\magnification 1200 

\tolerance 10000    \documentstyle{amsppt}

\define\id{\operatorname{id}} \define\ad{\operatorname{ad}}

\define\Hom{\operatorname{Hom}}

\define\tr{\operatorname{tr}} \define\gr{\operatorname{gr}} 

\define\co{\operatorname{co}}

\define\card{\operatorname{card}}

\define\ord{\operatorname{ord}}  \define\SkPr{\Cal P}

\define\End{\operatorname{End}} 

\define\Alg{\operatorname{Alg}}  \define\Ss{\operatorname{\Cal

S}} \define\varep{\varepsilon} \define\hq{\bold

h(q,m)}\define\pq{\bold p(q,C,J)}

\define\trr{\underline{\operatorname{tr}}\;} 



\topmatter \title\nofrills {On the coradical filtration

of Hopf algebras whose coradical is a

Hopf subalgebra} \endtitle  \author  {Nicol\'as Andruskiewitsch and

Hans-J\"urgen Schneider} \endauthor \address N.

Andruskiewitsch:  FAMAF. UNC. (5000) C. Universitaria. C\'ordoba. 

Argentina \endaddress \email  andrus\@mate.uncor.edu

\endemail  \address H.-J. Schneider: Mathematisches Seminar der

Universit\"at  M\"unchen. Theressienstr. 39. (80333) M\"unchen.

Germany \endaddress \email

hanssch\@rz.mathematik.uni-muenchen.de\endemail \thanks{(N.A.)

Forschungsstipendiat der Alexander von

  Humboldt-Stiftung. Also partially supported by CONICET,

CONICOR, SeCYT (UNC), TWAS (Trieste)  and

  FAMAF (Rep\'ublica Argentina).} \endthanks \keywords{Hopf

Algebras}  \endkeywords 

\abstract We offer a new proof of a Theorem of Taft and Wilson on the

coradical filtration of pointed Hopf algebras. We conjecture a

generalization of this result for Hopf Algebras whose coradical is a

Hopf subalgebra; we prove this conjecture in the case of coradically graded

Hopf algebras. \endabstract

\rightheadtext{The coradical filtration}

\leftheadtext{N. Andruskiewitsch and H.-J. Schneider}

\endtopmatter 



\subhead \S 1. Introduction\endsubhead



A basic Theorem of Taft and Wilson (\cite{TW}, \cite{M, Thm. 5.4.1})

describes the coradical filtration of a pointed coalgebra. This

Theorem is the key point in the proof of the following result (see

{\it e. g.} \cite{N, p. 1545}, \cite{AS1, Prop. 3.1}): If $A$ is a pointed

non-cosemisimple Hopf algebra, with coradical  $k(\Gamma)$

where $\Gamma$ is a finite abelian group, then there exist  $g\in \Gamma$, a 

$k$-character $\chi$ of $\Gamma$ such that $\chi(g) \ne 1$, and

$x\in A$, $x \notin k\Gamma$ such that  $$hxh^{-1} = \chi(h)x

\quad \forall h \in \Gamma,\quad \Delta(x) = x\otimes g +

1\otimes x. \tag 1.1$$  



The preceding statement is the starting point in several  classification

results on pointed Hopf algebras. See \cite{AS2}, \cite{CD}, \cite{SvO},

\cite{CDR}.  The proof of the result by Taft and Wilson is not very difficult

but somewhat involved \cite{M, Thm. 5.4.1}. In this article we present an

alternative natural proof of this Theorem for pointed Hopf algebras and

propose a (conjectural) generalization for Hopf algebras whose

coradical is a Hopf subalgebra. We prove the validity of the generalization

assuming that the  Hopf algebra is  coradically graded.



\subsubhead Conventions\endsubsubhead We shall work over a

fixed field $k$; all the algebras, coalgebras, tensor products and

homomorphisms are over $k$ unless explicitly stated.  If $C$ is a

coalgebra, we denote  by $G(C)$ the set of group-like elements of

$C$. If $g$, $h \in G(C)$, then we denote $P_{g,h}(C) = P_{g,h} =

\{x\in C: \Delta(x) = x\otimes h + g \otimes x\}$; the elements of

$P_{g,h}$ are called skew-primitives.  When $B$ is a bialgebra,

$P_{1,1}(B)$ is just the space $P(B)$ of primitive elements.



If $A$ is an algebra, $\Alg(A, k)$ denotes the set of all algebra

maps from $A$ to $k$.





\bigpagebreak

\subhead \S 2.  Hopf algebras whose coradical is a

Hopf subalgebra\endsubhead



We recall the principle proposed in \cite{AS2}, see also \cite{AS3}.



Let $A$ be a Hopf algebra over a field $k$. Let $A_{0} \subseteq A_{1}

\subseteq \dots \subseteq A$ be the coradical filtration of $A$ and let $\gr

A = \oplus_{n \ge 0} \gr A(n) = \oplus_{n \ge 0} A_n/A_{n-1}$ (with $A_{-1}

= 0$) be the associated graded coalgebra. If the coradical $A_{0}$ of $A$ is

a Hopf subalgebra, then the coradical filtration is in fact a Hopf algebra 

filtration and  $\gr A$ is a graded Hopf algebra.  See \cite{M, 5.2.8},

\cite{Sw, Section 11.2}. 



Let $\pi: \gr A\to A_{0}$ be the unique graded projection; it has a Hopf

algebra section $\gamma$, namely the inclusion of $A_{0}$ in $\gr A$.  Let $R = \gr

A^{\co \pi} = \{a \in \gr A: (\id \otimes \pi) \Delta(a) = a\otimes 1\}$ be the

algebra of coinvariants of $\pi$. By a result of Radford \cite{R},

interpreted in categorical terms by Majid \cite{Mj}, $R$ is a braided Hopf

algebra and $\gr A$ is the bosonization of $R$ and $A_{0}$: $\gr A \simeq 

R\# A_{0}$. In \cite{AS2}, we proposed the following principle to study $A$:

first to study $R$, then to transfer the information to $\gr A$ via

bosonization, and finally to lift to $A$.   We recall the relevant formulas

within the conventions of \cite{AS1}. 



\medpagebreak

The action of $A_{0}$  on $R$ is given by the adjoint representation

composed with $\gamma$. The coaction is $\delta_{R} = (\pi\otimes

\id)\Delta$.  These two structures are related by the

Yetter-Drinfeld condition: $$\delta_{R}(h.r) = h_{(1)} r_{(-1)}

\Ss(h_{(3)}) \otimes h_{(2)} .r_{(0)}.   $$ Hence, $R$ is an object  of

the category ${}_{A_{0}}^{A_{0}}\Cal{YD}$ of Yetter-Drinfeld modules over

$A_{0}$.   Moreover, $R$ is a subalgebra of $\gr A$ and a coalgebra with

comultiplication  $\Delta_{R}(r) = r_{(1)} \gamma\pi\Ss(r_{(2)})

\otimes r_{(3)};$ the counit is the restriction of the counit of $\gr A$.  To

avoid confusions, we denote here the comultiplication of $R$ in the

following way: $\Delta_{R}(r) = \sum r^{(1)}\otimes r^{(2)},$ or even we

omit sometimes the summation sign.  The multiplication $m$ and the

comultiplication $\Delta$ of $R$ satisfy $\Delta m =  (m \otimes m)(\id

\otimes c \otimes \id) (\Delta\otimes \Delta).$ Here $c$ is the

commutativity constraint  of ${}_{A_{0}}^{A_{0}}\Cal{YD}$; explicitly $c_{M,

N}(m\otimes n) = m_{(-1)}. n \otimes m_{(0)},$ for $M, N \in

{}_{A_{0}}^{A_{0}}\Cal{YD}$, $m\in M$, $n\in N$.



\medpagebreak It is not difficult to see that $R$ is a

braided Hopf algebra in  ${}_{A_{0}}^{A_{0}}\Cal{YD}$.

Note that $\gr A$ can be reconstructed from $R$ and $A_{0}$ in the

following way:  the multiplication gives rise to an isomorphism between

$\gr A$ and tensor product $R\otimes A_{0}$ and the Hopf algebra structure

is given via the smash product and smash coproduct: $$\aligned (r\#  

h)(s\#   f) &= r(h_{(1)}.s) \#   h_{(2)}f, \\ \Delta(r\#   h) &= r^{(1)}\# 

(r^{(2)})_{(-1)}h_{(1)} \otimes (r^{(2)})_{(0)} \# h_{(2)}. \endaligned \tag

2.1$$



Note also that $R$ inherits the grading from $\gr A$: $R = \oplus_{n\ge 0}

R(n)$, where $R(n) = R \cap \gr A(n)$. Moreover, it is a braided graded Hopf

algebra with respect to this grading. Hence $\gr A(n) = R(n)\# A_{0}$ and

$R_{0} = R(0) = k1$. Following \cite{CM}, we say that a  graded coalgebra $C

= \oplus_{n\ge 0} C(n)$ is {\it coradically graded} if the coradical filtration

coincides with the filtration deduced from the grading:  $C_m =  \oplus_{n

\le m} C(n)$. It is known that $\gr A$ is coradically graded \cite{AS2,

Lemma 2.3}. It follows that $$R \text{ is also a  coradically graded

coalgebra; } \tag 2.3$$ in particular  $$R_{0} = k1 = R(0) \text{ and }

P(R) = R(1). \tag 2.4$$ Therefore,  $\gr A_{1} = A_{0} \oplus

\left[ P(R)\# A_{0}\right]$. 



\bigpagebreak

\subhead \S 3. The Theorem of Taft and Wilson for pointed Hopf

algebras\endsubhead 



The Theorem of Taft and Wilson gives information about the

coradical filtration of a pointed coalgebra. 



\proclaim{Theorem 3.1} (\cite{TW}; see \cite{M, Thm. 5.4.1}). Let

$C$ be a pointed coalgebra.  \roster \item"(i)" If $n \ge 1$, the

$n$-th term of the coradical filtration can be decomposed as 

$$C_{n} = \sum_{g,h\in G(C)} C_{n}(g,h), \tag 3.1$$ where 

$$C_{n}(g,h) = \{x\in C: \Delta(x) = x\otimes h + g \otimes x + u,

\text{ for some }u \in C_{n-1}\otimes C_{n-1}\}.$$  \item"(ii)" The

first term of the coradical filtration can be expressed  as $$C_1 =

kG(C) + (\oplus_{g,h \in G(C)} P_{g,h}). \qed \tag 3.2$$ \endroster

\endproclaim 



The inclusions $\supseteq$ in (3.1) and (3.2) are clear; the point of

the Theorem is the other inclusions.



\bigpagebreak

Let $A$ be a Hopf algebra  whose coradical $A_0$ is a Hopf

subalgebra. Let $  \gr A$ be the associated graded Hopf algebra 

and $R$  the associated braided graded Hopf

algebra as in \S 2. 

\proclaim{Lemma 3.2} Theorem 3.1 holds for $R$. \endproclaim

\demo{Proof} Let $r\in R_{n}$. By \cite{AS2, Lemma 2.4}, $R_{n} = \oplus_{0

\le j \le n}R(j)$; so, we can assume $r\in R(n)$. Since $R$ is a

graded coalgebra and $R(0) = k1$, $\Delta_{R}(r) = r_{1}\otimes 1

+ 1\otimes r_{2} + u$, where $r_{1}, r_{2} \in R(n)$, $u \in \oplus_{0

<s < n} R(s)\otimes R(n-s)$. Applying $\id \otimes \varep$,

resp. $\varep \otimes \id$,we see that $r = r_{1}$,

resp. $r= r_{2}$. The equality (3.1) follows; the equality (3.2) is

part of \cite{AS2, Lemma 2.4} as explained above-- see (2.4). \qed\enddemo



{\it We assume now that $A$ is pointed}. Let us denote $\Gamma =

G(A)$, $G = \gr A$. So $A_{0} = k\Gamma$ and  $G$ is coradically

graded. 



\proclaim{Lemma 3.3}  Theorem 3.1 holds for $\gr

A$.\endproclaim  \demo{Proof} Let $x\in G_{n}$.

By\cite{AS2,  Lemma 2.4}, we can asume that $x\in G(n) = R(n)\#

k\Gamma$.  Now we can decompose $R(n)$ in isotypical

components as left comodule over $k\Gamma$: $$R(n) =

\oplus_{g\in \Gamma}R(n)_{g}, \text{ where }R(n)_{g} = \{r\in R(n):

\delta(r) = g\otimes r\}.$$ We can assume that $x = r\# h$, where

$r\in R(n)_{g}$ and $g, h \in \Gamma$.  We know by Lemma 3.2 that 

$\Delta_{R}(r) = r \otimes 1 + 1\otimes  r + u$, where  $u \in

\oplus_{0 <s < n} R(s)\otimes R(n-s)$.  We apply (2.1) to get

$$\align \Delta_{G}(x) &= r^{(1)}\#  (r^{(2)})_{(-1)}h \otimes

(r^{(2)})_{(0)}\#  h

 \\ &= r\# h \otimes 1\# h + 1\# gh \otimes r\#h + \tilde u

\\ &= x \otimes h +  gh \otimes x + \tilde u,

\endalign $$

where $\tilde u \in \oplus_{0 <s < n} R(s)\# k\Gamma\otimes

 R(n-s) \# k\Gamma$. This proves (3.1). The proof of (3.2) is

exactly the same; just use that $R(1) = P(R)$. \qed\enddemo



\proclaim{Lemma 3.4} Theorem 3.1 holds for a pointed Hopf

algebra $A$. \endproclaim

\demo{Proof} The proof of (3.1) for $A$ is a direct consequence of

Lemma 3.3. Indeed, let  $x \in A_{n}$ and let $\overline{x}$ be its

class in $\gr A(n) = A_{n}/A_{n-1}$. We can assume, by Lemma 3.3,

that  $$\Delta(\overline{x}) = \overline{x}\otimes h + g\otimes

\overline{x} + u,$$

where $g, h\in \Gamma$, $u \in \oplus_{i+j < n} \gr A(i)\otimes \gr

A(j)$. Hence there exist $a,b \in A_{n-1}$, $v \in A_{n-1}\otimes

A_{n-1}$ such that 

$$\Delta(x) = (x  + a)\otimes h + g\otimes

 (x  + b) +  v = x  \otimes h + g\otimes

 x +  \left(a\otimes h + g\otimes b + v\right);$$

(3.1) follows. 



To prove (3.2), we have to lift some obstruction. Let $g, h \in\Gamma$ and

let $A_{g,h}$ be the space of all $x\in A$ such that  $$\Delta(x) =

x\otimes h + g \otimes x + u,  \tag 3.3$$ for some $u \in k\Gamma

\otimes k\Gamma$. By (3.1), which we already proved,  $$A_{1} =

\sum_{g, h \in \Gamma}A_{g,h}.$$ Therefore, we have to prove the

somewhat more precise result $$A_{g,h} = P_{g,h} + k\Gamma. \tag

3.4$$ Clearly, the inclusion $\supseteq$ is true.  Let $x\in A_{g,h}$

and let $u$ be as in (3.3). By the coassociativity condition, (3.3)

implies the equality $$u\otimes h + (\Delta \otimes \id)(u) =  g

\otimes u + (\id \otimes \Delta)(u).  \tag 3.5$$ 

 We look for $a\in k\Gamma$ such that  $ x-a \in P_{g,h}.$ This is

equivalent to 

$$u= a\otimes h + g\otimes a - \Delta(a). \tag 3.6$$

The Lemma will follow from the next result. \qed\enddemo



\proclaim{Lemma 3.5}Given $u \in k\Gamma \otimes k\Gamma$

satisfying (3.5), there exists $a\in k\Gamma$ such that (3.6) holds.

\endproclaim

\demo{Proof} This is a  particular case of Lemma A.1, {\it cf.} the Appendix.

\qed\enddemo



\bigpagebreak

\subhead \S 4. The first term of the coradical filtration of a Hopf

algebra\endsubhead

Now we propose a generalization of the Theorem of Taft and Wilson for

arbitrary Hopf algebras. We assume in this section that the ground field $k$

is algebraically closed.  Let $A$ be a Hopf algebra and let $\widehat{A}$ be

the set of isomorphy classes of irreducible left comodules. The space

$C_{\tau}$ of matrix coefficients of $\tau \in \widehat{A}$ is a simple

subcoalgebra of $A$ and $A_{0} = \oplus_{\tau \in \widehat{A}} C_{\tau}$.



\proclaim{Conjecture 4.1} The first term of the coradical filtration of $A$

is given by 

$$\align

A_{1} &= A_{0} + \sum_{\tau \in \widehat{A}} (C_{\tau} \wedge k1).A_{0}

\tag 4.1\\ &= A_{0} + \sum_{\tau, \mu \in \widehat{A}} C_{\tau}.C_{\mu}

\wedge C_{\mu}. \tag 4.2 \endalign$$\endproclaim





We observe that (4.1) implies (4.2): it is clear that

$$ A_{1} \supseteq A_{0} + \sum_{\tau, \mu \in \widehat{A}}

C_{\tau}.C_{\mu} \wedge C_{\mu}  \supseteq A_{0} + \sum_{\tau \in

\widehat{A}} (C_{\tau} \wedge k1).A_{0}; $$

the second inclusion follows from a direct computation. 









We first check that the conjecture, for pointed Hopf algebras, is nothing but

the second part of the Theorem of Taft and Wilson. Indeed, if $A$ is pointed,

then $\widehat{A}$ is identified with $G(A)$ and the conjecture reads

$$\align A_{1} &= A_{0} + \sum_{g \in G(A)} (kg \wedge k1).A_{0} \tag 4.3\\

&= A_{0} + \sum_{g, h \in G(A)} kg \wedge kh, \tag 4.4 \endalign$$ after a

suitable change of variables in (4.4). Now we claim that  $$kg \wedge kh =

P_{g,h} + kg = P_{g,h} + kh. \tag 4.5$$ The second equality is evident since

$g-h \in P_{g,h}$. Also, the inclusion $\supseteq$ in the first equality

follows by definition. Now, if $x \in kg \wedge kh$, then

$$\Delta(x) = g\otimes x + h\otimes x - \epsilon(x) g\otimes h.$$

Therefore $x -  \epsilon(x) g \in P_{g,h}$ and (4.5) follows. We see, via

(4.5), that (4.4) is equivalent to (3.2).



\bigpagebreak

We assume now that the coradical $A_{0}$ is a Hopf subalgebra of $A$.  Let

$  \gr A$, $R$ be as in \S 2. 

\proclaim{Lemma 4.2} The conjecture 4.1 holds for $\gr A$.\endproclaim

\demo{Proof} By Lemma 2.3, $\gr A_{1} = A_{0}\oplus \gr A_{1} =

A_{0}\oplus P(R)\# A_{0}$.  Now let $P(R) = \oplus_{\tau \in \widehat{A}}

P(R)_{\tau}$ be the decomposition of $P(R)$ into isotypical components as

$A_{0}$-comodule.  If $r \in P(R)_{\tau}$, then by (2.1)

$$\Delta_{\gr A} (r) = r \otimes 1 + r_{(-1)} \otimes r_{(0)} \in \gr A

\otimes 1 + C_{\tau} \otimes \gr A;$$

that is, $r\in C_{\tau} \wedge k1$. Hence (4.1) holds. \qed\enddemo







\bigpagebreak

\subhead Appendix. The co-Hochschild cohomology for Hopf algebras

\endsubhead

In this Appendix we prove a result needed in the text, in the context of 

co-Hochschild cohomology for Hopf algebras; {\it cf. } \cite{GS}. See also

\cite{S} for applications to finiteness results for semisimple Hopf algebras. 



\bigpagebreak

Let  $C$ be a coalgebra and let $V$ be a $(C, C)$-bicomodule, with structure

maps $\sigma: V\to V\otimes C$, $\rho: V\to C\otimes V$. We denote as

usual $\sigma(v) = v_{(-1)}\otimes v_{(0)}$, $\rho(v) = v_{(0)}\otimes

v_{(1)}$; then the compatibility condition between the two coactions

justifies the notation

$$v_{(-1)}\otimes v_{(0)}\otimes v_{(1)}: = v_{(-1)}\otimes

\rho\left(v_{(0)}\right) = \sigma\left(v_{(0)}\right) \otimes v_{(1)}. $$



Analogous to the usual Hochschild complex of an algebra with coefficients

in a bimodule, the co-Hochschild complex of the coalgebra $C$ with

coefficients in the bicomodule $V$ is

$$

0@>>> \Hom(V, k) @>\delta^{0}>> \Hom(V, C) @>\delta^{1}>> \Hom(V,

C\otimes C) @>\delta^{2}>> \Hom(V, C\otimes C\otimes C) @>\delta^{3}>> 

\dots $$

where, if for $f\in \Hom(V, C^{\otimes n})$ we denote $f(v) = v^{1}\otimes

v^{2}\otimes \dots \otimes v^{n}$, then

$$\multline

\delta^{n}(f) (v) := v_{(-1)} \otimes f\left(v_{(0)}\right) \\ + \sum_{i = 1}^{n}

(-1)^{i}  v^{1}\otimes v^{2}\otimes \dots \otimes v^{i-1} \otimes

\Delta(v^{i})\otimes v^{i+1}\otimes  \dots\otimes v^{n} \\+ (-1)^{n+1}

f(v_{(0)}) \otimes v_{(1)}.

\endmultline

\tag A.1$$

It is routinary to verify that $\delta^{n +1}\delta^{n} = 0$ for all $n \ge 0$. 



The coalgebra $C$ itself is a $(C, C)$-bicomodule with $\sigma = \rho =

\Delta$. We regard $C\otimes C$ as a $(C, C)$-bicomodule with left

coaction in the first factor and right coaction in the second factor.



\definition{Definition} A coalgebra $C$ is {\it coseparable} iff the

comultiplication $\Delta: C\to C\otimes C$ has a retraction $\pi: C\otimes 

C\to C$ of $(C, C)$-bicomodules. Equivalently, if $C$ is an injective $(C,

C)$-bicomodule. \enddefinition



For instance, let $\Cal X$ be a set and let $C = k\Cal X$ be the "set coalgebra"; it

has a basis $(x)_{x \in \Cal X}$ and the comultiplication is given by

$\Delta(x) = x\otimes x$. Let $\pi: C\otimes C \to C$ be the map 

$$\pi(x\otimes y) = \delta_{x,y} x, \qquad x,y\in X.$$

Then $\pi$ is a bicomodule retraction of $\Delta$. That is, $C$ is

coseparable.





Let $C$ be coseparable and let $\pi$ be as in the preceding definition. Let

$$s_{n}: \Hom(V, C^{\otimes n}) \to \Hom(V, C^{\otimes n-1}), \qquad n\ge

1$$ be the map given by

$$s_{n}(f)(v) := \epsilon \pi\left(v_{(-1)} (v_{(0)})^{1}\right) (v_{(0)})^{2}

\otimes \dots \otimes (v_{(0)})^{n},$$

with the same convention as in (A.1). Then

$$\delta^{n-1} s_{n} + s_{n+1} \delta^{n} = \id, \qquad \forall n\ge 1.\tag

A.2$$

In particular, the $n$-th cohomology groups $H^{n}(C, V)$ of the

co-Hochschild complex are 0 for $n\ge 1$. 



\proclaim{Lemma A.1} Let $\Cal X$ be a set, $C = k\Cal X$ the set

coalgebra, $g,h \in \Cal X$. Given $u \in k\Cal X \otimes k\Cal X$ satisfying

(3.5), there exists $a\in k\Cal X$ such that (3.6) holds. \endproclaim

\demo{Proof} Let $V$ be a one dimensional vector space with base $\{v\}$.

We regard $V$ as a $(C, C)$-bicomodule via

$$\sigma(v) = g\otimes v, \qquad \rho(v) = v\otimes h.$$

Given $u \in k\Cal X \otimes k\Cal X$, $a\in k\Cal X$, we define $f \in

\Hom(V, C^{\otimes 2})$, $g \in \Hom(V, C)$ by 

$$f(v) = u, \qquad g(v) = a.$$

Then (3.5) means that  $\delta^{2}(f) = 0$,  (3.6)  that $\delta^{1}(g) =

f$, and the Lemma  that $H^{2}(C, V) = 0$. This follows from the above

considerations. One could check, explicitly, that $a = (\epsilon \pi\otimes

\id) (g\otimes u)$ satisfies (3.6) whenever $u$ satisfies (3.5). \qed\enddemo











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 \enddocument



  


