Representations of the super Jordan plane
Nicolás Andruskiewitsch, Dirceu Bagio, Saradia Della Flora, Daiana Flôres
aa r X i v : . [ m a t h . QA ] J u l REPRESENTATIONS OF THE SUPER JORDAN PLANE
NICOL ´AS ANDRUSKIEWITSCH, DIRCEU BAGIO, SARADIA DELLA FLORA,DAIANA FL ˆORES
To Antonio Paques.
Abstract.
It is shown that the finite-dimensional simple representa-tions of the super Jordan plane B are one-dimensional. The indecom-posable representations of dimension 2 and 3 of B are classified. Twofamilies of indecomposable representations of B of arbitrary dimensionare presented. Introduction
Nichols algebras are graded connected algebras with a comultiplicationin a braided sense. In particular, the Jordan plane and the super Jordanplane are two Nichols algebras that play an important role in the clas-sification of pointed Hopf algebras with finite Gelfand-Kirillov dimension[AAH1, AAH2].The Jordan plane was first defined in [G] and considered in many papers,e.g. [AS], see also the references in [AAH2, I]. Its representation theory wasstudied in [I].The purpose of this note is to begin the study of the representation the-ory of the super Jordan plane B : we classify the simple finite-dimensional B -modules (all of dimension 1, Theorem 2.6) and the indecomposable B -modules of dimension 2 (Theorem 3.2) and 3 (Theorem 3.11). We alsoobserve that one of the generators of B has at most two eigenvalues in ev-ery indecomposable B -module (Theorem 2.11) and describe two families ofindecomposable modules in every dimension.2. Basic facts
Notations and conventions.
Fix an algebraically closed field k ofcharacteristic 0; all vector spaces, tensor products, Hom spaces, algebras areover k . All algebras are associative and all modules are left, unless explicitlystated. Let A be a k -algebra; then [ , ] denotes the Lie bracket given by thecommutator. As customary we use indistinctly the languages of modules Mathematics Subject Classification. and representations. Denote by A M the category of finite dimensional A -modules. Given a k -vector space V , gl ( V ) denotes the Lie algebra of alllinear operators on V . The Jacobson radical of an algebra A it will bedenoted by Jac A.2.2. The Jordan plane.
The
Jordan plane is the free associative algebra A in generators y and y subject to the quadratic relation y y − y y − y . The algebra A is a Nichols algebra, GKdim A = 2 and { y a y b : a, b ∈ N } isa basis of A . By Proposition 3.4 of [I], A is a Koszul algebra.2.3. The super Jordan plane.
Let x = x x + x x in the free associa-tive algebra in generators x and x . Let B be the algebra generated by x and x with defining relations x , (2.1) x x − x x − x x . (2.2)The algebra B (which is graded by deg x = deg x = 1) was introducedin [AAH1, AAH2] and is called the super Jordan plane . Since B is not aquadratic algebra, it follows that B is not Koszul; see e. g. § . Proposition 2.1. [AAH2]
The algebra B is a Nichols algebra, GKdim B = 2 and { x a x b x c : a ∈ { , } , b, c ∈ N } is a basis of B . (cid:3) The following identities are valid in B : x x = x x , (2.3) x x = x x + x x x , (2.4) x x = ( x − x ) x . (2.5)Indeed, in presence of (2.1), (2.2) is equivalent to (2.4).By (2.5) and Proposition 2.1, the subalgebra of the super Jordan plane B generated by x and x , is isomorphic to the Jordan plane via y x and y x .It is convenient to introduce s = x and t = x . By (2.5), st = ts − s and whence[ t, s n ] = ns n +1 , n ≥ x s = sx ; x t = tx ; tx = x ( t + s ) . (2.6) Lemma 2.2.
Given b, c ∈ N , we have that x b x c ∗ = ( x − bx ) x b x c − , x x b x c ♥ = x x x b x c − . Proof.
We prove ∗ by induction. For b = c = 1, the relation is valid by (2.2).Suppose that ∗ is valid for b − > c = 1. Then x b x = x b − x x = x b − ( x x − x x )= ( x − ( b − x ) x b − x − x b x = ( x − bx ) x b . EPRESENTATIONS OF THE SUPER JORDAN PLANE 3
Fix b ∈ N and assume that the relation is true for c −
1, with c >
1. Thus x b x c = ( x b x c − ) x = ( x − bx ) x b x c − x = ( x − bx ) x b x c − . The proof of ♥ is similar. (cid:3) The next result follows immediately from Proposition 2.1 and Lemma 2.2.
Proposition 2.3.
The set { , x , x , x x } generates B as a right A -module. (cid:3) Simple modules.
Let (
V, ρ ) be a finite-dimensional representation of B ; set X = ρ ( x ), X = ρ ( x ), S = ρ ( s ) and T = ρ ( t ) and V = ker X . Then V is always = 0 and it is stable under S and T by (2.6). In fact, let E ( n ) ∈ gl ( k n ) (or E if n is clearly from de context) the matrix whosethe entry 1 × X consists of r blocks like E (2) and s blocks of size 1 filledby 0. Hence dim V = 2 r + s ; r = 0 ⇐⇒ V = V . Lemma 2.4.
Assume the previous notations. Then: (i) S and T have a simultaneous eigenvector in V . (ii) W = X V ∩ V is a submodule of V . (iii) U = X V + V is a submodule of V .Proof. (i): The subspace of gl ( V ) generated by T and S n , n ∈ N , is asolvable Lie subalgebra by (2.6); then Lie Theorem applies.(ii): Clearly X W ⊆ X V = { } ⊆ W . It remains to show that X W ⊆ W . In fact, let w ∈ W , this is, w ∈ V and w = X v for some v ∈ V .Clearly X w ∈ X V . Moreover, X ( X w ) = X X v (2.4) = ( X X − X X X ) v = 0 = ⇒ X w ∈ W. (iii): Since B · V ⊆ X V and B · ( X V ) ⊆ V , the claim follows. (cid:3) Lemma 2.5. If V ∈ B M is simple, then V = V .Proof. Assume that V = V . By Lemma 2.4 we have that W = X V ∩ V = 0and V = X V + V , so that V = X V ⊕ V . By Lemma 2.4 (i), there existsa simultaneous eigenvector v ∈ V of S and T , i.e. there exist α, τ ∈ k suchthat Sv = αv , T v = τ v .Now M = span { v, X v } 6 = 0 is a B -submodule of V and by simplicityof V , M = V . By our assumption, X v / ∈ V ; hence Λ = { v, X v } is abasis of V . Note that [ X ] Λ = (cid:18) α (cid:19) and [ X ] Λ = (cid:18) τ (cid:19) . The relation X X = X X + X X X is satisfied if and only if τ α = ατ + α . Therefore α = 0 and V = V , a contradiction. (cid:3) Let A ∈ End( k n ). Denote by k nA the B -module defined by X = 0 and X = A . Every B -module V with V = V is isomorphic to k nA for some A .If B ∈ End( k m ), then k nA ≃ k mB iff n = m and A and B are similar matrices. ANDRUSKIEWITSCH, BAGIO, DELLA FLORA, FL ˆORES
Theorem 2.6.
Every simple B -module is isomorphic to k a for a unique a ∈ k .Proof. This follows from Lemma 2.5 and the preceding considerations. (cid:3)
Corollary 2.7.
Let ρ : B →
End V a finite dimensional representation of B and B = ρ ( B ) . Then there exists an integer s such that B/ Jac B ≃ k s and Jac B = { x ∈ B : x is nilpotent } .Proof. Since B/ Jac B is semisimple and k is algebraically closed, there arepositive integers n , . . . , n s such that B/ Jac B = M n ( k ) × · · · × M n s ( k ).The composition B ρ / / / / B π / / / / B/ Jac B π j / / / / M n j ( k )is a finite dimensional simple representation of B . Hence, by Theorem (2.6), n = · · · = n s = 1. Thus, B/ Jac B ≃ k s . Let x ∈ B a nilpotent element.Then π ( x ) is a nilpotent element of B/ Jac B . Since B/ Jac B is commutative,we obtain that π ( x ) ∈ Jac ( B/ Jac B ) = { } . Hence, x ∈ Jac B . On theother hand, B finite dimensional implies that Jac B is a nilpotent ideal.Consequently, Jac B = { x ∈ B : x is nilpotent } . (cid:3) We also remark:
Proposition 2.8. If V is an indecomposable B -module with V = V , thenthere exist n ∈ N and λ ∈ k such that V is isomorphic to k nA where A is theJordan block of size n with eigenvalue λ . (cid:3) If A is the Jordan block of size n with eigenvalue λ , then denote A λ = k nA .2.5. Indecomposable modules.
Throughout this subsection, V , X , X , T and S are as in § V is indecomposable, we will prove that T hasa unique eigenvalue. In order to do this, the following relations are useful. Lemma 2.9.
Let λ ∈ k , z := t − λ id ∈ B and n ∈ N . Then z n x ♣ = x n X j =0 n !( n − j )! s j z n − j , z n x x ♦ = x x n X j =0 n !( n − j )! s j z n − j . Proof.
We prove ♣ by induction on n ; the proof of ♦ is similar. We will usethat x zs n = x x n z + nx s n +1 , which can be verified easily. Note that zx = x z + x x x = x z + x s = x ( z + s ) , EPRESENTATIONS OF THE SUPER JORDAN PLANE 5 and whence the formula is true for n = 1. Denote ζ n,j := n !( n − j )! , 0 ≤ j ≤ n .Consider n > n −
1. Then z n x = ( zx ) n − X j =0 ζ n − ,j s j z n − − j (2.2) = ( x z + x s ) n − X j =0 ζ n − ,j s j z n − − j = n − X j =0 ζ n − ,j ( x zs j ) z n − − j + n − X j =0 ζ n − ,j x s j +1 z n − − j = x n X j =0 ζ n,j s j z n − j . (cid:3) Let λ be an eigenvalue of T . Denote by V Tλ the generalized eigenspace of V associated to λ , i. e. V Tλ := ∪ j ≥ ker ( T − λ id) j Lemma 2.10. V Tλ is a B -submodule of V , for all eigenvalue λ of T .Proof. Clearly V Tλ = ker ( T − λ id) r = ker (cid:0) X − λ id (cid:1) r , where r is the max-imal size of λ -blocks in the Jordan normal form of T . Thus V Tλ is stable by X . It remains to show that it stable by X . By Lemma 2.9, if u ∈ V Tλ then( T − λ id) n X u = X n X j =0 ζ j,n S j ( T − λ id) n − j u. By Lemma 2.1 of [I], S is nilpotent. Taking n big enough, it follows that( T − λ id) n X u = 0 and whence X u ∈ V Tλ . (cid:3) Now Lemma 2.10 implies the next result.
Theorem 2.11.
Let λ , . . . , λ t be the different eigenvalues of T . Then V decomposes into the direct sum of the B -submodules V Tλ i .In particular, if V is indecomposable then T has a unique eigenvalue.Hence either X has a unique eigenvalue or else the eigenvalues of X are λ and − λ , with λ ∈ k × . (cid:3) Given λ ∈ k , denote by B M λ the full subcategory of B M whose objectsare the B -modules V such that V = ker ( T − λ id) m , for some m ∈ N . Withthis notation, the next result follows immediately from Theorem 2.11. Corollary 2.12. B M ≃ Q λ ∈ k B M λ . (cid:3) The next result will be useful in § Lemma 2.13.
Let
Λ = { v , · · · , v n } be a basis of V such that [ X ] Λ = E and W a one-dimensional B -submodule of V . Then: (i) If L is a complement (as a B -module) of W in V then L ∩ V = h v i . (ii) W = h v i does not have a complement (as a B -submodule) in V .Proof. (i): Assume that W = h w i and { u , u , · · · , u n − } is a basis of L .Since W = W ∩ V = 0, it follows that W ⊂ V . Using that v is alinear combination of w, u , u , · · · , u n − we see that v = X v ∈ L ; hence v ∈ V ∩ L .(ii): It follows at once from (i). (cid:3) ANDRUSKIEWITSCH, BAGIO, DELLA FLORA, FL ˆORES Indecomposable representations of dimension and Dimension . In this subsection we describe all 2-dimensional inde-composable representations of B . Fix ( V, ρ ) a 2-dimensional representationof B . Lemma 3.1. If V = V then V is indecomposable.Proof. Suppose that V is decomposable, i.e. there are non-trivial submod-ules U and W such that V = U ⊕ W . Then V = U ⊕ W = U ⊕ W = V . (cid:3) Define representations of B on the vector space k given by X = E andthe following action of x : ⋄ X = (cid:18) a b a (cid:19) , a, b ∈ k . This is denoted by U a,b . ⋄ X = (cid:18) a − a (cid:19) , a ∈ k × . This is denoted by V a .It is easy to check that these are indecomposable modules pairwise non-isomorphic. Theorem 3.2.
Every 2-dimensional indecomposable representation of B isisomorphic either to U a,b , or to V a , or to k λ for unique a, b, λ ∈ k . This confirms Theorem 2.11.
Proof. If V = V , then Proposition 2.8 applies. Assume that V = 0; thenthere exists a basis Λ = { v , v } of V such that [ X ] Λ = E . Let [ X ] Λ = (cid:18) a bc d (cid:19) . Then (2.4) is satisfied if and only if c ( a + d ) = 0 and d + c = a . Suppose that c = 0. Then by the first equation it follows that d = − a . Re-placing in the second equation we have that c = 0, which is a contradiction.Therefore c = 0 and consequently d = a or d = − a .If d = a then V ≃ U a,b . Assume that d = − a = 0 and take w = v and w = − b a v + v . Then Ω = { w , w } is a basis of V such that [ X ] Ω = E and [ X ] Ω = (cid:18) a − a (cid:19) . Thus V ≃ V a . (cid:3) Corollary 3.3. If Ext ( k a , k b ) = 0 then a = ± b . (cid:3) Dimension . Let V be a B -module of dimension 3 such that V = V .Throughout this subsection, Λ = { v , v , v } denotes a basis of V such that[ X ] Λ = E . We define four families of representations of B on the vector EPRESENTATIONS OF THE SUPER JORDAN PLANE 7 space V determined by the following action of [ X ] Λ , for all a, b, c, d, e ∈ k :Θ : a b c d e a − d e − d , e ∈ k × ; Θ : a b c a d e ;Θ : a b c − a d c d e − a , d ∈ k × ; Θ : a b c − a d e , a ∈ k × . Lemma 3.4.
The families Θ , Θ , Θ and Θ contain all -dimensionalrepresentations of B , up to isomorphism.Proof. Let [ X ] Λ = α β γδ ǫ ζη θ ι . Then (2.4) is valid if and only if δ ( α + ǫ ) = − ζηζ ( ǫ + ι ) = − γδη ( α + ι ) = − δθǫ − α = − δ + γη − ζθ (3.1) Claim:
If the system (3.1) has solution then δ = 0.Assume that δ = 0. If ζ = 0 then γ = 0 and ǫ = − α . Thus, δ = 0 which isa contradiction. If ζ = 0 then γ = − ζ ( ǫ + ι ) δ , η = − δ ( α + ǫ ) ζ and θ = ( α + ǫ )( α + ι ) ζ . From the last equation of (3.1), δ = 0 which is again a contradiction.Assume δ = 0. Thus ζη = 0. If ζ = 0 then η = 0, ι = − ǫ and θ = α − ǫ ζ .Hence V belongs to the family Θ . When ζ = 0 and η = 0, it follows that ι = − α and γ = ǫ − α η . Thus, V belongs to the family Θ . If ζ = 0 and η = 0 then ǫ = | α | . In this case, V belongs to the families Θ or Θ . (cid:3) Remark . Let L a B -submodule of V of dimension 2 such that L ∩ V isone-dimensional. Fix L := L/ ( L ∩ V ) = h u i . Since u / ∈ V , we can supposethat u = αv + v + γv ∈ L , with α, γ ∈ k . Proposition 3.6.
Let V be a B -module. Then: (i) the representations in the family Θ are always indecomposable; (ii) a representation in the family Θ is indecomposable if and only if c = 0 and e = a or d = 0 and e = a ; (iii) the representations in the family Θ are always indecomposable; (iv) a representation in the family Θ is indecomposable if and only if c = 0 and e = a or d = 0 and e = − a .Proof. (i): The unique one-dimensional B -submodule of V is h v i which doesnot have complement by Lemma 2.13 (ii). ANDRUSKIEWITSCH, BAGIO, DELLA FLORA, FL ˆORES (ii): Let V be a representation of B of the type Θ . Suppose that W = h w i is a one-dimensional B -submodule of V . Since W ⊂ V , see § . w = αv + βv , with α, β ∈ k . Note that X w = γw , γ ∈ k , if and only if β ( γ − e ) = 0 and α ( γ − a ) = βc . Consequently, the one-dimensional B -submodules of V are: ⋄ h v i , h v i , c = 0, e = a , ⋄ h αv + βv i , c = 0, e = a , ⋄ h v i , h v + e − ac v i , c = 0, e = a ⋄ h v i , c = 0, e = a .Assume e = a . If c = 0, V = h v + e − ac v i ⊕ h v , v + v + da − e v i . If c = 0, V = h v i ⊕ h v , v + v + da − e v i . If c = d = 0, V = h v , v i ⊕ h v i . Hence, V is decomposable.Conversely, suppose e = a and c = 0. Then the unique one-dimensional B -submodule of V is h v i which does not have complement. Suppose that e = a and d = 0. Assume that W is a one-dimensional B -submodule of V which admits a complement L = h u , u i . Then by Lemma 2.13 (ii), v ∈ L ∩ V . By Remark 3.5, L = h u i where u = αv + v + βv . Thus X u = γu , γ ∈ k , if and only if γ = a and β ( a − e ) = d . Since d = 0 and e = a then W does not have complement in V .(iii): Suppose that W = h w i is a one-dimensional B -submodule of V which admits a complement L . Then by Lemma 2.13 (ii), h v i = L ∩ V ,which is a contradiction because d = 0.(iv): Analogous to item (ii). (cid:3) Isomorphism classes in Θ . Assume V in the family Θ . We distin-guish: for all a, b, c, d, e ∈ k ⋄ X = a b c d e a − d e − d , e ∈ k × . This is denoted by Y a,b,c,d,e . ⋄ X = a b a
10 0 − a . This is denoted by U a,b . By Proposition 3.6 (i), these representations are indecomposable. Note that U a,b = Y a,b, ,a, . Proposition 3.7.
Every -dimensional indecomposable representation V of B in Θ is isomorphic either to U a,b , or to Y a,b,c,d,e . Moreover, Y a,b,c,d,e ≃ Y a,b ′ ,c ′ ,d ′ ,e ′ if and only if ( a − d ′ ) ce ′ − c ′ ee ′ = e ( b ′ − b ) + c ( d ′ − d ) . In particular, U a,b ≃ U a,b ′ if and only if b = b ′ .Proof. Since h X v i = Im X , we obtain that a is invariant. Consider theindecomposable representation Y a,b ′ ,c ′ ,d ′ ,e ′ of B . If d ′ = a , taking the basis { v , c ′ e ′ v + v , e ′ v } we conclude that Y a,b ′ ,c ′ ,d ′ ,e ′ ≃ U a,b ′ . EPRESENTATIONS OF THE SUPER JORDAN PLANE 9
Note that Y a,b,c,d,e and Y a,b ′ ,c,d ′ ,e ′ are isomorphic if and only if there exists abasis { w , w , w } of V such that X w = X w = 0, X w = w , X w = aw , X w = b ′ w + d ′ w + a − d ′ e ′ w and X w = c ′ w + e ′ w − d ′ w .Since h v i = Im X and V has dimension 2, then we can consider w = v , w = λ v + λ v + λ v and w = β v + β v , λ , λ , λ , β , β ∈ k . Then, Y a,b,c,d,e ≃ Y a,b ′ ,c,d ′ ,e ′ if and only if ( a − d ′ ) ce ′ − c ′ ee ′ = e ( b ′ − b ) + c ( d ′ − d ). (cid:3) Isomorphism classes in Θ . Consider V in the family Θ and thefollowing distinguish representations: for all a ∈ k ⋄ X = a a
00 1 a . This is denoted by R a . ⋄ X = a a
00 0 a . This is denoted by S a . ⋄ X = a b a c a , b ∈ k × or c ∈ k × . This is denoted by T a,b,c .By Proposition 3.6 (ii), these are indecomposable representations. Noticethat R a = T a, , and S a = T a, , . Proposition 3.8.
Every -dimensional indecomposable representation V of B in Θ is isomorphic either to R a , or to S a or to T a,b,c . Moreover, T a,b,c and T a,b ′ ,c ′ are isomorphic if and only if bc = b ′ c ′ .Proof. Let V ′ the representation of B given by[ X ] Λ = a d ′ b ′ a c ′ a . If b ′ = 0, then by Proposition 3.6 (ii) we have that c ′ = 0. In this case, takingthe basis { v , v , d ′ v + c ′ v } of V ′ , we conclude that V ′ ≃ R a . Similarly, if c ′ = 0 then b ′ = 0. Taking the basis { v , v − d ′ b ′ v , b ′ v } of V ′ , we obtainthat V ′ ≃ S a . If b, b ′ , c, c ′ ∈ k × , taking the basis { v , v , d ′ c ′ v + v } , it followsthat V ′ ≃ T a,b ′ ,c ′ .Finally, notice that T a,b,c ≃ T a,b ′ ,c ′ if and only if there exists a basis { w , w , w } of k such that X w = X w = 0, X w = w , X w = aw , X w = aw + cw and X w = bw + aw . We can assume w = v , w = λ v + λ v + λ v and w = β v + β v , λ , λ , λ , β , β ∈ k . Notethat X w = w if and only if λ = 1. Moreover, X w = aw + cw and X w = bw + aw if and only if bc = b ′ c ′ . (cid:3) Isomorphism classes in Θ . Consider V in the family Θ and thefollowing distinguished representations: for all a, b, c, d, e ∈ k ⋄ X = a b c − a d c d e − a , d ∈ k × . This is denoted by W a,b,c,d,e . ⋄ X = a b a
01 0 − a . This is denoted by U a,b .By Proposition 3.6 (iii), these representations are indecomposable. Observethat U a,b = W a,b,a, , . Proposition 3.9.
Every -dimensional indecomposable representation V of B in Θ is isomorphic either to U a,b , or to W a,b,c,d,e . Moreover, W a,b,c,d,e ≃ W a ′ ,b ′ ,c,d ′ ,e ′ if and only if ae − bd − ced = a ′ e ′ − b ′ d ′ − ce ′ d ′ . In particular, U a,b ≃ U a,b ′ iff b = b ′ .Proof. Since the characteristic polynomial of X is ( t − c ) ( t + c ), c is aninvariant. Let the indecomposable representation W a ′ ,b ′ ,c,d ′ ,e ′ of B . If c = a ,taking the basis { v , − ed v + v , dv } we conclude that W a ′ ,b ′ ,c,d ′ ,e ′ ≃ U a ′ ,b ′ .Note that W a,b,c,d,e ≃ W a ′ ,b ′ ,c,d ′ ,e ′ if and only if there is a basis { w , w , w } of k such that X w = X w = 0, X w = w , X w = aw , x w = aw + cw and X w = bw + aw . We can assume w = v , w = λ v + λ v + λ v and w = β v + β v , where λ , λ , λ , β , β ∈ k . However X w = a ′ w + d ′ w if and only if β = a − a ′ d ′ e β = dd ′ . With this choose of β and β we havethat X w = c − a ′ d ′ w − a ′ w . Finally, X w = b w + cw + e ′ w if andonly if ae − bd − ced = a ′ e ′ − b ′ d ′ − ce ′ d ′ . (cid:3) Isomorphism classes in Θ . Consider V in Θ and the following dis-tinguish representations: for all a ∈ k × ⋄ X = a − a
00 0 a . This is denoted by V a . ⋄ X = a − a
00 1 − a . This is denoted by V a .By Proposition 3.6 (iv), these are indecomposable representations pairwisenon-isomorphic. Proposition 3.10.
Every -dimensional indecomposable representation V of B in Θ is isomorphic either to V a or to V a for unique a ∈ k × .Proof. Let V ′ be a 3-dimensional indecomposable representation of B suchthat [ X ] Λ = a b c − a d e , a ∈ k × . EPRESENTATIONS OF THE SUPER JORDAN PLANE 11
Since V ′ is indecomposable, by Proposition 3.6 (iv) we have that c = 0and e = a or d = 0 and e = − a . If c = 0 and e = a , taking the basis { v , cd − ab a v + v − d a v , v + c v } of V ′ , we obtain V ′ ≃ V a . If d = 0 and e = − a , taking the basis { v , − ab + cd a v + v , − dc a v + dv } of V ′ , it followsthat V ′ ≃ V a . (cid:3) Classification of indecomposable 3-dimensional B -modules. Theorem 3.11.
Every -dimensional indecomposable B -module is isomor-phic either to k λ for a unique λ , or else to a representation in one of thefamilies Θ j , j = 1 , , , , with the constraints described in Proposition 3.6.The isomorphism classes are described in Propositions 3.7, 3.8, 3.9 and3.10. (cid:3) Again, this agrees with Theorem 2.11.
Remark . It is straightforward to verify that two 3-dimensional indecom-posable representations of B that belong to different families Θ i , i = 1 , , , Families of indecomposable B -modules Throughout this section (
V, ρ ) is an n -dimensional representation of B ,Λ = { v , . . . , v n } is a basis of V , X = ρ ( x ), X = ρ ( x ) and [ X ] Λ = E .4.1. The family U a . Let a ∈ k . Consider the following action of X on V :[ X ] Λ = a . . . a . . . a . . . a . . . . . . a
00 0 0 . . . a . Clearly V with this action is a B -module which will be denoted by U a . Lemma 4.1.
Let W be a proper B -submodule of U a . Then: (i) v / ∈ W ; (ii) If v = P ni =1 λ i v i ∈ W then λ = 0 .Proof. (i): Suppose v ∈ W . Then v = X v ∈ W and X v = av + v ∈W . Hence v ∈ W . Again, X v = av + v ∈ W and consequently v ∈ W .With this procedure, we obtain that Λ ⊂ W . Thus, W = U a and we have acontradiction.(ii): Assume λ = 0 and fix w = λ − v . Thus w = α v + v + . . . + α n v n ,where α i = λ − λ i , for all 1 ≤ i ≤ n . Consider the following elements of V : w j := v j +1 + α v j +2 + . . . + α n − j +1 v n , for all 2 ≤ j ≤ n − . By a straightforward calculation, we obtain that X w j = aw j + w j +1 , forall 1 ≤ j ≤ n −
2. Thus, w , . . . , w n − ∈ W and X w n − = aw n − + v n .Therefore, v n ∈ W . But w n − = v n − + α v n and whence v n − ∈ W . Bythis procedure, it follows that v , . . . , v n ∈ W . From v = X w ∈ W , itfollows that v ∈ W which contradicts (i). (cid:3) Theorem 4.2. U a is an indecomposable B -module, for all n ≥ .Proof. Suppose U a decomposable. Let W , f W be nontrivial B -submodules of U a such that U a = W ⊕ f W . Consider { w , . . . , w r } and { w r +1 , . . . , w n } basisof W and f W respectively. By Lemma 4.1, w i = λ i v + λ i v + . . . + λ in v n ,for all 1 ≤ i ≤ n . Since v ∈ U a , there exist α , . . . , α n ∈ k such that v = α w + . . . + α n w n , a contradiction. (cid:3) The family V a . Let a ∈ k × . Consider the following action of X on V : [ X ] Λ = a . . . − a . . . − a . . . − a . . . . . . − a
00 0 0 . . . − a . Notice that V is a B -module which will be denoted by V a . Since a = 0, U a and V a are not isomorphic. Theorem 4.3. V a is an indecomposable B -module, for all n ≥ .Proof. Let W a proper B -submodule of V a . As in Lemma 4.1 (i), we canshow that v / ∈ W . Let v ∈ W such that v = P ni =1 λ i v i . Assume that λ = 0and consider u := λ − v ∈ W . Then v = X u ∈ W . Take w := u − λ − λ v and note that w = α v + . . . + α n v n , where α i = λ − λ i , for all 2 ≤ i ≤ n .Considering the following elements of Vw j := v j +1 + α v j +2 + . . . + α n − j +1 v n , for all 2 ≤ j ≤ n − , it follows X w j = − aw j + w j +1 , for all 1 ≤ j ≤ n −
2. As in Lemma 4.1,this implies that v ∈ W which is a contradiction. Thus, the result followsas in Theorem 4.2. (cid:3) References [AAH1] N. Andruskiewitsch, I. Angiono and I. Heckenberger.
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E-mail address : [email protected] Departamento de Matem´atica, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil
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