Introduction . Zhang embed U q ( sl 2 ) into Assume q is not a root - PowerPoint PPT Presentation

Two Dimensional YD-modules over U q ( sl 2 ) are trivial Emine Yildirim University of New Brunswick June 27th, 2014 Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 1 / 21

Introduction Introduction . Zhang embed U q ( sl 2 ) into Assume q is not a root of unity, X. W. Chen and P the path coalgebra of the Gabriel quiver D of the coalgebra of U q ( sl 2 ) . They also describe the category of U q ( sl 2 ) -comodules in terms of representations of the quiver D . I will present examples of comodules over U q ( sl 2 ) , and show that all YD-modules over U q ( sl 2 ) are trivial. Throughout this presentation, k denotes a field of characteristic zero. Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 2 / 21

Some Basic Definitions The Algebra U q ( sl 2 ) We define U q ( sl 2 ) as the algebra generated by the four variables E , F , K , K − 1 with the relations; KK − 1 = K − 1 K = 1 KEK − 1 = q 2 E , KFK − 1 = q − 2 F , and [ E , F ] = K − K − 1 q − q − 1 Note that the algebra U q is Noetherian and has no zero divisors. The set { E i F j K l } i , j ∈ N ; l ∈ Z is a basis of U q . Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 3 / 21

Some Basic Definitions The Hopf Algebra Structure on U q ( sl 2 ) U q ( sl 2 ) has a Hopf structure with ∆ ( E ) = 1 ⊗ E + E ⊗ K , ∆ ( K ) = K ⊗ K ∆ ( F ) = K − 1 ⊗ F + F ⊗ 1 , ∆ ( K − 1 ) = K − 1 ⊗ K − 1 ε ( K ) = ε ( K − 1 ) = 1 ε ( E ) = ε ( F ) = 0 , S ( E ) = − EK − 1 , S ( K ) = K − 1 , S ( K − 1 ) = K . S ( F ) = − KF , Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 4 / 21

Some Basic Definitions The Path Coalgebra kQ c A quiver Q = ( Q 0 , Q 1 , s , t ) is a datum, where Q is an oriented graph with Q 0 the set of vertices and Q 1 the set of arrows, s and t are two maps from Q 1 to Q 0 , such that s ( a ) and t ( a ) are respectively the starting vertex and terminating vertex of a ∈ Q 1 . A path p of length l in Q is a sequence p = a l ... a 2 a 1 of arrows a i , 1 ≤ i ≤ l . A vertex is regarded as a path of length 0. Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 5 / 21

Some Basic Definitions Given a quiver Q , one defines the path coalgebra kQ c as follows: the underlying space has as basis the set of all paths in Q , the coalgebra structure is given by ∆ ( p ) = ∑ β ⊗ α , βα = p ε ( p ) = 0 if l ≥ 1 , ε ( p ) = 1 if l = 0 for each path p of length l . Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 6 / 21

The path coalgebra kQ c By a graded coalgebra one means a coalgebra C with decomposition � C = C ( n ) of k -space such that n ≥ 0 ∆ ( C ( n )) ⊆ ∑ C ( i ) ⊗ C ( j ) i + j = n ε ( C ( n )) = 0 for all n ≥ 1 . Note that a path coalgebra kQ c is graded with length grading, and it is coradically graded, and kQ c ≃ Cot kQ 0 ( kQ 1 ) Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 7 / 21

The path coalgebra kQ c Proposition : � Let C = C ( n ) be a graded coalgebra. Then n ≥ 0 (i) There is a unique graded coalgebra map θ : C → Cot C ( 0 ) C ( 1 ) such that θ | C ( i ) = id for i = 0 , 1 . (ii) θ ( x ) = π ⊗ n + 1 ◦ ∆ n ( x ) for all x ∈ C ( n + 1 ) and n ≥ 1 , where π : C → C ( 1 ) is the projection, and ∆ n = ( Id ⊗ ∆ n − 1 ) ◦ ∆ for all n ≥ 1 , with ∆ 0 = id . Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 8 / 21

U q ( sl 2 ) as a Subcoalgebra of a Path Coalgebra U q ( sl 2 ) as a Subcoalgebra of a Path Coalgebra � U q ( sl 2 ) = C ( n ) is a graded coalgebra with n ≥ 0 kK l � C ( 0 ) = l ∈ Z and C ( 1 ) has a basis { K l E , K l F | l ∈ Z } One has in C ( 1 ) ∆ ( K l − 1 E ) = K l − 1 ⊗ K l − 1 E + K l − 1 E ⊗ K l ∆ ( K l F ) = K l − 1 ⊗ K l F + K l F ⊗ K l Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 9 / 21

� U q ( sl 2 ) as a Subcoalgebra of a Path Coalgebra The quiver D of U q ( sl 2 ) is of the form � •··· �� • �� • ···• Vertices are labelled by integers, i.e., D 0 = { e l | l ∈ Z } . Write v as v = ( v 1 ,..., v n ) , where v j = 1 or − 1 for each j . Define P ( v ) = a | v | ... a 2 a 1 l to be the concatenated path in D starting at e l of lenght | v | . Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 10 / 21

U q ( sl 2 ) as a Subcoalgebra of a Path Coalgebra For instance, P ( 0 ) = e l , l � • , P ( 1 ) = • l � • P ( 1 , − 1 ) = • � • with starting at the vertex e l in D . l One can write the set of all paths in D as follows: ( v | v | ) { P ( v ) l −| v | + 1 ... P ( v 2 ) l − 1 P ( v 1 ) = P | l ∈ Z , v ∈ I } l l Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 11 / 21

U q ( sl 2 ) as a Subcoalgebra of a Path Coalgebra Lemma : There is a unique graded coalgebra map θ : U q ( sl 2 ) → kD c such that θ ( K l ) = e l θ ( K l − 1 E ) = P ( 1 ) l θ ( K l F ) = P ( − 1 ) l for each integer l . Theorem : Assume that q is not a root of unity. Then as a coalgebra U q ( sl 2 ) is isomorphic to the subcoalgebra of kD c with the basis { b ( l , n , i ) | 0 ≤ i ≤ n , n ∈ N 0 , l ∈ Z } Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 12 / 21

U q ( sl 2 ) as a Subcoalgebra of a Path Coalgebra χ ( v ) P ( v ) ∑ ∈ kD c b ( l , n , i ) : = l v ∈{± 1 } n , | T v | = i where T v : = { t | 1 ≤ t ≤ n , v t = 1 } , and χ ( v ) : = q 2 ∑ t ∈ Tv t , if n ≥ 1 , T v � = / 0 ; χ ( v ) : = 1 , otherwise. For instance, b ( l , 1 , 0 ) = P ( − 1 ) b ( l , 0 , 0 ) = e l l b ( l , 1 , 1 ) = q 2 P ( 1 ) b ( l , 2 , 0 ) = P ( − 1 , − 1 ) l l b ( l , 2 , 2 ) = q 6 P ( 1 , 1 ) b ( l , 2 , 1 ) = q 2 P ( 1 , − 1 ) + q 4 P ( − 1 , 1 ) l l l Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 13 / 21

Comodules of U q ( sl 2 ) Comodules of U q ( sl 2 ) Representations of Quivers A k -representation of Q is a datum V = ( V e , f a ; e ∈ Q 0 , a ∈ Q 1 ) , V e is a k -space for each vertex e ∈ Q 0 , f a : V s ( a ) → V t ( a ) is a k -linear map for each arrow a ∈ Q 1 . Set f p : = f a l ◦···◦ f a 1 for each path p = a l ... a 1 , where each a i is an arrow, 1 ≤ i ≤ l , and f e : = id for e ∈ Q 0 The standard comodule structure on a quiver representation Let V = ( V e , f a ; e ∈ Q 0 , a ∈ Q 1 ) be a representation of a quiver Q , one defines a kQ c -comodule structure ρ : V → V ⊗ kQ c as follows; ρ ( m ) = ∑ f p ( m ) ⊗ p for every m ∈ V e s ( p )= e Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 14 / 21

Comodules of U q ( sl 2 ) Theorem : Assume that q is not a root of unity. Then there is an equivalence between the category of the right U q ( sl 2 ) -comodules and the full subcategory of representation of D with the standard comodule structures that satisfy the following conditions: (i) f ( 1 ) l − 1 ◦ f ( − 1 ) = q 2 f ( − 1 ) l − 1 ◦ f ( 1 ) l l (ii) For any m ∈ V l , f ( m ) = 0 for all but finitely many paths. l Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 15 / 21

Comodules of U q ( sl 2 ) Example Let l be an integer and n a non-negative integer. For each λ ∈ k , one can define a representation V of quiver D as follows: V j : = k , l � j � l + n if V j : = 0 , otherwise ; f ( 1 ) : = 1 , l + 1 � j � l + n if j f ( 1 ) : = 0 , otherwise ; j f ( − 1 ) : = λ q − 2 ( l + n − j ) , if l + 1 � j � l + n j f ( − 1 ) : = 0 , otherwise . j Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 16 / 21

� � � Comodules of U q ( sl 2 ) And V has an induced right U q ( sl 2 ) comodule structure ρ ( m ) = ∑ f p ( m ) ⊗ p s ( p )= e which is denoted by M ( l , n , λ ) . Let’s write these explicitly for n = 1 ; K l E � • v l • v l + 1 K l + 1 K l K l + 1 F ρ ( v l ) = v l ⊗ K l ρ ( v l + 1 ) = v l + 1 ⊗ K l + 1 + v l ⊗ K l E + λ v l ⊗ K l + 1 F Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 17 / 21

Comodules of U q ( sl 2 ) Theorem : The comodules M ( l , n , λ ) give a complete list of all non-isomorphic, indecomposable Schurian right U q ( sl 2 ) comodules, where l ∈ Z , n ∈ N 0 , λ ∈ ( k ∪{ ∞ } ) . A finite-dimensional right U q ( sl 2 ) comodule ( M , ρ ) is said to be Schurian, if dim k M j = 1 or 0 for each integer j , where M j : = { m ∈ M | ( Id ⊗ π 0 ) ρ ( m ) = m ⊗ e j } and π 0 is the projection from kD c to kD 0 . Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 18 / 21

Module over U q ( sl 2 ) An Example of Two Dimensional YD module over U q ( sl 2 ) Let V : = M ( l , 1 , λ ) be a two dimensional comodule over U q ( sl 2 ) and also take a two dimensional module V is generated by w 1 and w − 1 with the following structure: K ± 1 w 1 = q ± 1 w 1 , K ± 1 w − 1 = q ∓ 1 w − 1 Ew 1 = 0 , Ew − 1 = w 1 Fw 1 = w − 1 , Fw − 1 = 0 Now let us try to match the module and comodule structures... Emine Yildirim (University of New Brunswick) Hopf Algebras in (co)action June 27th, 2014 19 / 21