Monadic Decompositions and Classical Lie Theory

0.19 MB·English

by  Alessandro Ardizzoni

MONADIC DECOMPOSITIONS AND CLASSICAL LIE THEORY
ALESSANDROARDIZZONI,JOSE´ GO´MEZ-TORRECILLAS,ANDCLAUDIAMENINI
3
1 Abstract. We show that the functor from bialgebras to vector spaces sending a bialgebra to
0 itssubspaceofprimitiveshasmonadiclengthatmost2.
2

n
a
J
Introduction
8
1 Given a functor R : A→B with left adjoint L :B →A we get, following [AHW, MS], and
0 0 0 0

under suitable hypotheses, a sequence of adjoint pairs of functors
]
T
C Aoo IdA Aoo IdA Aoo IdA ...
OO OO OO
.
h
L0 R0 L1 R1 L2 R2
t
ma B0(cid:15)(cid:15) oo U0,1 B1(cid:15)(cid:15) oo U1,2 B2(cid:15)(cid:15) oo U2,3 ...
[ wherefori≥0,Bi+1istheEilenberg-Moorecategoryofthemonad(Li,Ri),Ri+1isthecomparison

2 functor,andUi,i+1isthecorrespondingforgetfulfunctor. Itisnaturaltoinquirewetherthisprocess
stops, as was done in [AHW, MS]. To be more specific, the monadic length of R is the first N
v 0
1 such that UN,N+1 is an isomorphism of categories. In many basic examples, the functor R0 is
8 monadic and, therefore, it has monadic length at most 1. In this note, we show that the functor
8 P from bialgebras to vector spaces sending a bialgebra to its subspace of primitives has monadic
2
length 2 (Theorem 2.4).
.
3 Section 1 contains some remarks on the monadic decompositions of functors studied in [AHW,
0
MS] and their relationship with idempotent monads ([AT]). The basic case of the adjoint pair
2
encoded by a bimodule over unital rings is described in Remark 1.14, with an eye on descent
1
theory for modules. We also study the existence of comonadic decompositions under separability
:
v
conditions (Proposition 1.16).
i
X Section 2 contains the aforementioned monadic decomposition of monadic length at most 2 of
r the functor P from bialgebras to vector spaces.
a
1. Monadic decompositions
Consider categories A and B. Let (L:B →A,R:A→B) be an adjunction with unit η and
counit ǫ, and consider the monad (RL,RǫL,η) generated on B. By B we denote its Eilenberg-
1
Moore category of algebras. Hence we can consider the so-called comparison functor of the ad-
junction (L,R) i.e. the functor
K :A→B , KX :=(RX,RǫX), Kf :=Rf.
1
Recall that the functor R : A → B is called monadic (tripleable in Beck’s terminology [Be,
Definition3’,page8])wheneverthecomparisonfunctorK :A→B isanequivalenceofcategories.
1
2010 Mathematics Subject Classification. Primary18C15;Secondary 16S30.
Key words and phrases. Monads,Liealgebras.
This paper was written while the first and the last authors were members of GNSAGA. The second author
thanks the members of the Dipartimento di Matematica at the University of Ferrara for their warm hospitality
duringhisvisitinFebruary2010, when theworkonthispaper was initiated. Researchpartiallysupported bythe
grant MTM2010-20940-C02-01 from the MICINN and FEDER. The authors are grateful to the referee for several
usefulcomments,andtoBachukiMesablishviliforpointingtousthebibliographyonmonadicdecompositions.
1
2 ALESSANDROARDIZZONI,JOSE´ GO´MEZ-TORRECILLAS,ANDCLAUDIAMENINI
1.1. Idempotentmonadsandmonadicdecompositions. Thenotionofanidempotentmonad
is, as we will see below, tightly connected with the monadic length of a functor.
Definition 1.1. [AT, page 231] A monad (Q,m,u) with multiplication m and unit u is called
idempotent whenever m is an isomorphism. An adjunction (L,R) is called idempotent whenever
the associated monad is idempotent.
There are several basic characterizations of idempotent adjunctions (see [AT, MS]). In partic-
ular, idempotency of an adjunction means equivalently that any of the four factors appearing in
the two triangular identities of the adjoint pair (L,R) is an isomorphism ([MS, Proposition 2.8]).
In the following we will denote by U :B →B the forgetful functor and by F :B →B the free
1 1
functor associated to an adjunction (L,R).
Remark 1.2. Note that the adjunctions (F,U) and (L,R) have the same associated monad so
that (L,R) is idempotent if and only if (F,U) is if and only if, by [MS, Proposition 2.8], one has
ηU is an isomorphism.
Proposition 1.3. For and adjunction (L:B →A,R:A→B) with unit η and counit ǫ, the fol-
lowing assertions are equivalent.
(a) (L,R) is idempotent,
(b) the structure map of every object in B is an isomorphism,
1
(c) LU is a left adjoint of the comparison functor K : A → B of (L,R), and ηU = Uη , where
1 1
η is the unit of the new adjunction (being the new counit still ǫ).
1
Moreover, if one of these conditions holds, then LU is full and faithful.
Proof. (a) ⇒ (b). By [MS, Proposition 2.8], we have that ηRL = RLη. Now, for any algebra
(X,µ) ∈ B , we have that µ◦ηX = 1 . Moreover, by naturality of η, we know that ηX ◦µ =
1 X
RLµ◦ηRLX =RLµ◦RLηX =1 . Therefore, µ is an isomorphism.
RLX
(b) ⇒ (c). If (X,µ) is an algebra over RL, then µ : RLX → X is an homomorphism of RL–
algebras. Now, if µ is an isomorphism, then, necessarily, µ=(ηX)−1. We get easily that ηX is a
morphism in B . Therefore, η will serve as the unit for an adjunction (LU,K) (being the counit
1
ǫ).
(c) ⇒ (a). Given an object Y of A, consider its free RL–algebra(RY,RǫY). Since, by assump-
tion,ηRY isahomomorphismofRL–algebras,wegettheidentityRǫLRY◦RLηRY =ηRY◦RǫY,
which implies, by the triangular identities for the adjunction (L,R), that ηRY ◦RǫY = 1 ,
RLRY
whence RǫY is an isomorphism. With Y =LX for any object X of B, one obtains that RǫLX is
an isomorphism. Hence, (L,R) is idempotent.
Let us prove the last part of the statement. By Remark 1.2, condition (a) is equivalent to ηU
isomorphism. From ηU =Uη and the fact that U reflects isomorphisms we deduce that η is an
1 1
isomorphism so that LU is full and faithful. (cid:3)
Remark 1.4. By [MS, Proposition 2.8], if L is full and faithful, then the adjunction (L,R) is
idempotent. On the other hand, since the units of the adjunctions (L,R)and (F,U) are equal,we
getfromProposition1.3that L is full andfaithful if and only if U is anisomorphismofcategories
if and only if U is an equivalence of categories.
Before defining monadic decompositions and monadic length, we derive the following conse-
quence of Proposition 1.3, which is interpreted in the classical setting of epimorphisms of rings in
Example 1.6.
Proposition 1.5. Let (L,R) be an adjunction. The following are equivalent.
(1) R is full and faithful.
(2) (L,R) is idempotent and R is monadic.
Proof. Let η be the unit and ǫ be the counit of the adjunction (L,R).
(1) ⇒ (2). By assumption ǫ is an isomorphism so that RǫL is an isomorphism i.e. (L,R) is
idempotent. By Proposition1.3, weknowthat Λ:=LU is a leftadjointofthe comparisonfunctor
MONADIC DECOMPOSITIONS AND CLASSICAL LIE THEORY 3
K : A → B of (L,R). Moreover Λ is full and faithful and ǫA = ǫ A where ǫ is the counit of
1 1 1
the adjunction (Λ,K). Thus ǫ is an isomorphism i.e. K is full and faithful too. Hence K is an
1
equivalence.
(2)⇒(1). Since(L,R)isidempotent,byProposition1.3,wegetǫA=ǫ A.SinceRismonadic,
1
the comparison functor is an equivalence and hence ǫ A is an isomorphism. Hence ǫA is an
1
isomorphism so that R is full and faithful. (cid:3)
Example 1.6. Let us consider a morphism of rings π : B → A, and its canonical associated
adjunction
L:Mod-B →Mod-A, R:Mod-A→Mod-B.
By [St, Proposition1.2, page 226], π is an epimorphism if and only if the counit of the adjunction
is an isomorphism. This is equivalent to say that R is full and faithful. Thus, by Proposition 1.5,
π is an epimorphism if and only if (L,R) is idempotent and R is monadic. Note that L needs not
to be full and faithful. Thus, when π is an epimorphism, since R is monadic, it has a monadic
decomposition of monadic length 1 in the sense of Definition 1.7 but a monadic decomposition of
(essential) length 0 in the sense of [AHW, Definition 2.1].
Definition 1.7. (See [AHW, Definition 2.1] and [MS, Definitions 2.10 and 2.14]) Fix a N ∈ N.
We say that a functor R has a monadic decomposition of monadic length N whenever there exists
a sequence (R ) of functors R such that
n n≤N n
1) R =R;
0
2) for 0≤n≤N, the functor R has a left adjoint functor L ;
n n
3) for 0 ≤ n ≤ N −1, the functor R is the comparison functor induced by the adjunction
n+1
(L ,R ) with respect to its associated monad;
n n
4) L is full and faithful while L is not full and faithful for 0≤n≤N −1.
N n
Compare with the construction performed in [Ma, 1.5.5, page 49].
Note that for functor R : A → B having a monadic decomposition of monadic length N, we
have a diagram
(1) Aoo IdA Aoo IdA Aoo IdA ··· ···oo IdA A
OO OO OO OO
L0 R0 L1 R1 L2 R2 LN RN
(cid:15)(cid:15) U0,1 (cid:15)(cid:15) U1,2 (cid:15)(cid:15) U2,3 UN−1,N (cid:15)(cid:15)
B oo B oo B oo ··· ···oo B
0 1 2 N
where B =B and, for 1≤n≤N,
0
• B is the category of (R L )-modules B ;
n n−1 n−1 Rn−1Ln−1 n−1
• U :B →B is the forgetful functor U.
n−1,n n n−1 Rn−1Ln−1
We will denote by η : Id → R L and ǫ : L R → Id the unit and counit of the
n Bn n n n n n A
adjunction(L ,R ) respectivelyfor 0≤n≤N. Note thatone canintroduce the forgetfulfunctor
n n
U :B →B for all m≤n with 0≤m,n≤N.
m,n n m
Remarks 1.8. 1) Assume that R fits into a diagramsuch as (1). If R is monadic i.e. R is a
N−1 N
categoryequivalence,thenobviouslyL isfullandfaithfulsothatR hasamonadicdecomposition
N 0
of monadic length at most N. Nevertheless if R has monadic length N, then R needs not to be
0 N
an equivalence.
2) The notion of comonadic decomposition of comonadic length N can be easily introduced. In
this case we will use the notations (Ln,Rn) with superscripts and require that RN is full and
n∈N
faithful.
Proposition 1.9. Let (L:B →A,R:A→B) be an idempotent adjunction. Then R : A → B
has a monadic decomposition of monadic length at most 1.
Proof. By Proposition 1.3, L =L U is full and faithful. (cid:3)
1 0 0,1
4 ALESSANDROARDIZZONI,JOSE´ GO´MEZ-TORRECILLAS,ANDCLAUDIAMENINI
Remark 1.10. It follows from Remark 1.4 that condition 4) in Definition 1.7 is equivalent to the
requirement that the forgetful functor U : B → B is an isomorphism of categories.
N,N+1 N+1 N
Thus, if R:A→B has a monadic decomposition of monadic length N ∈N, then we can consider
the comparison functor R : A → B of (L ,R ). Moreover, still by Remark 1.4, L
N+1 N+1 N N N
full and faithful implies that the adjunction (L ,R ) is idempotent. Hence, by Proposition 1.3,
N N
L := L U is a left adjoint of R (and L is full and faithful too). Note that the
N+1 N N,N+1 N+1 N+1
fact that R is a right adjoint is assumed from the very beginning in [AHW, Definition 2.1].
N+1
By Proposition 1.3 again, we deduce that η U = U η and ǫ A = ǫ A where
N N,N+1 N,N+1 N+1 N N+1
η is the unit and ǫ is the counit of the adjunction (L ,R ) for all n ≤ N +1. Iterating this
n n n n
process we get that for all M ≥ N, the tower in (1) can be extended with adjoints (L ,R )
M M
where L is full andfaithful so thatU :B →B is a categoryisomorphism. Moreover
M M,M+1 M+1 M
η U =U η and ǫ A=ǫ A. By the foregoing we have that
M M,M+1 M,M+1 M+1 M M+1
R=R =U ◦U ◦···◦U ◦R
0 0,1 1,2 N−1,N N
where U ,U ,··· ,U are monadic functors but not categoryisomorphisms. Moreoverthis
0,1 1,2 N−1,N
is a maximal decomposition of this form. This is essentially [AHW, Remarks 2.2].
Remark 1.11. If R:A→B has a monadic decomposition of length N, then, since L :B →A
N N
isfullandfaithful, the dualofProposition1.5givesthatL isacomonadicfunctor and(L ,R )
N N N
is coidempotent. Thus, the comparison functor C :B →AN, where AN denotes the category of
N
L R –coalgebras,is an equivalence of categories.
N N
1.2. Essentially surjective. The following result determines the objects which are images of
rightadjointfunctorsundersuitableassumptions. Thiscanberegardedasasortofdescenttheory
for these functors.
Notation 1.12. Let R : A → B. We will denote by ImR the full subcategory of B consisting of
those objects B ∈B such that there is an object A∈A and an isomorphism B ∼=RA in B.
Recall that a functor R:A→B is essentially surjective if ImR=B.
Proposition 1.13. Suppose that R : A → B has a monadic decomposition of monadic length
N ∈N. Let n∈{0,...,N}. Then
1) ImR⊆ImU .
0,n
2) ImR=ImU whenever R is essentially surjective.
0,n n
3) ImR=ImU .
0,N
Proof. It follows from the equalities U R =R =R. (cid:3)
0,n n 0
Remark 1.14. Proposition 1.13 can be considered as a “general dual descent theory” result. In
fact the theorem states that the objects of B = B which are isomorphic to objects of the form
0
RA, for some A∈A, are exactly those of the form U B where B ∈B . In particular, when
0,N N N N
N = 1, i.e. L is full and faithful, we have that the objects of B which are isomorphic to objects
1
of the form RA, for some A ∈ A, are exactly those of the form U B where B ∈ B . This is
0,1 1 1 1
exactly the dual form of classicaldescent theory for (bi)modules. In fact, let S, T be rings and let
M be a bimodule. Consider the following adjunction
S T
L:M →M , R:M →M
S T T S
LX =X⊗ M, RY =Hom (M,Y),
S T
betweenthe categoryM ofrightS-modulesandthe categoryM ofrightT-modules. The cate-
S T
goryM has (co)equalizers. By (dual) Beck’sTheorem[Be, Proofof Theorem1], the comparison
T
functors R and L1 have a left adjoint L and a right adjoint R1 respectively.
1 1
Assume that M is flat as a left S-module. Then L = L0 is exact so that, the dual of Beck’s
Theorem ensures that R1 is full and faithful. Therefore, L admits a comonadic decomposition of
comonadiclengthatmost1. ThuswehavethattheobjectsofM whichareisomorphictoobjects
T
of the form LX, for some X ∈ M , are exactly those of the form U0,1X1 where X1 is an object
S
of the category of LR–coalgebras (M )1. Hence the category (M )1 solves the descent problem
T T
for modules. When M is finitely generatedand projective,we have anisomorphismof comonads
T
MONADIC DECOMPOSITIONS AND CLASSICAL LIE THEORY 5
LR∼=−⊗T M∗⊗SM where M∗⊗SM is the comatrixcoring associatedto SMT (see [EGT], and
[GT,GTV]formoregeneralbimodules). CoalgebrasoverLRarepreciselythecomodulesoverthe
T–coring M∗⊗ M.
S
Assume that M is projective as a right T-module. Then R = R is exact so that, Beck’s
0
Theorem ensures that L is full and faithful, and R has a comonadic decomposition of length at
1
most 1.
1.3. Separability. Let (Q,m,u) be a monad on a category B, with multiplication m and unit
u. A right module functor on (Q,m,u) is a pair (W,µ) where W : B → A is a functor and
µ:WQ→W is a natural transformation such that
µ◦µQ=µ◦Wm and µ◦Wu=Id .
Q
Amorphismf :(W,µ)→(W′,µ′)ofrightmodulefunctorsisanaturaltransformationf :W →W′
such that µ′◦fQ=f ◦µ.
It is clear that (WQ,Wm) is a right module functor on (Q,m,u) and that µ : (WQ,Wm) →
(W,µ) is morphism of right module functors. We will say that (W,µ) is relatively projective
whenever µ : (WQ,Wm) → (W,µ) splits as a morphism of right module functors. Explicitly
this means that there is a morphism γ :(W,µ) →(WQ,Wm) of right module functors such that
µ◦γ = Id i.e. that there is a natural transformation γ : W → WQ such that µ◦γ = Id
(W,µ) W
and Wm◦γQ=γ◦µ.
Let (L:B →A,R:A→B) be an adjunction with unit η and counit ǫ. Then (L,ǫL) is a right
module functor on (RL,RǫL,η). In fact ǫL◦ǫLRL=ǫL◦LRǫL and ǫL◦Lη =Id .
RL
The notion of a separable functor was introduced in [NVV]. This concept is motivated by
various examples, being perhaps the most fundamental the following. Given a homomorphism of
ringsR→S, then the restrictionof scalarsfunctor M →M is separablein the sense of [NVV]
S R
if and only if the extension R→S is separable (i.e., the multiplication map S⊗ S →S splits as
R
anS–bimodule epimorphism). In general,if (L,R) is anadjunction, then R is a separablefunctor
if and only if its counit is a split natural epimorphism ([Ra, Theorem 1.2]).
Lemma 1.15. Let (L:B →A,R:A→B) be an adjunction. If R is separable, then (L,ǫL) is
relatively projective as a right module functor on (RL,RǫL,η)
Proof. By assumption, there is a natural transformation σ : Id → LR such that ǫ◦σ = Id .
A IdA
Setγ :=σL. Thenγ is anaturaltransformationsuchthatǫL◦γ =Id andLRǫL◦γRL=γ◦ǫL.
L
Then ǫL:(LRL,LRǫL)→(L,ǫL) splits as a morphism of right module functors. (cid:3)
In the following result, part 3) may be compared, in its dual version, with [Me, Proposition
3.16] and the results quoted therein.
Proposition 1.16. Let (L:B →A,R:A→B) be an adjunction with unit η and counit ǫ.
1) If R is a separable functor then the comparison functor R :A→B is full and faithful.
1 1
2) Supposethatthecomparison functorR :A→B has aleft adjoint L . If(L,ǫL)isrelatively
1 1 1
projective as a right module functor on (RL,RǫL,η), then L is full and faithful.
1
3) Suppose that the comparison functor R : A→ B has a left adjoint L . If R is a separable
1 1 1
functor, then R is monadic.
Proof. 1) By assumption there is a natural transformation σ :Id →LR such that ǫ◦σ =Id .
A IdA
Let f,g :X →Y morphisms in A such that R f =R g. Since R=UR , we get
1 1 1
σY ◦f =LRf ◦σX =LRg◦σX =σY ◦g
Now, σY is a monomorphism, whence f = g. Thus, R is faithful. To check that R is full,
1 1
consider a morphism h : R X → R Y in B , and put h′ = ǫY ◦Lh◦σX. Since h is a morphism
1 1 1
of algebras, we get
Rh′ =RǫY ◦RLh◦RσX h∈=B1 h◦RǫX◦RσX =h.
6 ALESSANDROARDIZZONI,JOSE´ GO´MEZ-TORRECILLAS,ANDCLAUDIAMENINI
2) By [Be, Proof of Theorem 1], since L exists, there exists a morphism π such that
1
Lµ
(2) LRLB ⇉ LB −π→L (B,µ)
1
ǫLB
is a coequalizer for all (B,µ)∈B . Moreover L is full and faithful whenever
1 1
RLµ
RLRLB ⇉ RLB −R→π RL (B,µ)
1
RǫLB
is a coequalizer too. By assumption there is a natural transformation γ : L → LRL such that
ǫL◦γ =Id and LRǫL◦γRL=γ◦ǫL. Clearly, we have ǫLB◦γB =Id . Moreover
L LB
natγ
(Lµ◦γB)◦Lµ = Lµ◦γB◦Lµ = Lµ◦LRLµ◦γRLB
= Lµ◦(LRǫLB◦γRLB)
= Lµ◦(γB◦ǫLB)
= (Lµ◦γB)◦ǫLB
so that there is a unique morphism p:L (B,µ)→LB such that p◦π =Lµ◦γB. We have
1
π◦p◦π =π◦Lµ◦γB =π◦ǫLB◦γB =π
so that, since π is an epimorphism, we get π ◦p = Id . We have so proved that (2) is a
L1(B,µ)
contractible coequalizer. Thus it is preserved by any functor, in particular by R. Thus L is full
1
and faithful too.
3) It follows from 1), 2) and Lemma 1.15 that both L and R are full and faithful. (cid:3)
1 1
2. Examples
Let us fix a field k. Vector spaces and algebras are meant to be over k. From any vector space
V we can construct its tensor algebra TV = k⊕V ⊕V⊗2⊕···. In fact, this is the object part
of a functor T : Vectk → Algk from the category Vectk of vector spaces to the category Algk of
(associative and unital) algebras. By Ω:Algk →Vectk we denote the forgetful functor.
2.1. Vector spaces and algebras.
Example 2.1. If A is an algebra, and V a vector space, then the universal property of TV gives
a bijection
(3) Algk(TV,A)∼=Vectk(V,ΩA),
which is natural in both variables. In other words, the functor T :Vectk →Algk is left adjoint to
the forgetfulfunctor Ω:Algk →Vectk. Itisverywell-knownthatΩis amonadicfunctor (cf. [Bo,
Proposition 4.6.2]). Next, we check that T is a comonadic functor.
In fact, given V ∈Vectk, consider the canonical projection π =πV :ΩTV →V on degree one.
Letuscheckthatit isnaturalinV. Letf :V →V′ be a morphisminVectk. Forall z ∈V⊗n with
n6=1,
(πV′◦ΩTf)(z)=πV′(cid:0)f⊗n(z)(cid:1)=0=(f ◦πV)(z).
For v ∈V, we have
(πV′◦ΩTf)(v)=πV′(f(v))=f(v)=(f ◦πV)(v).
so that πV′◦ΩTf = f ◦πV and πV is natural in V. Moreover, we have πV ◦i = Id , where
V V
iV :V →ΩTV is the canonical inclusion map for every V ∈Vectk. Since iV gives the unit of the
adjunctionatV,wecanapplyRafael’sTheorem[Ra,Theorem1.2],toobtainthatT isaseparable
functor. By the dual version of Proposition 1.16, in order to prove that T is comonadic it suffices
to check that T1 has a right adjoint. This follows by Beck’s Theorem [Be, Proofof Theorem 1] as
Vectk has equalizers.
MONADIC DECOMPOSITIONS AND CLASSICAL LIE THEORY 7
2.2. Vector spaces and bialgebras.
Example 2.2. Let Bialgk be the category of bialgebras and Ω : Bialgk → Vectkbe the forgetful
functor. ByP :Bialgk →Vectk wedenote thefunctorthatsendsabialgebraAtoitsspacePAof
primitive elements. Obviously, P is a subfunctor of Ω, let j :P →Ω denote the inclusion natural
transformation. We know that the tensor algebra TV of a vector space V is already a bialgebra.
Therefore, the bijection (3) gives, by restriction, a bijection
Bialgk(TV,A)∼=Vectk(V,PA)
which, of course, is natural.
Inthisway,weseethatT isleftadjointtoP. WewillprovethatP hasamonadicdecomposition
of monadic length at most 2. First we need to prove a technical result.
Lemma 2.3. Let (L:B →A,R:A→B) be an adjunction and let (B,µ)∈B . Let ζ :B →Z be
1
a morphism in B. Then
ζ◦Lµ=ζ◦ǫLB ⇔Rζ◦ηB◦µ=Rζ.
Proof. Consider the canonical isomorphism Φ(X,Y) : Hom (LX,Y) → Hom (X,RY) defined
A B
by Φ(X,Y)f =Rf ◦ηX. Then
ζ◦Lµ = ζ◦ǫLB ⇔
Φ(RLB,Z)[ζ◦Lµ] = Φ(RLB,Z)[ζ◦ǫLB]⇔
R[ζ◦Lµ]◦ηRLB = R[ζ◦ǫLB]◦ηRLB ⇔
Rζ◦RLµ◦ηRLB = Rζ◦RǫLB◦ηRLB ⇔
Rζ◦ηB◦µ = Rζ.
(cid:3)
Theorem 2.4. The functor P has a monadic decomposition of monadic length at most 2. Keep
the notations of Definition 1.7 (so, in particular, B0 =Vectk, R0 =P, and L0 =T).
1) The functor L is given, for all (V ,µ )∈B , by
1 0 0 1
L V
L (V ,µ )= 0 0 .
1 0 0 (Im(Id −η V ◦µ ))
R0L0V0 0 0 0
2) The adjunction (L ,R ) is idempotent.
1 1
4) For all V :=((V ,µ ),µ )∈B , we have the following cases.
2 0 0 1 2
– chark = 0. Then, for all x,y ∈ V we have that xy −yx ∈ R L V . Define a map
0 0 0 0
[−,−]:V ⊗V →V bysetting[x,y]:=µ (xy−yx).Then(V ,[−,−])isanordinary
0 0 0 0 0
Lie algebra and L V is the universal enveloping algebra
2 2
TV
UV := 0 .
0 (xy−yx−[x,y]|x,y ∈V )
0
– chark=p, a prime. Then, for all x,y ∈V wehave that xy−yx,xp ∈R L V . Define
0 0 0 0
two maps [−,−] : V ⊗V → V and −[p] : V → V by setting [x,y] := µ (xy−yx)
0 0 0 0 0 0
and x[p] :=µ0(xp). Then (cid:0)V0,[−,−],−[p](cid:1) is a restricted Lie algebra and L2V2 is the
restricted enveloping algebra
TV
uV := 0 .
0 (cid:0)xy−yx−[x,y],xp−x[p] |x,y ∈V0(cid:1)
Proof. NotethatA=Bialgkhascoequalizers(seee.g. [Ag,page1478]). Thus,usingthenotations
of Definition 1.7, by Beck’s Theorem [Be, Proof of Theorem 1], the functors L and L exist. By
1 2
construction, for every V := (V ,µ :R L V →V ) ∈ B we have that L V is given by the
1 0 0 0 0 0 0 1 1 1
coequalizer in A of the diagram
L0µ0
L R L V ⇉ L V .
0 0 0 0 0 0
ǫ0L0V0
8 ALESSANDROARDIZZONI,JOSE´ GO´MEZ-TORRECILLAS,ANDCLAUDIAMENINI
We want to compute explicitly this coequalizer. To this aim, we set
L V
T V := 0 0,
1 1 (S)
whereS :=Im(Id −η V ◦µ ),andletuscheckitisabialgebra. Itisenoughtocheckthat
R0L0V0 0 0 0
∆ S ⊆ (S)⊗L V +L V ⊗(S),
L0V0 0 0 0 0
ε S = 0.
L0V0
Both equalities follow trivially since S ⊆R L V =PTV. Hence T V ∈A. Let us check that
0 0 0 1 1
L R L V L⇉0µ0 L V −π→T V
0 0 0 0 0 0 1 1
ǫ0L0V0
is a coequalizer in A, where π is the canonical projection. Let ζ :L V →Z be a morphism in A.
0 0
By Lemma 2.3
ζ◦L µ =ζ◦ǫ L V ⇔R ζ◦η V ◦µ =R ζ ⇔ζ vanishes on S.
0 0 0 0 0 0 0 0 0 0
Hence we can take L V :=T V .
1 1 1 1
We need to describe L V in a different wayfor every V :=(V ,µ )∈B . Note that R L V =
1 1 1 0 0 1 0 0 0
V ⊕EV whereEV denotesthesubspaceofΩL V spannedbyprimitiveelementsofhomogeneous
0 0 0 0 0
degree greater than one. Let x = η V :V →R L V and x : EV →R L V be the canonical
1 0 0 0 0 0 0 2 0 0 0 0
injections andset b:=µ ◦x :EV →V . Let c:V ⊗V →V ⊗V be the canonicalflip. Then b
0 2 0 0 0 0 0 0
is a bracket for the braided vector space (V ,c) in the sense of [Ar1, Definition 3.2]. We compute
0
(Id −η V ◦µ )◦x =x −η V ◦µ ◦x =η V −η V ◦µ ◦η V =0
R0L0V0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0
so that
S =Im(Id −η V ◦µ )=Im[(Id −η V ◦µ )◦x ]=Im(x −η V ◦b)
R0L0V0 0 0 0 R0L0V0 0 0 0 2 2 0 0
and hence
L V L V L V
L V = 0 0 = 0 0 = 0 0 .
1 1 (S) (Im(x −η V ◦b)) (z−b(z)|z ∈EV )
2 0 0 0
Therefore L V =U(V ,c,b) in the sense of [Ar1, Definition 3.5].
1 1 0
Let now V := (V ,µ ) ∈B . Then V is of the form (V ,µ ). By construction, the unit of the
2 1 1 2 1 0 0
adjunction is the unique map η V :V →R L V such that
1 1 1 1 1 1
U η V =R π◦η V .
0,1 1 1 0 0 0
Consider the canonical map i :V →U(V ,c,b) i.e.
U 0 0
i =Ωπ◦jL V ◦η V =jL V ◦R π◦η V =jL V ◦U η V
U 0 0 0 0 1 1 0 0 0 1 1 0,1 1 1
so that i corestricts to U η V . Now
U 0,1 1 1
U µ ◦U η V =U (µ ◦η V )=Id
0,1 1 0,1 1 1 0,1 1 1 1 V0
so that U η V is injective. Therefore i is injective. This means that (V ,c,b) is a braided Lie
0,1 1 1 U 0
algebra in the sense of [Ar1, Definition 4.1]. Let S denote the class of braided vector spaces of
combinatorialrank at most one. Then (V ,c)∈S
0
(see [Ar2, Example 6.10], if char(k)=0, and [Ar3, Example 3.13], if char(k)6=0).
By [Ar1, Corollary 5.5], we have that U η V is an isomorphism. Since U reflects isomor-
0,1 1 1 0,1
phism, we getthat η V is an isomorphism. We haveso provedthat η U is anisomorphism. By
1 1 1 1,2
Remark 1.2, we have that the adjunction (L ,R ) is idempotent. By Proposition 1.9, the functor
1 1
R hasamonadicdecompositionofmonadiclengthatmost1sothatRhasmonadicdecomposition
1
of monadic length at most 2.
We have observed that (V ,c,b) is a braided Lie algebra in the sense of [Ar1, Definition 4.1].
0
The last part of the statement follows by [Ar1, Remark 6.4] in case chark=0 and by the same
argument as in [Ar3, Example 3.13] in case chark=p. (cid:3)
MONADIC DECOMPOSITIONS AND CLASSICAL LIE THEORY 9
Remark 2.5. In the setting of Theorem 2.4, R = P : A → B has a monadic decomposition of
monadic length at most 2. Thus, by Theorem 1.13, we have that
ImR=ImU .
0,2
Note, since (L ,R ) is idempotent, we can apply Proposition 1.3 to get that an object in ImU
1 1 0,2
is isomorphicto anobjectofthe formU (V ,µ )=U V forsome V ∈B suchthatη V is an
0,2 1 1 0,1 1 1 1 1 1
isomorphism.
Remark 2.6. Let (L,R)be the adjunction considered in 2.2. For a moment let L′ denote the left
adjoint L of Example 2.1. Let W be the forgetful functor from the category of bialgebras to the
categoryof algebras. Then W ◦L=L′. Hence, in view of [NVV, Lemma 1.1], from separabilityof
L′ we deduce separability of L. Since B has all equalizers, by the dual version of Beck’s Theorem
[Be, Proof of Theorem 1], we have that the comparison functor L1 : B → A1 has a right adjoint
R1. Thus, by the dual version of Proposition 1.16, we have that L is comonadic.
Now, as observed in [Ag, Theorem 2.3], in view of [Sw, page 134], the functor W has a right
adjoint, say Γ. Explicitly ΓA is the cofree bialgebra associated to A, for any algebra A. Now
(WL,RΓ) is an adjunction as composition of adjunctions. Since W ◦L = L′ and (L′,R′) is an
adjunction, we deduce that RΓ is functorially isomorphic to R′.
2.3. Pretorsion theories.
Example2.7. LetAbearingandletT beafullsubcategoryofMod-Aclosedundersubmodules,
quotients and direct sums i.e. T is an hereditary pretorsion class. Let t : Mod-A → T be the
associated left exact preradical ([St, Corollary 1.8 page 138]). Then R = t is a right adjoint of
the inclusion functor L = i : T → Mod-A. Note that RL = Id and η = Id so that L is full
T IdT
and faithful. Hence, R has a monadic decomposition of monadic length 0. By Remark 1.11, the
comparison functor C :T →(Mod-A)1 is a category equivalence.
As a particular example we consider the case when A = C∗ for some coalgebra C over a
field k and T is the class of rational right C∗-modules i.e. the image of the canonical functor
C-CoMod→Mod-C∗.
References
[Ag] A.L.Agore,CategoricalConstructionsforHopfAlgebras.Comm.Algebra,1532-4125,Vol.39(4),(2011),
1476-1481.
[Ar1] A.Ardizzoni,OnPrimitivelyGeneratedBraidedBialgebras,Algebr.Represent.Theory,Vol.15(4)(2012),
639-673.
[Ar2] A. Ardizzoni, A Milnor-Moore Type Theorem for Primitively Generated Braided Bialgebras, J. Algebra,
Vol.327(1)(2011), 337-365.
[Ar3] A.Ardizzoni,Universal Enveloping Algebras of PBW Type,Glasg.Math.J.,Vol.54(1)(2012), 9-26.
[AHW] J. Ada´mek, H. Herrlich, W. Tholen, Monadic decompositions. J. Pure Appl. Algebra 59 (1989), no. 2,
111-123.
[AT] H.Appelgate,M.Tierney,Categorieswithmodels.1969Sem.onTriplesandCategoricalHomologyTheory
(ETH,zu¨rich,1966/67) pp.156–244Springer,Berlin.
[Be] J.M.Beck,Triples,algebrasandcohomology,ReprintsinTheoryandApplicationsofCategories2(2003),
1–59.
[Bo] F.Borceux, Handbook of categorical algebra. 2. Categories and structures. Encyclopedia of Mathematics
anditsApplications,51.CambridgeUniversityPress,Cambridge,1994.
[EGT] L.ElKaoutitandJ.G´omez-Torrecillas,Comatrixcorings: Galoiscorings,descenttheory,andastructure
theorem for cosemisimple corings,Math.Z.244(4),(2003)887–906.
[GT] J.G´omez-Torrecillas,Comonads and Galois corings,Appl.Categ.Structures 14(5-6), (2006)579–598.
[GTV] J.G´omez-Torrecillasand J.Vercruysse, Comatrix corings and Galois comodules over firm rings, Algebr.
Represent.Theory10(3),(2007) 271–306.
[Ma] E.G.Manes,ATRIPLEMISCELLANY:SOMEASPECTSOFTHETHEORYOFALGEBRASOVER
A TRIPLE.Thesis(Ph.D.)–Wesleyan University.1967.
[Me] B.Mesablishvili,Monads of effectivetype and comonadicity.TheoryAppl.Categ.16(2006), No.1,1–45
[MS] J.L.MacDonald,A.Stone, The towerand regular decomposition. CahiersTopologieG´eom.Diff´erentielle
23(1982), no.2,197-213.
[NVV] C. N˘ast˘asescu, M. van den Bergh, and F. Van Oystaeyen, Separable functors applied to graded rings, J.
Algebra123(1989), 397413.
10 ALESSANDROARDIZZONI,JOSE´ GO´MEZ-TORRECILLAS,ANDCLAUDIAMENINI
[Ra] M.D.Rafael,Separable Functors Revisited,Comm.Algebra18(1990), 1445–1459.
[St] B. Stenstr¨om, Rings of quotients Die Grundlehren der Mathematischen Wissenschaften, Band 217. An
introductiontomethodsofringtheory.Springer-Verlag,NewYork-Heidelberg,1975.
[Sw] M.Sweedler,Hopf Algebras,Benjamin,NewYork,1969.
UniversityofTurin,DepartmentofMathematics”G.Peano”,viaCarloAlberto10,I-10123Torino,
Italy
E-mail address: [email protected]
URL:www.unito.it/persone/alessandro.ardizzoni
Universidad de Granada,Departamentode A´lgebra. Facultadde Ciencias,E-18071-Granada,Spain
E-mail address: [email protected]
URL:http://www.ugr.es/local/gomezj
University of Ferrara, DepartmentofMathematics,Via Machiavelli35,Ferrara,I-44121,Italy
E-mail address: [email protected]
URL:http://www.unife.it/utenti/claudia.menini