Representation Theory o H Algebroids Atsushi Yamaguchi
Contents o this slide 1. Internal categories and H algebroids (7p) 2. Fibered category o modules (6p) 3. Representations o H algebroids (7p) 4. Restrictions o representations (4p) 5. Let regular representations (3p) 6. Construction o let induced representations (12p)
1. Internal categories and H algebroids Let C be a category with inite limits. Deinition 1.1 An internal category in C consists o the ollowing data. (1) A pair (C 0, C 1 ) o objects o C. (2) Four morphisms,: C 1 C 0,: C 0 C 1, :C 1 C 1 C C0 1 pr 1 pr 2 o C, where C 1 C 1 C 1 C 1 is a limit o C 1 C 0 C 1, such that = = 1 C0 commute. C 0 and the ollowing diagrams pr C 1 pr 1 C 1 C 2 1 C C0 1 C 0 C 1 C 0 1 C 1 C 1 C 1 C 1 C 1 C 0 C 0 1 C 0 C 1 C 1 C 1 C 0 1 C 1 C 1 C0 (1, ) (, 1) C 1 C 1 C 1 1
We also have a notion o internal unctors between internal categories. Deinition 1.2 Let C = (C 0, C 1 ;,,, ) and D = (D 0, D 1 ;,,, ) be internal categories in C. An internal unctor rom C to D is a pair ( 0, 1 ) o morphisms 0 : C 0 D 0 and 1 : C 1 D 1 which make the ollowing diagrams commute. C 0 C 1 C 0 C 1 C 1 C 1 C 0 C 0 1 0 D 0 D 1 D 0 0 1 1 1 0 D 1 D 1 D 1 D 0 D 0
Let k be a commutative ring. We denote by Alg k the category o commutative graded k-algebras and homomorphisms between them. For objects A and B o Alg k, we deine maps i 1 : A A k B and i 2 : B A k B by i 1 (x) = x 1 and i 2 (y) = 1 y, respectively. Then, a diagram A i 1 i A k B 2 B is a croduct o A and B in Alg k. For morphisms, g : A B o Alg k, let I be the ideal o B generated by {(x) - g(x) x A }. Then the quotient map p : B B /I is a coequalizer o and g. Hence, Alg k is a category with inite colimits.
In other words, the posite category Alg k o Alg k is a category with inite limits. Thus we can consider the notion o internal categories in Alg k. Deinition 1.3 We call an internal category in Alg k a H algebroid. Namely, a H algebroid consists o the ollowing data. (1) A pair (A,) o objects o Alg k. (2) Four morphisms, : A, : A, : A o A A i 1 i A 2 A id A A A A Alg k, where A is a colimit o A, such that i 1 i 2 = = id A and the ollowing diagrams in Alg k are commutative. A id id id (1, ) (, 1) A
Deinition 1.4 Let = (A,;,,, ) and = (B, ;,,, ) be H algebroids in Alg k. An internal unctor rom to is make the ollowing diagrams commute. A called a morphism o H algebroids rom to. Namely, a morphism o H algebroids rom to is a pair ( 0, 1 ) o morphisms 0 : A B and 1 : which A 0 1 B B A A 0 1 1 1 0 B B
For a category C with inite limits and an object A o C, pr 1 pr 2 id A id A let A A A A A be a limit o A A A and d : A A A A the unique morphism that satisies pr 1 d = pr 2 d = id A. Then, d is an isomorphism and -1 (A, A ; id A, id A, id A, d ) is an internal category in C. Deinition 1.5 We call (A, A ; id A, id A, id A, d -1 ) the trivial internal category in C with object A o objects. We denote this by. Remark 1.6 For an internal category C = (C 0, C 1 ;,,, ) in C, (id C, ) : C is unique internal unctor rom C to 0 C 0 C0 whose irst component is the identity morphism o C 0. A
Deinition 1.7 For an object A o Alg k, the trivial internal category in Alg k with object A o objects is called the trivial H 0 algebroid over A and denote it by. Remark 1.8 A For a H algebroid = (A,;,,, ) in Alg k, it is clear that (id A, ) : A is unique morphism o H algebroids whose irst component is the identity map o A. We denote this morphism by e A. A Let : A A A A be the isomorphism that maps 1 to 1 1. Then, we have = (A, A ; id A, id A, id A, ). 0
2. Fibered category o modules For a commutative ring k, we denote by Mod k the category o graded k-modules and homomorphisms preserving degrees. We deine a category MOD as ollows. Ob MOD consists o triples (R, M,) where R Ob Alg k, M Ob Mod k and : M k R M is a right R -module structure o M. o morphisms Alg k (R, S ) the right diagram commutes. A morphism rom (R, M,) to (S, N,) is a pair (, ) and Mod k (M, N ) such that M k R N k S M k Composition o (, ) :(R, M,) (S, N,) and (, ):(S, N,) (T, L,) is deined to be (, ). N
Deine a unctor p : MOD Alg k by p(r, M,) = R and p(,) = For a morphism : S R o Alg k and an object (S, N,) o MOD, let : (N S R ) k R N S R be the ollowing composition. Here mdenotes the multiplication o R. morphisms o the orm (id R, ). (N S R ) k R N S (R k R ) N S R For an object R o Alg k, let MOD R be a subcategory o MOD consisting o objects o the orm (R, M,) and id N m
For a morphism : S R o Alg k, we deine unctors : MOD S MOD R : MOD R MOD S We can show the ollowing. Prosition 2.1 (Fibered category o modules) by (S, N,) = (R, N S R, ), (id S,) = (id R, S id R ) p : MOD Alg k is a ibered category. and (R, M,) = (S, M,(id M K )), (id S,) = (id R, ). Prosition 2.2 is a let adjoint o : MOD S MOD R. is called the inverse image unctor. For any morphism : R S o Alg k, : MOD R MOD S
Remark 2.3 For an object M = (R, M,) o MOD R and an object N = (S, N,) o MOD S, the natural bijection ad : MOD R ( (N), M) MOD S (N, (M)) maps (id R, ) :(N) M to (id S, i N ): N (M). Here, : N S R M and i N : N N S R is gven by i N (x) = x 1.
For morphisms : R S We also deine an isomorphism o MOD T, : S T o Alg k and an object M = (R, M,) o MOD R, we deine a morphism c c (M): ( )(M) (M), (M) : M R T (M R S ) S T, by c (M)(m t) = (m 1) t. Then, c (M) is an isomorphism.,, by c (M) = (id T, c (M)).,,
Let : R S,: T S, : S U be morphisms o Alg k and M = (R, M,) an object o MOD R, N = (T, N,) an object o MOD T. c For a morphism :(M) (N) o MOD S, we denote by (M), ( ) (M) (M) (N) ( ) (N) then, : M R U N T U is the ollowing composition. : ( ) (M) ( ) (N) the ollowing composition. ( ) c (N) c (N) N T U, -1 I = (id S,) or: M R S N T S and = (id U, ), c, (M) S id U M R U (M R S ) S U (N T S ) S U, -1
3. Representations o H algebroids Let = (A,;,,, ) be a H algebroid in Alg k. Deinition 3.1 A pair (M, ) o an object M o MOD A and a morphism : (M) (M) o MODis called a representation o on M i the ollowing conditions (A) and (U) are satisied. (A) The ollowing diagram in MODis commutative. i2 ( )(M) = (i 2 )(M) (i 2 )(M) = (i 1 )(M) (U) i1 ( )(M) = (i 1 )(M) : M = ( )(M) ( )(M) = M coincides with the identity morphism o M.
Prosition 3.2 For a morphism : (M) (M) o MOD, let : ˆ M (M) be the image o by the natural bijection ad : MOD A ( (M), (M)) MOD A (M, (M)). M We put = ˆ (id A, ) or : M M A. (1) satisies (A) o (3.1) i and only i the ollowing let diagram is commutative. (2) satisies (U) o (3.1) i and only i the ollowing right diagram is commutative. M A id M M A M A A M M A id id M M A A
Deinition 3.3 A pair (M, ) o right A -module M and a homomorphism : M Let M A o right A -modules which makes the : M ollowing diagrams commute is called a right M M A M M A id id M M A M A A Remark 3.4 e -comodule. id M M A A M A A be a map which maps x to x 1. Then, it is an isomorphism and (M, ) is a right -comodule. We call this a trivial -comodule. A e A
We deine the category Rep( ) o representations o a H algebroid as ollows. Objects o Rep( ) are representations o. For representations (M, ) and (N, ) o, the set : N M o MOD A commute. Rep( )((M, ), (N, )) o morphisms rom (M, ) to (N, ) consists o morphims which make the ollowing diagram (N) (M) (N) ( ) ( ) (M)
We deine the category Comod( ) o right -comodules as ollows. Objects o Comod( ) are right -comodules. For right -comodules (M, ) and (N, ), the set Comod( )((M, ), (N, )) o morphisms rom (M, ) to (N, ) consists o homo- morphisms : M N o right A -modules which make the ollowing diagram commute. M N M A id N A
Prosition 3.5 Let M = (A, M,) and N = (A, N,) be objects o MOD A. For representations (M, ) and (N, ) o, we put ad ( ) = (id A, ) and ad ( ) = (id A, ) or : M M A and : N N A. a morphism A homomorphism : N (id A,:(M, ) (N, ) o representations o i and only i is a morphism o right M o right A -modules deines : (N, ) (M, ) -comodules.
We deine a unctor as ollows. : Rep( ) Comod( ) For an object (M, ) o Rep( ), i M = (A, M,) and ad ( ) = (id A, ) or : M M A, we put (M, ) = (M, ). For a morphism (id A, :(M, ) (N, ) o Rep( ), i ad ( ) = (id A, ) or : M M A, we put (id A,) = ( : (N, ) (M, )). (M, ). Combining results so ar, we have the ollowing. We call (M, ) the right -comodule associated with Prosition 3.6 : Rep( ) Comod( ) is an isomorphism o categories.
4. Restrictions o representations Let = (A,;,,, ) and = (B, ;,,, ) be H algebroids in Alg k. For a morphism = ( 0, 1 ) : o H algebroids and a representation (M, ) o, we deine a morphism o MOD to be the ollowing composition. c (M) : ( 0 (M)) ( 0 (M)) 0, ( 0 (M)) ( 0 ) (M) = ( 1 ) (M) ( 1 ) (M) 1 c (M) -1 0, = ( 0 ) (M) ( 0 (M))
Prosition 4.1 (1) ( 0 (M), ) is a representation o on 0 (M). (2) I o We call ( 0 (M), ) the restriction o (M, ) along. deine a unctor : (M, ) (N, ) is a morphism o representations, 0 ( ) :( 0 (M), ) ( 0 (N), ) is a morphism o representations o. Deinition 4.2 For a morphism = ( 0, 1 ) : o H algebroids, we. : Rep( ) Rep( ). by (M, ) = ( 0. (M), ) and ( ) = 0 ( ) or an object (M, ) and a morphism : (M, ) (N, ) o Rep( ).
Prosition 4.3 For a representation (M, ) o on M = (A, M,), let (M, ) be the right -comodule associated with (M, ).. Then, the right -comodule ( (M, )) associated with. (M, ) is given as ollows. Deine a map by : (M A ) A B (M A B ) B ((x y) b) = (x 1) y (b) and let : M A B (M A B ) B M A B (M A ) A B (M A ) A B A id B be the ollowing composition. (id M A 1 ) A id B. Then, we have ( (M, )) = (M A B, ). (M A B ) B
Remark 4.4 I is the morphism e A = (id A, ) : A given in (1.8), 0 (M) = id A (M) = (A, M A A, ) is isomorphic to M. Since (id M A ) : M M A A maps x to x 1 and coincides with the identity morphism,. (e A (M, )) = (M A A, ) (3.4). : (M A A is isomorphic to a trivial -comodule (M, ) given in id A ) A A (M A A ) A A A e A A e
5. Let regular representations Let = (A,;,,, ) be a H algebroids in Alg k. We denote by U : Rep( ) MOD A the orgetul unctor. That is, U is given by U (M, ) = M and U ( ) =. Deine a unctor F id M A M A M A ( A ) (M A ) A F () = A id. : MOD A Comod( ) as ollows. For an object M = (A, M,) o MOD A, let : M A (M A ) A We put F (M) = (M A, ). For a morphism = (id A, ) : M N o MOD A, we put be the ollowing composition. M M
Let F : MOD A Rep( ) be the ollowing composition. Lemma 5.1 F MOD A Comod( ) Rep( ). (M, ) be the right -comodule associated with (M, ). -1 For a representation (M, ) o on M = (A, M,), let Then, : M M A deines a morphism o representations o. (id A, ) : F (M) (M, )
Prosition 5.2 For a representation (M, ) o and N an object o MOD A, let (M, ) be the right with (M, ) and we deine a map N ad : MOD A (N, U (M, )) Rep( )(F (N),(M, )) (M, ) N (M, ) by ad (id A, ) = (id A, F ( )). Then, ad is bijective. Hence F is a let adjoint o U. Deinition 5.3 For an object M o MOD A, we call F (M) the let regular regular representation associated with M. -comodule associated N (M, )
6. Construction o let induced representations Let = (A,;,,, ) and = (B, ;,,, ) be H algebroids in Alg k and = ( 0, 1 ) : a morphism o H algebroids. We deine maps by 0 (x) = 1 x and 0 : B A and : B B A (b) = b 1, respectively. Then, is cocartesian. A B 0 0 B A
Let M = (B, M,) an object o MOD B. Then, we have ( 0 ) (M) = (A, M B (B A ), (id M (B ) k( 0 ))). We regard M as a right A -module by on M A by ((x y) (b z)) = (-1) We also deine a map Lemma 6.1 deine a right B A -module strcture (x y) = x (1 y). The ollowing morphism is an isomorphism in MOD A. (id A, ):(A, M A, (id M k ( 0 ))) ( 0 ) (M) A B A :(M A ) k (B A ) M A : M A M B (B A ) by deg(y)deg(b) (id M k 0 ) and (x b) yz.
We put = (id M k ( 0 )) and identiy ( 0 ) (M) with (A, M A A, ) by (id A, ) below. Deine a homomorphism (M): M A (M A ) A Thus we have a unctor MOD B Comod( ) by assigning an object M = (B, M,) o MOD B to (M A, ˆ (M)). We put (M A, ˆ (M)) = (( 0 ) (M), (M)). ˆ o right A -module to be the ollowing composition. (M A id M A M A M A ( A ) (M A ) A Prosition 6.2, ˆ (M)) is a right -comodule. -1 l
For a representation (M, ) o on M = (B, M,), we put (M, ) = (M, ). Here is a homomorphism 1 2 is the ollowing composition. A id M A (M B id M A (( 1 A id ) ) ) A M B ( A ) M A M A ( A ) M B ( A ) 0 : M M B o right B -modules. We deine homomorphisms o right A -modules 1, : M A M B ( A ) as ollows. is the ollowing composition. 2 0 Here, is the quotient map induced by 0 : A B.
Let us denote by : K(M,; ) M A the kernel o - : M A M B ( A ). 1 2 (M, ) We assume that : A is lat. Then, is the kernel o A id - A id : (M A ) A (M B ( A )) A. 1 2 (M, ) A id : K(M,; ) A (M A ) A Then, the ollowing diagram is commutative. (M, ) K(M,; ) (M, ) ˆ(M) M A M A (M A ) A ˆ(M) (M A ) A 1 A id (M B ( A )) A 2 A id
There exists unique homomorphism o right A -modules l (M): K(M,; ) K(M,; ) A that makes the ollowing diagram commute. Prosition 6.3 K(M,; ) (M, ) M A l (M) K(M,; ) A ˆ(M) (K(M,; ), (M)) is a right (M, ) l l (M A ) A (M, ) A id -comodule and :(K(M,; ), (M)) (M A, (M)) ˆ is a morphism o right -comodules.
Let M = (B, M,) and N = (B, N,) be objects o MOD B. For a morphism (id A, :(M, ) (N, ) be o representa- tions o, we put (M, ) = (M, ) and (N, ) = (N, ). :(N, ) (M, ) is a morphism o right -comodules. Since is natural in morphism o right 1 : K(N,; ) K(M,; ) K(N,; ) N A A id -comodules and is natural in homomorphism o right B -modules, there 2 exists unique homomorphism o right A -modules that makes the ollowing diagram commute. K(M,; ) (N, ) (M, ) M A
Prosition 6.4 l :(K(N,; ),(N)) (K(M,; ),(M)) is a morphism o right We deine a unctor -1 -comodules.! : Rep( ) Rep( ) l by! (M, ) = (K(M,; ),(M)) and! (id B, ) = (id A, ) or an object (M, ) and a morphism (id B, ) :(M, ) (N, ) o Rep( ). Theorem 6.5! is a let adjoint o.. l We call! (M, ) the let induced representation o (M, ) associated with.
For an object M = (B, M,) o MOD B, we deine a map M :(M A ) A B M ( ) : K(M,; ) A B M K(M,; ) A B (M A ) A B M We put (M, ) = (M, ). Then, by M ((x y) b) = (x 0 ( (y))b) and i (M, ) is an object o Rep( ), let us deine (M, ) to be the ollowing composition. Prosition 6.6 is a morphism o right (M, ) A id B -comodules. M ( ) :(K(M,; ) A B,((M)) ) (M, ) (M, ) l
Let (N, ) be a representation o and (M, ) a represen- tation o ad (M, ) (N, ). Put M = (B, M,) and N = (A, N,). We also right -comodules, ad put (M, ) = (M, ) and (N, ) = (N, ). We deine a map : Comod( )((N, ), (K(M,; ),(M))) Comod( )((N A B, ),(M, )) below. For a morphism : (N, ) (K(M,; ),(M)) o (M, ) (N, ) ( ) is the ollowing composition. A id B ( ) N A B K(M,; ) A B M l (M, ) l
Prosition 6.7 ad (M, ) (N, ) is bijective. Since we have the ollowing equalities, the above result shows that we have the ollowing natural bijection. Rep( )(! (M, ), (N, )) Rep( )((M, ), (N, )). This shows Theorem 6.5. -1! (M, ) = (K(M,; ),(M)) l. -1 (N, ) = (N A B, )
Remark 6.8 For an object M = (A, M,) o MOD A, we consider the the trivial right -comodule (M, ) and denote by A e (M, ) the representation which is mapped to (M, ) by M induced representation e A! (M, ) associated with e A = (id A, ) : isomorphic to the let regular e For a H algebroid in Alg k, we claim that the let representation o on M. In act, or a representation. (N, ) o, the A -comodule associated with e A (N, ) is isomorphic to the trivial right -comodule by (4.4), U : Rep( A A the orgetul unctor U : Rep( ) Comod( ). A A A A M )((M, ), e A (N, )) MOD A (M, N). A. A gives a bijection M
Thank you or listening and your patience.