 , [ x , y ] = xy − yx . sl 2 has a basis � 0 � � 0 � � 1 � 1 0 0 e = f = h = , , . 0 0 1 0 0 − 1 Observe that [ h , e ] = 2 e , [ h , f ] = − 2 f , [ e , f ] = h . Alison Gordon Lynch (Wisconsin) June 5, 2014

The algebras U ( sl 2 ) and U q ( sl 2 ) The universal enveloping algebra U ( sl 2 ) is the associative algebra defined by generators e , f , h and relations he − eh = 2 e , hf − fh = − 2 f , ef − fe = h . Alison Gordon Lynch (Wisconsin) June 5, 2014

The algebras U ( sl 2 ) and U q ( sl 2 ) The universal enveloping algebra U ( sl 2 ) is the associative algebra defined by generators e , f , h and relations he − eh = 2 e , hf − fh = − 2 f , ef − fe = h . The quantum enveloping algebra U q ( sl 2 ) is the associative algebra defined by generators e , f , k , k − 1 and relations kk − 1 = k − 1 k = 1 , kek − 1 = q 2 e , kfk − 1 = q − 2 f , ef − fe = k − k − 1 q − q − 1 . Alison Gordon Lynch (Wisconsin) June 5, 2014

Equitable presentation for U q ( sl 2 ) In 2006, Ito, Terwilliger, and Weng showed that U q ( sl 2 ) has a presentation in generators x , y ± 1 , z and relations yy − 1 = y − 1 y = 1 , qxy − q − 1 yx = 1 , q − q − 1 qyz − q − 1 zy = 1 q − q − 1 qzx − q − 1 xz = 1 . q − q − 1 This presentation is called the equitable presentation for U q ( sl 2 ). Alison Gordon Lynch (Wisconsin) June 5, 2014

Connections with U q ( sl 2 ) U q ( sl 2 ) and its equitable presentation have connections with: Q -polynomial distance regular graphs (Worawannotai, 2012), Leonard pairs (Alnajjar, 2011), Tridiagonal pairs (Ito/Terwilliger, 2007), the q -Tetrahedron algebra (Ito/Terwilliger 2007, Funk-Neubauer 2009, Miki 2010), the universal Askey-Wilson algebra (Terwilliger, 2011). Alison Gordon Lynch (Wisconsin) June 5, 2014

A basis for U q ( sl 2 ) Lemma (Terwilliger, 2011) The following is a basis for the F -vector space U q ( sl 2 ) : x r y s z t r , t ∈ N , s ∈ Z . Alison Gordon Lynch (Wisconsin) June 5, 2014

The algebra A Define A to be the F -subspace of U q ( sl 2 ) spanned by x r y s z t r , s , t ∈ N , r + s + t even . Alison Gordon Lynch (Wisconsin) June 5, 2014

The algebra A Define A to be the F -subspace of U q ( sl 2 ) spanned by x r y s z t r , s , t ∈ N , r + s + t even . Lemma (Bockting-Conrad and Terwilliger, 2013) A is a subalgebra of U q ( sl 2 ) . Alison Gordon Lynch (Wisconsin) June 5, 2014

The elements ν x , ν y , ν z The relations from the equitable presentation for U q ( sl 2 ) can be reformulated as: q (1 − xy ) = q − 1 (1 − yx ) , q (1 − yz ) = q − 1 (1 − zy ) , q (1 − zx ) = q − 1 (1 − xz ) . Alison Gordon Lynch (Wisconsin) June 5, 2014

The elements ν x , ν y , ν z The relations from the equitable presentation for U q ( sl 2 ) can be reformulated as: q (1 − xy ) = q − 1 (1 − yx ) , q (1 − yz ) = q − 1 (1 − zy ) , q (1 − zx ) = q − 1 (1 − xz ) . We denote these elements ν x , ν y , ν z respectively. Observe that ν x , ν y , ν z ∈ A . Alison Gordon Lynch (Wisconsin) June 5, 2014

Generators for A Proposition (Bockting-Conrad and Terwilliger, 2013) The F -algebra A is generated by ν x , ν y , ν z . Alison Gordon Lynch (Wisconsin) June 5, 2014

Generators for A Proposition (Bockting-Conrad and Terwilliger, 2013) The F -algebra A is generated by ν x , ν y , ν z . In the same paper, Bockting-Conrad and Terwilliger posed the problem of finding a presentation for A in generators ν x , ν y , ν z . Alison Gordon Lynch (Wisconsin) June 5, 2014

Relations involving ν x , ν y , ν z Proposition In U q ( sl 2 ) , the elements ν x , ν y , ν z satisfy x = ( q 2 − q − 2 )( q − q − 1 ) ν x , q 3 ν 2 x ν y − ( q + q − 1 ) ν x ν y ν x + q − 3 ν y ν 2 y = ( q 2 − q − 2 )( q − q − 1 ) ν y , q 3 ν 2 y ν z − ( q + q − 1 ) ν y ν z ν y + q − 3 ν z ν 2 z = ( q 2 − q − 2 )( q − q − 1 ) ν z , q 3 ν 2 z ν x − ( q + q − 1 ) ν z ν x ν z + q − 3 ν x ν 2 and y = ( q 2 − q − 2 )( q − q − 1 ) ν y , q − 3 ν 2 y ν x − ( q + q − 1 ) ν y ν x ν y + q 3 ν x ν 2 z = ( q 2 − q − 2 )( q − q − 1 ) ν z , q − 3 ν 2 z ν y − ( q + q − 1 ) ν z ν y ν z + q 3 ν y ν 2 x = ( q 2 − q − 2 )( q − q − 1 ) ν x . q − 3 ν 2 x ν z − ( q + q − 1 ) ν x ν z ν x + q 3 ν z ν 2 Alison Gordon Lynch (Wisconsin) June 5, 2014

Relations involving ν x , ν y , ν z Proposition (AGL) In U q ( sl 2 ) , the elements ν x , ν y , ν z satisfy q ν y ν z − q − 1 ν z ν y = ν x − q − 2 ν y − q 2 ν z + q 2 ν y ν z − q − 2 ν z ν y ν x , q − q − 1 q − q − 1 q ν y ν z − q − 1 ν z ν y ν x = ν x − q 2 ν y − q − 2 ν z + q 2 ν y ν z − q − 2 ν z ν y , q − q − 1 q − q − 1 and the relations obtained from these by cyclically permuting ν x → ν y → ν z → ν x . Alison Gordon Lynch (Wisconsin) June 5, 2014

A presentation for A Theorem The F -algebra A is isomorphic to the F -algebra defined by generators ν x , ν y , ν z and the 12 relations from the previous two propositions. Alison Gordon Lynch (Wisconsin) June 5, 2014

Representation theory of U q ( sl 2 ) We now turn our attention to the representation theory of A . First, we recall the representation theory of U q ( sl 2 ). Alison Gordon Lynch (Wisconsin) June 5, 2014

Representation theory of U q ( sl 2 ) For n ∈ N , ε ∈ { 1 , − 1 } , there exists an irreducible U q ( sl 2 )-module L ( n , ε ) of dimension n which has a basis { v i } n i =0 such that ε x . v i = q 2 i − n v i + ( q n − q 2 i − 2 − n ) v i − 1 (1 ≤ i ≤ n ) , ε x . v 0 = q − n v 0 , ε y . v i = q n − 2 i v i (0 ≤ i ≤ n ) , ε z . v i = q 2 i − n v i + ( q − n − q 2 i +2 − n ) v i +1 (0 ≤ i ≤ n − 1) , ε z . v n = q n v n . Moreover, every finite-dimensional irreducible U q ( sl 2 )-module is isomorphic to some L ( n , ε ). Alison Gordon Lynch (Wisconsin) June 5, 2014

Induced modules of A Observe that L ( n , ǫ ) has an induced A -module structure. For n ∈ N , the A -modules L ( n , 1) and L ( n , − 1) are isomorphic. We denote by L ( n ) the common A module structure of L ( n , 1) and L ( n , − 1). Alison Gordon Lynch (Wisconsin) June 5, 2014

Facts about L ( n ) L ( n ) is irreducible as an A -module. The actions of ν x , ν y , ν z on L ( n ) are nilpotent. The actions of x 2 , y 2 , z 2 on L ( n ) are diagonalizable. Alison Gordon Lynch (Wisconsin) June 5, 2014

Facts about finite-dimensional irreducible A -modules What about arbitrary finite-dimensional irreducible A -modules? Alison Gordon Lynch (Wisconsin) June 5, 2014

Facts about finite-dimensional irreducible A -modules What about arbitrary finite-dimensional irreducible A -modules? Lemma (AGL) Let V be a finite-dimensional irreducible A -module. Then the actions of ν x , ν y , ν z on V are nilpotent. Alison Gordon Lynch (Wisconsin) June 5, 2014

Facts about finite-dimensional irreducible A -modules What about arbitrary finite-dimensional irreducible A -modules? Lemma (AGL) Let V be a finite-dimensional irreducible A -module. Then the actions of ν x , ν y , ν z on V are nilpotent. Lemma (AGL) Let V be a finite-dimensional irreducible A -module. Then the actions of x 2 , y 2 , z 2 on V are diagonalizable. Alison Gordon Lynch (Wisconsin) June 5, 2014

Representation theory of A Theorem (AGL) Let V be a finite-dimensional irreducible A -module. Then V is isomorphic to L ( n ) for some n ∈ N . Alison Gordon Lynch (Wisconsin) June 5, 2014

Future work Investigate the induced A -modules from the U q ( sl 2 ) modules related to tridiagonal pairs, the q -tetrahedron algebra, etc. Are there any naturally arising A -modules other than those induced by an existing U q ( sl 2 )-module? Alison Gordon Lynch (Wisconsin) June 5, 2014

Thank you! Alison Gordon Lynch (Wisconsin) June 5, 2014