Create a right quiver \(q\) with a single vertix \(o\) and a single arrow \(m\).
gap> q := RightQuiver( "q(o)[m:o->o]" ); q(o)[m:o->o] gap> o := q.o; (o) gap> m := q.m; (m)
Create a monoid from \(q\).
gap> N := FreeCategory( q ); FreeCategory( RightQuiver( "q(o)[m:o->o]" ) ) gap> IsCommutative( N ); true gap> N.o; <(o)> gap> N.m; (o)-[(m)]->(o) gap> SetOfObjects( N ); [ <(o)> ] gap> SetOfGeneratingMorphisms( N ); [ (o)-[(m)]->(o) ] gap> SetOfGeneratingMorphisms( N, N.o, N.o ); [ (o)-[(m)]->(o) ] gap> ObjectInFpCategory( N, o ) = N.o; true gap> N.o = o / N; true gap> IdentityMorphism( N.o ); (o)-[(o)]->(o) gap> MorphismInFpCategory( N, m ) = N.m; true gap> N.m = m / N; true
We can compute in the algebroids. For instance we can form compositions.
gap> PreCompose( N.m, N.m ); (o)-[(m*m)]->(o) gap> unit := Unit( N ); Functor from FreeCategory( RightQuiver( "*(1)[]" ) ) -> FreeCategory( RightQuiver( "q(o)[m:o->o]" ) ) gap> mul := Multiplication( N ); Functor from FreeCategory( RightQuiver( "qxq(oo)[om:oo->oo,mo:oo->oo]" ) ) / relations -> FreeCategory( RightQuiver( "q(o)[m:o->o]" ) )
Create a right quiver \(q\) with a single vertix \(o\) and two arrows \(m\), \(i\).
gap> q := RightQuiver( "q(o)[m:o->o,i:o->o]" ); q(o)[m:o->o,i:o->o] gap> o := q.o; (o) gap> m := q.m; (m) gap> i := q.i; (i) gap> L := [ [ m * i, o ], [ i * m, o ] ]; [ [ (m*i), (o) ], [ (i*m), (o) ] ]
Create a monoid from \(q\) and the list \(L\) of relations.
gap> zz := Category( q, L ); FreeCategory( RightQuiver( "q(o)[m:o->o,i:o->o]" ) ) / [ m*i = o, i*m = o ] gap> IsCommutative( zz ); true gap> zz.o; <(o)> gap> IsWellDefined( zz.o ); true gap> m := zz.m; (o)-[(m)]->(o) gap> i := zz.i; (o)-[(i)]->(o) gap> SetOfObjects( zz ); [ <(o)> ] gap> SetOfGeneratingMorphisms( zz ); [ (o)-[(m)]->(o), (o)-[(i)]->(o) ] gap> SetOfGeneratingMorphisms( zz, zz.o, zz.o ); [ (o)-[(m)]->(o), (o)-[(i)]->(o) ] gap> ObjectInFpCategory( zz, o ) = zz.o; true gap> zz.o = o / zz; true gap> IdentityMorphism( zz.o ); (o)-[(o)]->(o) gap> MorphismInFpCategory( zz, q.m ) = zz.m; true gap> zz.m = q.m / zz; true
We can compute in the algebroids. For instance we can form compositions.
gap> m2 := PreCompose( [ m, i, m, m, i, m, i, m ] ); (o)-[(m*m)]->(o) gap> IsWellDefined( m2 ); true gap> unit := Unit( zz ); Functor from FreeCategory( RightQuiver( "*(1)[]" ) ) -> FreeCategory( RightQuiver( "q(o)[m:o->o,i:o->o]" ) ) / [ m*i = o, i*m = o ] gap> mul := Multiplication( zz ); Functor from FreeCategory( RightQuiver( "qxq(oo)[om:oo->oo,oi:oo->oo,mo:oo->oo,io:oo->oo]" ) ) / relations -> FreeCategory( RightQuiver( "q(o)[m:o->o,i:o->o]" ) ) / [ m*i = o, i*m = o ]
Create a left quiver \(ql\) with 3 vertices u,v,w and 4 edges a,b,c,d.
gap> ql := LeftQuiver( "ql(u,v,w)[a:u->v,b:v->u,c:v->u,d:v->w]" ); ql(u,v,w)[a:u->v,b:v->u,c:v->u,d:v->w]
Create a right quiver \(qr\) with 3 vertices u,v,w and 4 edges a,b,c,d.
gap> qr := RightQuiver( "qr(u,v,w)[a:u->v,b:v->u,c:v->u,d:v->w]" ); qr(u,v,w)[a:u->v,b:v->u,c:v->u,d:v->w]
Create f.p. categories from these quivers.
gap> Cl := FreeCategory( ql ); FreeCategory( LeftQuiver( "ql(u,v,w)[a:u->v,b:v->u,c:v->u,d:v->w]" ) ) gap> Cr := FreeCategory( qr ); FreeCategory( RightQuiver( "qr(u,v,w)[a:u->v,b:v->u,c:v->u,d:v->w]" ) ) gap> Cl.u; <(u)> gap> Cr.u; <(u)> gap> Cl.a; (v)<-[(a)]-(u) gap> Cr.a; (u)-[(a)]->(v) gap> SetOfObjects( Cl ); [ <(u)>, <(v)>, <(w)> ] gap> SetOfGeneratingMorphisms( Cl ); [ (v)<-[(a)]-(u), (u)<-[(b)]-(v), (u)<-[(c)]-(v), (w)<-[(d)]-(v) ] gap> SetOfGeneratingMorphisms( Cl, Cl.u, Cl.v ); [ (v)<-[(a)]-(u) ] gap> SetOfObjects( Cr ); [ <(u)>, <(v)>, <(w)> ] gap> SetOfGeneratingMorphisms( Cr ); [ (u)-[(a)]->(v), (v)-[(b)]->(u), (v)-[(c)]->(u), (v)-[(d)]->(w) ] gap> SetOfGeneratingMorphisms( Cr, Cr.u, Cr.v ); [ (u)-[(a)]->(v) ] gap> ObjectInFpCategory( Cl, ql.u ) = Cl.u; true gap> Cl.u = ql.u / Cl; true gap> MorphismInFpCategory( Cl, ql.a ) = Cl.a; true gap> Cl.a = ql.a / Cl; true gap> IdentityMorphism( Cl.u ); (u)<-[(u)]-(u) gap> IdentityMorphism( Cr.u ); (u)-[(u)]->(u)
We can compute in the algebroids. For instance we can form compositions.
gap> PreCompose( Cl.a, Cl.d ); (w)<-[(d*a)]-(u) gap> PreCompose( Cr.a, Cr.d ); (u)-[(a*d)]->(w)
We can construct quotient categories
gap> q:= RightQuiver( "q(1)[a:1->1]" ); q(1)[a:1->1] gap> Fq := FreeCategory( q ); FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) gap> C := Fq / [ [ Fq.a^3, Fq.a^2 ] ]; FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a*a ] gap> UnderlyingQuiverAlgebra( C ); (Q * q) / [ 1*(a*a*a) - 1*(a*a) ] gap> Size( C ); 3 gap> D := C / [ [ C.a^9, C.1 ] ]; FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a*a, a*a = 1 ] gap> UnderlyingQuiverAlgebra( D ); (Q * q) / [ 1*(a) - 1*(1) ] gap> Size( D ); 1
The full subcategory of the simplicial category \(\Delta\) on the objects \([0], [1], [2]\)
gap> LoadPackage( "Algebroids" ); true gap> Delta2 := SimplicialCategoryTruncatedInDegree( 2 ); FreeCategory( RightQuiver( "Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1, is:C2->C1,it:C2->C1, ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations gap> RelationsOfFpCategory( Delta2 ); [ [ (s*id), (C0) ], [ (t*id), (C0) ], [ (ps*is), (C1) ], [ (pt*it), (C1) ], [ (is*id), (it*id) ], [ (pt*is), (id*t) ], [ (ps*it), (id*s) ], [ (s*pt), (t*ps) ], [ (s*mu), (s*ps) ], [ (t*mu), (t*pt) ], [ (mu*is), (C1) ], [ (mu*it), (C1) ] ] gap> Size( Delta2 ); 31 gap> Delta2_op := OppositeFiniteCategory( Delta2 ); Opposite( FreeCategory( RightQuiver( "Delta(C0,C1,C2)[id:C1->C0,s:C0->C1,t:C0->C1, is:C2->C1,it:C2->C1, ps:C1->C2,pt:C1->C2,mu:C1->C2]" ) ) / relations ) gap> IsIdenticalObj( OppositeFiniteCategory( Delta2_op ), Delta2 ); true
Create a left quiver \(ql\) with 3 vertices u,v,w and 4 edges a,b,c,d.
gap> ql := LeftQuiver( "ql(u,v,w)[a:u->v,b:v->u,c:v->u,d:v->w]" ); ql(u,v,w)[a:u->v,b:v->u,c:v->u,d:v->w]
Create a right quiver \(qr\) with 3 vertices u,v,w and 4 edges a,b,c,d.
gap> qr := RightQuiver( "qr(u,v,w)[a:u->v,b:v->u,c:v->u,d:v->w]" ); qr(u,v,w)[a:u->v,b:v->u,c:v->u,d:v->w]
Create algebroids from these quivers over \(\mathbb{Q}\).
gap> Q := HomalgFieldOfRationals( ); Q gap> Al := Algebroid( Q, ql ); Algebroid( Q, FreeCategory( LeftQuiver( "ql(u,v,w)[a:u->v,b:v->u,c:v->u,d:v->w]" ) ) ) gap> algebra_Al := UnderlyingQuiverAlgebra( Al ); Q * ql gap> Ar := Algebroid( Q, qr ); Algebroid( Q, FreeCategory( RightQuiver( "qr(u,v,w)[a:u->v,b:v->u,c:v->u,d:v->w]" ) ) ) gap> algebra_Ar := UnderlyingQuiverAlgebra( Ar ); Q * qr gap> Al.u; <(u)> gap> Ar.u; <(u)> gap> Al.a; (v)<-[1*(a)]-(u) gap> Ar.a; (u)-[1*(a)]->(v) gap> SetOfObjects( Al ); [ <(u)>, <(v)>, <(w)> ] gap> SetOfGeneratingMorphisms( Al ); [ (v)<-[1*(a)]-(u), (u)<-[1*(b)]-(v), (u)<-[1*(c)]-(v), (w)<-[1*(d)]-(v) ] gap> SetOfGeneratingMorphisms( Al, Al.u, Al.v ); [ (v)<-[1*(a)]-(u) ] gap> SetOfObjects( Ar ); [ <(u)>, <(v)>, <(w)> ] gap> SetOfGeneratingMorphisms( Ar ); [ (u)-[1*(a)]->(v), (v)-[1*(b)]->(u), (v)-[1*(c)]->(u), (v)-[1*(d)]->(w) ] gap> SetOfGeneratingMorphisms( Ar, Ar.u, Ar.v ); [ (u)-[1*(a)]->(v) ] gap> CommutativeRingOfLinearCategory( Al ) = Q; true gap> ObjectInAlgebroid( Al, ql.u ) = Al.u; true gap> Al.u = ql.u / Al; true gap> MorphismInAlgebroid( Al, algebra_Al.a ) = Al.a; true gap> Al.a = algebra_Al.a / Al; true gap> IdentityMorphism( Al.u ) = algebra_Al.u / Al; true
We can compute in the algebroids. For instance we can form compositions.
gap> PreCompose( Al.a, Al.d ); (w)<-[1*(d*a)]-(u) gap> PreCompose( Ar.a, Ar.d ); (u)-[1*(a*d)]->(w)
We also have addition and subtraction of morphisms and scalar multiplication with elements from the ground ring \(\mathbb{Q}\).
gap> Al.b+Al.c; (u)<-[1*(c) + 1*(b)]-(v) gap> Ar.b+Ar.c; (v)-[1*(c) + 1*(b)]->(u) gap> Al.b-Al.c; (u)<-[-1*(c) + 1*(b)]-(v) gap> Ar.b-Ar.c; (v)-[-1*(c) + 1*(b)]->(u) gap> Al.d - Al.d; (w)<-[0]-(v) gap> Ar.d - Ar.d; (v)-[0]->(w) gap> 1/2 * Al.d; (w)<-[1/2*(d)]-(v) gap> 1/2 * Ar.d; (v)-[1/2*(d)]->(w)
We can construct quotient categories
gap> q:= RightQuiver( "q(1)[a:1->1]" ); q(1)[a:1->1] gap> Fq := FreeCategory( q ); FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) gap> Q := HomalgFieldOfRationals( ); Q gap> A := Q[Fq]; Algebra( Q, FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) ) gap> UnderlyingQuiverAlgebra( A ); Q * q gap> B := A / [ 2*A.a^3 - A.a^2 ]; Algebra( Q, FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) ) / relations gap> UnderlyingQuiverAlgebra( B ); (Q * q) / [ 2*(a*a*a) - 1*(a*a) ] gap> Dimension( B ); 3 gap> C := B / [ B.a^2 + 2*B.a ]; Algebra( Q, FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) ) / relations gap> UnderlyingQuiverAlgebra( C ); (Q * q) / [ 2*(a*a*a) - 1*(a*a), 1*(a*a) + 2*(a) ] gap> Dimension( C ); 1
Create a right quiver \(q\) with two vertices \(u\) and \(v\) and two edges \(a\) and \(b\).
gap> q := RightQuiver( "q(u,v)[a:u->v,b:v->u]" ); q(u,v)[a:u->v,b:v->u]
Create the path algebra \(\mathbb{Q}q\) of \(q\) over \(\mathbb{Q}\).
gap> q := RightQuiver( "q(u,v)[a:u->v,b:v->u]" ); q(u,v)[a:u->v,b:v->u] gap> Q := HomalgFieldOfRationals( ); Q gap> Qq := PathAlgebra( Q, q ); Q * q
Form a quotient of \(\mathbb{Q}q\) by the relations \(ab - u\) and \(ba - v\).
gap> B := Algebroid( Qq, [ Qq.a * Qq.b - Qq.u, Qq.b * Qq.a - Qq.v ] ); Algebroid( Q, FreeCategory( RightQuiver( "q(u,v)[a:u->v,b:v->u]" ) ) ) / relations gap> RelationsOfAlgebroid( B ); [ (u)-[1*(a*b) - 1*(u)]->(u), (v)-[1*(b*a) - 1*(v)]->(v) ]
gap> IsCommutative(B); false
Next we want to define a counit and a comultiplication on \(B\) to make it a bialgebroid. The following record will be used to define the counit of \(B\).
gap> counit := rec( a := 1, b := 1 ); rec( a := 1, b := 1 )
We compute the tensor product \(B \otimes_\mathbb{Q} B\).
gap> B2 := B^2; Algebroid( Q, FreeCategory( RightQuiver( "qxq(uu,uv,vu,vv)[\ ua:uu->uv,ub:uv->uu,va:vu->vv,vb:vv->vu,\ au:uu->vu,av:uv->vv,bu:vu->uu,bv:vv->uv]" ) ) ) / relations
The following record will be used to define the comultiplication \(B \to B \otimes_{\mathbb{Q}} B\).
gap> comult:= rec( a := PreCompose(B2.au, B2.va), b:= PreCompose(B2.bv, B2.ub) ); rec( a := (uu)-[{ 1*(ua*av) }]->(vv), b := (vv)-[{ 1*(vb*bu) }]->(uu) )
We use the records counit and comult to equip \(B\) with the induced morphisms \(B \to \mathbb{Q}\) and \(B \to B \otimes_{\mathbb{Q}} B\).
gap> AddBialgebroidStructure( B, counit, comult ); Bialgebroid( Q, FreeCategory( RightQuiver( "q(u,v)[a:u->v,b:v->u]" ) ) ) / relations
We extract comultiplication and counit from \(B\). They are now functors.
gap> counit := Counit( B ); Functor from Bialgebroid( Q, FreeCategory( RightQuiver( "q(u,v)[a:u->v,b:v->u]" ) ) ) / relations -> Algebra( Q, FreeCategory( RightQuiver( "*(1)[]" ) ) ) gap> comult := Comultiplication( B ); Functor from Bialgebroid( Q, FreeCategory( RightQuiver( "q(u,v)[a:u->v,b:v->u]" ) ) ) / relations -> Algebroid( Q, FreeCategory( RightQuiver( "qxq(uu,uv,vu,vv)[ua:uu->uv,ub:uv->uu,va:vu->vv,vb:vv->vu,\ au:uu->vu,av:uv->vv,bu:vu->uu,bv:vv->uv]" ) ) ) / relations
Apply comultiplication and counit.
gap> ApplyFunctor( counit, B.u ); <(1)> gap> ApplyFunctor( counit, B.v ); <(1)> gap> ApplyFunctor( counit, IdentityMorphism( B.u ) ); (1)-[1*(1)]->(1) gap> ApplyFunctor( counit, IdentityMorphism( B.v ) ); (1)-[1*(1)]->(1) gap> ApplyFunctor( counit, B.a ); (1)-[1*(1)]->(1) gap> ApplyFunctor( counit, B.b ); (1)-[1*(1)]->(1) gap> ApplyFunctor( comult, B.u ); <(uu)> gap> ApplyFunctor( comult, B.v ); <(vv)> gap> ApplyFunctor( comult, IdentityMorphism( B.u ) ); (uu)-[{ 1*(uu) }]->(uu) gap> ApplyFunctor( comult, IdentityMorphism( B.v ) ); (vv)-[{ 1*(vv) }]->(vv) gap> ApplyFunctor( comult, B.a ); (uu)-[{ 1*(ua*av) }]->(vv) gap> ApplyFunctor( comult, B.b ); (vv)-[{ 1*(vb*bu) }]->(uu) gap> IsCoassociative(B); true gap> IsCocommutative(B); true
gap> q := RightQuiver( "q(1)[g:1->1,h:1->1]" ); q(1)[g:1->1,h:1->1] gap> Q := HomalgFieldOfRationals( ); Q gap> Qq := PathAlgebra( Q, q ); Q * q gap> B := Algebroid( Qq, [ Qq.h * Qq.g - Qq.1, Qq.g * Qq.h - Qq.1 ] ); Algebra( Q, FreeCategory( RightQuiver( "q(1)[g:1->1,h:1->1]" ) ) ) / relations gap> B2 := B^2; Algebra( Q, FreeCategory( RightQuiver( "qxq(1x1)[1xg:1x1->1x1,1xh:1x1->1x1,gx1:1x1->1x1,hx1:1x1->1x1]" ) ) ) / relations gap> counit_rec := rec( g := 1, h := 1 );; gap> comult_rec := rec( g := PreCompose( B2.1xg, B2.gx1 ), > h := PreCompose( B2.1xh, B2.hx1 ) );; gap> AddBialgebroidStructure( B, counit_rec, comult_rec ); Bialgebra( Q, FreeCategory( RightQuiver( "q(1)[g:1->1,h:1->1]" ) ) ) / relations gap> counit := Counit( B ); Functor from Bialgebra( Q, FreeCategory( RightQuiver( "q(1)[g:1->1,h:1->1]" ) ) ) / relations -> Algebra( Q, FreeCategory( RightQuiver( "*(1)[]" ) ) ) gap> comult := Comultiplication( B ); Functor from Bialgebra( Q, FreeCategory( RightQuiver( "q(1)[g:1->1,h:1->1]" ) ) ) / relations -> Algebra( Q, FreeCategory( RightQuiver( "qxq(1x1)[1xg:1x1->1x1,1xh:1x1->1x1,gx1:1x1->1x1,hx1:1x1->1x1]" ) ) ) / relations gap> antipode_rec := rec( g := B.h, h := B.g );; gap> AddAntipode(B, antipode_rec ); gap> B; HopfAlgebra( Q, FreeCategory( RightQuiver( "q(1)[g:1->1,h:1->1]" ) ) ) / relations gap> Antipode(B); Contravariant functor from HopfAlgebra( Q, FreeCategory( RightQuiver( "q(1)[g:1->1,h:1->1]" ) ) ) / relations -> HopfAlgebra( Q, Fr eeCategory( RightQuiver( "q(1)[g:1->1,h:1->1]" ) ) ) / relations gap> ApplyFunctor( counit, B.g ); (1)-[1*(1)]->(1) gap> IsCommutative(B); true gap> IsCounitary(B); true gap> IsCocommutative(B); true gap> IsHopfAlgebroid(B); true
gap> # Hopf algebra generated by a primitive element > q := RightQuiver( "q(1)[t:1->1]" ); q(1)[t:1->1] gap> Q := HomalgFieldOfRationals( ); Q gap> Qq := PathAlgebra( Q, q ); Q * q gap> B := Algebroid( Qq ); Algebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) gap> Q = UnderlyingAlgebra( B ); true gap> B2 := B^2; Algebra( Q, FreeCategory( RightQuiver( "qxq(1x1)[1xt:1x1->1x1,tx1:1x1->1x1]" ) ) ) / relations gap> B0 := B^0; Algebra( Q, FreeCategory( RightQuiver( "*(1)[]" ) ) ) gap> counit_rec := rec( t := 0 );; gap> comult_rec := rec( t := B2.1xt + B2.tx1 );; gap> AddBialgebroidStructure( B, counit_rec, comult_rec ); Bialgebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) gap> antipode_rec := rec( t:= -B.t );; gap> AddAntipode( B, antipode_rec ); gap> counit := Counit( B ); Functor from Bialgebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) -> Algebra( Q, FreeCategory( RightQuiver( "*(1)[]" ) ) ) gap> comult := Comultiplication( B ); Functor from Bialgebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) -> Algebra( Q, FreeCategory( RightQuiver( "qxq(1x1)[1xt:1x1->1x1,tx1:1x1->1x1]" ) ) ) / relations gap> idB := IdentityFunctor( B ); Identity functor of Algebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) gap> ApplyFunctor( comult, B.t ); (1x1)-[{ 1*(tx1) + 1*(1xt) }]->(1x1) gap> ApplyFunctor( counit, B.t ); (1)-[0]->(1) gap> IsCommutative(B); true gap> IsCounitary(B); true gap> IsCocommutative(B); true gap> IsHopfAlgebroid(B); true gap> BB := CategoryOfAlgebroidsObject(B);; gap> IsCoassociative( BB ); true gap> IsCounitary( BB ); true
gap> q := RightQuiver( "q(1)[t:1->1]" ); q(1)[t:1->1] gap> Q := HomalgFieldOfRationals( ); Q gap> Qq := PathAlgebra( Q, q ); Q * q gap> B := Algebroid( Qq ); Algebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) gap> B2 := B^2; Algebra( Q, FreeCategory( RightQuiver( "qxq(1x1)[1xt:1x1->1x1,tx1:1x1->1x1]" ) ) ) / relations gap> B0 := B^0; Algebra( Q, FreeCategory( RightQuiver( "*(1)[]" ) ) ) gap> counit_rec := rec( t := 0 );; gap> comult_rec := rec( t := B2.1xt );; gap> AddBialgebroidStructure( B, counit_rec, comult_rec ); Bialgebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) gap> antipode_rec := rec( t:= -B.t );; gap> AddAntipode(B, antipode_rec ); gap> counit := Counit( B ); Functor from Bialgebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) -> Algebra( Q, FreeCategory( RightQuiver( "*(1)[]" ) ) ) gap> comult := Comultiplication( B ); Functor from Bialgebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) -> Algebra( Q, FreeCategory( RightQuiver( "qxq(1x1)[1xt:1x1->1x1,tx1:1x1->1x1]" ) ) ) / relations gap> IsCommutative(B); true gap> IsCoassociative(B); true gap> IsCounitary(B); false
gap> q := RightQuiver( "q(1)[x:1->1,g:1->1]" ); q(1)[x:1->1,g:1->1] gap> Q := HomalgFieldOfRationals( ); Q gap> Qq := PathAlgebra( Q, q ); Q * q gap> H := Algebroid( Qq, [ Qq.g * Qq.g - Qq.1, > Qq.x * Qq.x, > Qq.x * Qq.g + Qq.g * Qq.x, > ] ); Algebra( Q, FreeCategory( RightQuiver( "q(1)[x:1->1,g:1->1]" ) ) ) / relations gap> H2 := H^2; Algebra( Q, FreeCategory( RightQuiver( "qxq(1x1)[1xx:1x1->1x1,1xg:1x1->1x1,xx1:1x1->1x1,gx1:1x1->1x1]" ) ) ) / relations gap> counit_rec := rec( x := 0, > g := 1 > );; gap> comult_rec := rec( g := PreCompose(H2.gx1, H2.1xg), > x := H2.1xx + PreCompose(H2.xx1, H2.1xg) > );; gap> AddBialgebroidStructure( H, counit_rec, comult_rec ); Bialgebra( Q, FreeCategory( RightQuiver( "q(1)[x:1->1,g:1->1]" ) ) ) / relations gap> counit := Counit( H ); Functor from Bialgebra( Q, FreeCategory( RightQuiver( "q(1)[x:1->1,g:1->1]" ) ) ) / relations -> Algebra( Q, FreeCategory( RightQuiver( "*(1)[]" ) ) ) gap> comult := Comultiplication( H ); Functor from Bialgebra( Q, FreeCategory( RightQuiver( "q(1)[x:1->1,g:1->1]" ) ) ) / relations -> Algebra( Q, FreeCategory( RightQuiver( "qxq(1x1)[1xx:1x1->1x1,1xg:1x1->1x1,xx1:1x1->1x1,gx1:1x1->1x1]" ) ) ) / relations gap> antipode_rec := rec( g:= H.g, > x := - PreCompose( H.x, H.g ) > );; gap> AddAntipode(H, antipode_rec ); gap> antipode := Antipode( H ); Contravariant functor from HopfAlgebra( Q, FreeCategory( RightQuiver( "q(1)[x:1->1,g:1->1]" ) ) ) / relations -> HopfAlgebra( Q, FreeCategory( RightQuiver( "q(1)[x:1->1,g:1->1]" ) ) ) / relations gap> IsCommutative(H); false gap> ApplyFunctor(antipode, H.g); (1)-[{ 1*(g) }]->(1) gap> ApplyFunctor(antipode, H.x); (1)-[{ -1*(x*g) }]->(1) gap> IsEqualForMorphisms( > ApplyFunctor( antipode, H.x ), > -PreCompose( H.x, H.g ) ); true
Verify that the antipode of \(H\) has order 4.
gap> IsEqualForMorphisms( ApplyFunctor( antipode, H.g ), H.g ); true gap> IsEqualForMorphisms( ApplyFunctor( antipode, H.x ), H.x ); false gap> IsEqualForMorphisms( ApplyFunctor( > PreCompose( [ antipode, antipode ] ), H.x ), H.x ); false gap> IsEqualForMorphisms( ApplyFunctor( > PreCompose( [ antipode, antipode, antipode ] ), H.x ), H.x ); false gap> IsEqualForMorphisms( ApplyFunctor( > PreCompose( [ antipode, antipode, antipode, antipode ] ), H.x ), H.x ); true gap> IsCocommutative(H); false
We perform simple computations with the Homomorphism structure of the Beilison quiver of the projective space of dimension 3.
gap> q := RightQuiver( "q(4)[\ > x0:1->2,x1:1->2,x2:1->2,x3:1->2,\ > y0:2->3,y1:2->3,y2:2->3,y3:2->3,\ > z0:3->4,z1:3->4,z2:3->4,z3:3->4]" );; gap> F := FreeCategory( q );; gap> beilinson_P3 := F / > [ [ PreCompose( F.x0, F.y1 ), PreCompose( F.x1, F.y0 ) ], > [ PreCompose( F.x0, F.y2 ), PreCompose( F.x2, F.y0 ) ], > [ PreCompose( F.x0, F.y3 ), PreCompose( F.x3, F.y0 ) ], > [ PreCompose( F.x1, F.y2 ), PreCompose( F.x2, F.y1 ) ], > [ PreCompose( F.x1, F.y3 ), PreCompose( F.x3, F.y1 ) ], > [ PreCompose( F.x2, F.y3 ), PreCompose( F.x3, F.y2 ) ], > [ PreCompose( F.y0, F.z1 ), PreCompose( F.y1, F.z0 ) ], > [ PreCompose( F.y0, F.z2 ), PreCompose( F.y2, F.z0 ) ], > [ PreCompose( F.y0, F.z3 ), PreCompose( F.y3, F.z0 ) ], > [ PreCompose( F.y1, F.z2 ), PreCompose( F.y2, F.z1 ) ], > [ PreCompose( F.y1, F.z3 ), PreCompose( F.y3, F.z1 ) ], > [ PreCompose( F.y2, F.z3 ), PreCompose( F.y3, F.z2 ) ] ];; gap> Q := HomalgFieldOfRationals( );; gap> Beilinson_P3 := Q[beilinson_P3];; gap> objs := SetOfObjects( Beilinson_P3 );; gap> Length( objs ) = 4; true gap> maps := SetOfGeneratingMorphisms( Beilinson_P3.1, Beilinson_P3.2 );; gap> maps = SetOfGeneratingMorphisms( Beilinson_P3, 1, 2 ); true gap> Length( maps ) = 4; true gap> ForAll( [ 0 .. 3 ], k -> > RankOfObject( HomomorphismStructureOnObjects( objs[1], objs[1 + k] ) ) > = Binomial( 4 + k - 1, k ) > ); true gap> alpha := Beilinson_P3.x0;; gap> beta := ( Beilinson_P3.z1 + Beilinson_P3.z2 );; gap> gamma := PreCompose( [ alpha, (Beilinson_P3.y1 - Beilinson_P3.y3), beta ] );; gap> sol := SolveLinearSystemInAbCategory( > [ [ alpha ] ], [ [ beta ] ], [ gamma ] )[1];; gap> IsCongruentForMorphisms( PreCompose( [ alpha, sol, beta ] ), gamma ); true
We create an algebroid over \(\mathbb{Q}\) and its descent to \(\mathbb{Z}\) defined by basis paths.
gap> q := RightQuiver( "q(4)[a:1->2,b:1->3,c:3->4,d:2->4]" );; gap> Q := HomalgFieldOfRationals( );; gap> Qq := PathAlgebra( Q, q );; gap> Qq := Qq / [ Qq.a * Qq.d - Qq.b * Qq.c];; gap> A_Q := Algebroid( Qq );; gap> A_Z := DescentToZDefinedByBasisPaths( A_Q );;
We demonstrate the practical difference between the two algebroids by trying to solve one and the same linear system.
gap> alpha := 2 * A_Q.a;; gap> id := IdentityMorphism( A_Q.4 );; gap> gamma := PreCompose( A_Q.b, A_Q.c );; gap> sol := SolveLinearSystemInAbCategory( [ [ alpha ] ], [ [ id ] ], [ gamma ] ); [ (2)-[{ 1/2*(d) }]->(4) ]
Thus, a solution exists over the rationals.
gap> alpha := 2 * A_Z.a;; gap> id := IdentityMorphism( A_Z.4 );; gap> gamma := PreCompose( A_Z.b, A_Z.c );; gap> sol := MereExistenceOfSolutionOfLinearSystemInAbCategory( > [ [ alpha ] ], [ [ id ] ], [ gamma ] ); false
No solution exists over the integers.
gap> q := LeftQuiver( "ql(u,v,w)[a:u->v,b:v->u,c:v->u,d:v->w]" );; gap> Q := HomalgFieldOfRationals( );; gap> B := Algebroid( Q, q );; gap> B0 := B^0;; gap> lu := LeftUnitorAsFunctor(B);; gap> ru := RightUnitorAsFunctor(B);; gap> lui := LeftUnitorInverseAsFunctor(B);; gap> rui := RightUnitorInverseAsFunctor(B);; gap> AsCapCategory(Target(lu)) = B; true gap> AsCapCategory(Target(ru)) = B; true gap> AsCapCategory(Source(lu)) = TensorProductOnObjects( B0, B ); true gap> AsCapCategory(Source(ru)) = TensorProductOnObjects( B, B0 ); true gap> AsCapCategory(Source(lui)) = B; true gap> AsCapCategory(Source(rui)) = B; true gap> AsCapCategory(Target(lui)) = TensorProductOnObjects( B0, B ); true gap> AsCapCategory(Target(rui)) = TensorProductOnObjects( B, B0 ); true gap> l := Concatenation( [ > List( SetOfObjects(B), > o -> ApplyFunctor( PreCompose(lui, lu), o) = o and > ApplyFunctor( PreCompose(rui, ru), o) = o ), > List( SetOfGeneratingMorphisms(B), > m -> ApplyFunctor( PreCompose(lui, lu), m) = m and > ApplyFunctor( PreCompose(rui, ru), m) = m ), > List( SetOfObjects(B^0 * B), > o -> ApplyFunctor( PreCompose(lu, lui), o) = o ), > List( SetOfGeneratingMorphisms(B^0 * B), > m -> ApplyFunctor( PreCompose(lu, lui), m) = m) > ] );; gap> and_function := function(x,y) return x and y; end;; gap> Iterated( l, and_function ); true gap> # Consider B as an object in the category of algebroids over Q > BB := CategoryOfAlgebroidsObject(B);; gap> BB2 := TensorProductOnObjects(BB, BB);; gap> BB0 := TensorUnit( CapCategory( BB ) );; gap> ru_as_functor := RightUnitor( BB );; gap> lu_as_functor := LeftUnitor( BB );; gap> rui_as_functor := RightUnitorInverse( BB );; gap> lui_as_functor := LeftUnitorInverse( BB );; gap> IsCongruentForMorphisms( PreCompose( ru_as_functor, rui_as_functor ), > IdentityMorphism( TensorProductOnObjects( BB, BB0 ) ) ); true gap> IsCongruentForMorphisms( PreCompose( lu_as_functor, lui_as_functor ), > IdentityMorphism( TensorProductOnObjects( BB0, BB ) ) ); true gap> IsCongruentForMorphisms( PreCompose( rui_as_functor, ru_as_functor ), > IdentityMorphism( BB ) ); true gap> IsCongruentForMorphisms( PreCompose( lui_as_functor, lu_as_functor ), > IdentityMorphism( BB ) ); true
gap> q := RightQuiver( "q(1)[t:1->1]" ); q(1)[t:1->1] gap> Q := HomalgFieldOfRationals( ); Q gap> Qq := PathAlgebra( Q, q ); Q * q gap> B := Algebroid( Qq ); Algebra( Q, FreeCategory( RightQuiver( "q(1)[t:1->1]" ) ) ) gap> ass_ltr := AssociatorLeftToRightWithGivenTensorProductsAsFunctor( > (B*B)*B, B, B, B, B*(B*B) );; gap> ass_rtl := AssociatorRightToLeftWithGivenTensorProductsAsFunctor( > B*(B*B), B, B, B, (B*B)*B );; gap> Source(ass_ltr) = Target(ass_rtl); true gap> Target(ass_ltr) = Source(ass_rtl); true gap> # Consider B as an object in the category of algebroids over Q > BB := CategoryOfAlgebroidsObject(B);; gap> BB2 := TensorProductOnObjects(BB, BB);; gap> BB0 := TensorUnit( CapCategory( BB ) );; gap> ass_ltr_as_morphism := AssociatorLeftToRight(BB,BB,BB);; gap> ass_rtl_as_morphism := AssociatorRightToLeft(BB,BB,BB);; gap> IsCongruentForMorphisms( > PreCompose( ass_ltr_as_morphism, ass_rtl_as_morphism ), > IdentityMorphism( Source( ass_ltr_as_morphism ) ) ); true gap> IsCongruentForMorphisms( > PreCompose( ass_rtl_as_morphism, ass_ltr_as_morphism ), > IdentityMorphism( Source( ass_rtl_as_morphism ) ) ); true
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