This article was downloaded by: [University of New Mexico]On: 03 October 2014, At: 07:06Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20

BINOMIAL ALGEBRASJessica K. Sklar aa University of Oregon , Eugene, OR, 97403, U.S.A.Published online: 01 Sep 2006.

To cite this article: Jessica K. Sklar (2002) BINOMIAL ALGEBRAS, Communications in Algebra, 30:4, 1961-1978, DOI: 10.1081/AGB-120013226

To link to this article: http://dx.doi.org/10.1081/AGB-120013226

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

BINOMIAL ALGEBRAS

Jessica K. Sklar

University of Oregon, Eugene, OR 97403

ABSTRACT

Let K be a field. A split basic finite-dimensional K-algebrawith left quiver G is binomial if it can be represented as thepath algebra KG modulo relations of the form p ¼ lq, wherel 2 K and p and q are paths in G. We here characterize allbinomial algebras A as twisted semigroup algebras A ffi KxS,where S is an algebra semigroup, and where x 2 Z2ðS;K�Þ is atwo-dimensional cocycle of S with coefficients in the multi-plicative group of units K � of K. Subject to certain conditionson an algebra semigroup S, we classify the twisted semigroupalgebras of S up to isomorphism. Finally, subject to the sameconditions on S, we show that for each binomial algebraA ffi KxS there exists a short exact sequence

1 �! H1ðS;K �Þ �! OutA �! StabxðAutSÞ �! 1;

where H1ðS;K�Þ is the first cohomology group of S withcoefficients in K�, OutA is the group of outer automorphismsof A, AutS is the group of semigroup automorphisms of S,and StabxðAutSÞ is the stabilizer in AutS of ½x� under thenatural action of AutS on the second cohomology group,H2ðS;K�Þ, of S. Moreover, if x ¼ 1 (so that A ffi KS is un-twisted), then the above sequence splits, yielding OutA ffiH1ðS;K �Þ �jAutS.

1961

Copyright # 2002 by Marcel Dekker, Inc. www.dekker.com

COMMUNICATIONS IN ALGEBRA, 30(4), 1961–1978 (2002)

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

INTRODUCTION

It is well-known that if A is a split basic finite-dimensional algebra Aover a field K with left quiver G, then A is isomorphic to a quotient of itspath algebra KG by an ideal I � ðRadKGÞ2, said to be an adequate ideal forA. (See [6].) Monomial algebras (that is, algebras with adequate ideals thatare generated by paths) have been studied extensively. For instance, Guil-Asensio and Saorın describe their automorphism groups in [7], Saorın hasdetermined conditions under which they are isomorphic in [10], and Jianbohas found conditions under which a given algebra has an adequate mono-mial idea (see [9]).

We here consider another class of algebras, which includes allmonomial algebras. A split basic finite-dimensional K-algebra A with leftquiver G is binomial if there exists an adequate ideal for A which is gen-erated by relations of the form p� lq, where l 2 K, and where p and q arepaths in G. (See [5].) Binomial algebras are abundant. Clearly, any mono-mial algebra is binomial, though, in general, binomial algebras are notmonomial (see Example 2.1). Diagram algebras, introduced by Fuller in [4],constitute another class of binomial algebras; in fact, we show in Sec. 2 thatthe diagram algebras over a field K are precisely the untwisted binomial K-algebras. Yet another class of binomial algebras is comprised of basicsquare-free algebras (which arise as a generalization of incidence algebras),though it is easy to find binomial algebras that are not square-free (see,again, Example 2.1).

In [1], expanding on the results of Stanley [11], Clark [2] and Coelho[3], Anderson and D’Ambrosia characterize (not necessarily basic) square-free algebras as twisted semigroup algebras, and dissect their outer auto-morphism groups. In this paper we extend those results to the larger class ofbinomial algebras. In the first section we review the cohomology groups of asemigroup, and define the twisted semigroup algebra KxS of an algebrasemigroup S over a field K, where x is a two-dimensional cocycle of S withcoefficients in the multiplicative group of units K� of K. In Sec. 2 weintroduce binomial algebras, and characterize them as twisted semigroupalgebras of algebra semigroups. Further, given a field K and an almoststrongly acyclic locally square-free algebra semigroup S, we use a variationof Saorın’s change of variable method (see [10]) to classify the twisted K-semigroup algebras of S up to isomorphism. Finally, in Sec. 3, we use ourcharacterization of binomial algebras to examine the structures of theirautomorphism groups. In particular, let K be a field, let S be an almoststrongly acyclic locally square-free algebra semigroup, and let A ¼ KxS bethe K-semigroup algebra of S twisted by cocycle x. Sharpening Theorem 7 of[10] in this special case, we show that there is a short exact sequence

1962 SKLAR

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

1 �! H1ðS;K �Þ �! OutA �! StabxðAutSÞ �! 1;

where H1ðS;K �Þ is the first cohomology group of S with coefficients in K �,OutA is the group of outer automorphisms of A, AutS is the group ofsemigroup automorphisms of S, and StabxðAutSÞ is the stabilizer in AutSof ½x� under the natural action of AutS on the second cohomology group,H2ðS;K �Þ, of S. Moreover, when A is untwisted, we have thatStabxðAutSÞ ¼ AutS, and that our sequence splits, yielding OutA ffiH1ðS;K�Þ �jAutS.

1. PRELIMINARIES

Let S be a finite semigroup with zero y and set of pairwise orthogonalnonzero idempotents E such that

S ¼[

e; f2Ee � S � f

and such that

J ¼ SnE

is a nilpotent semigroup ideal of S. (For example, let S be the path semi-group of a finite quiver.) As in [4], we call such a semigroup an algebrasemigroup, and we let S � ¼ S nfyg. In [2], and later, more generally, in [1], acohomology was developed for special classes of algebra semigroups. Thiseasily extends to all algebra semigroups, and is defined as follows. LetSh0i ¼ E, and define each Shni for n > 0 by

Shni ¼ fðs1; s2; . . . ; snÞ 2 Snjs1 � s2 � � � sn 6¼ yg:

Let G be any abelian group written multiplicatively, and for each n � 0, letF nðS;GÞ be the set of all functions from Shni to G. These sets are abeliangroups under multiplication; define for each n � 0

@n : F nðS;GÞ �! F nþ1ðS;GÞ

by

ð@0jÞðsÞ ¼ jð f ÞjðeÞ�1

BINOMIAL ALGEBRAS 1963

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

for all s ¼ e � s � f 2 S � and all j : E �! G, and by

ð@njÞðsÞ ¼ jðs2; . . . ; snþ1Þ

�Yn

i¼1

jðs1; . . . ; si � siþ1; . . . ; snþ1Þð�1Þi !

jðs1; . . . ; snÞð�1Þnþ1

for each n > 0;j 2 F nðS;GÞ and s ¼ ðs1; s2; . . . ; snþ1Þ 2 Shnþ1i. It is straight-forward to verify that each @n is a group homomorphism, with @nþ1@n ¼ 1for every n. We define the groups of n-dimensional cocycles and cobound-aries, for each n � 0, to be

ZnðS;GÞ ¼ Ker @n and BnðS;GÞ ¼ Im @n�1;

respectively. Since @n is a differential, BnðS;GÞ � ZnðS;GÞ for each n; wedefine the n-dimensional cohomology group of S with coefficients in G to be

HnðS;GÞ ¼ Z nðS;GÞ=BnðS;GÞ:

Given x 2 ZnðS;GÞ, we denote the BnðS;GÞ coset of x by ½x�, and we say thatx; z 2 ZnðS;GÞ are cohomologous if ½x� ¼ ½z�.

We further say that a cocycle x 2 Z2ðS;GÞ is normal if we havexðe; eÞ ¼ 1 for every e 2 E. Note that if x is any two-dimensional cocycle,and we define Z 2 F1ðS;GÞ by

ZðsÞ ¼ xðs; sÞ�1; if s 2 E;1; otherwise,

�

then ð@ZÞx is easily seen to be a normal cocycle that is cohomologous with x.So every cohomology class ½x� 2 H2ðS;GÞ has a normal representative.

Now let K be a field with multiplicative group of units K �. For anycocycle x 2 Z2ðS;K�Þ we define the K-semigroup algebra of S twisted by x tobe the associative K-algebra KxS with basis S� whose multiplication isgiven by

st ¼ xðs; tÞs � t; if s � t 6¼ y;0; otherwise

�

for each s; t 2 S�. Note that for the identity element 1 of the group Z2ðS;GÞ,the algebra K1S is simply the untwisted K-semigroup algebra KS.

1964 SKLAR

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

We can use the group AutS of semigroup automorphisms of a givenalgebra semigroup S to classify the twisted semigroup K-algebras of S up toisomorphism. Given j 2 AutS and x 2 Z2ðS;K�Þ, we can, as in [1], define anew two-dimensional cocycle xj : Sh2i �! K � by

xjðs; tÞ ¼ xðjðsÞ;jðtÞÞ

for all s; t 2 S with s � t 6¼ y. It is easy to see that if x; z 2 Z2ðS;K �Þ with½x� ¼ ½z�, then ½xj� ¼ ½zj� for every j; therefore, AutS acts on H2ðS;K �Þ by

j : ½x� �! ½xj�

for every j 2 AutS and x 2 Z2ðS;K �Þ.We easily extend results from [1]:

Lemma 1.1. Let S be an algebra semigroup, and let x; z 2 Z2ðS;K�Þ. If

½xj� ¼ ½z�

for some j 2 AutS, then, as K-algebras, we have

KxS ffi KzS:

In particular, for each x 2 Z2ðS;K�Þ there exists a normal z 2 Z2ðS;K�Þ withKxS ffi KzS. j

Notice that given any algebra semigroup S and any normalx 2 Z2ðS;K�Þ, we have that KxS is a split basic finite-dimensional K-algebrawith complete set of idempotents E, and with RadKxS ¼ KJ, the K-linearsubspace of KxS spanned by the nonzero elements of J .

Next, let A be a split basic finite-dimensional K-algebra with leftquiver G. We let P be the path semigroup of G, with zero element y, vertexset E and set of arrows A; we denote the set of nonzero elements of P by P�,the K-path algebra of G by KG, and the radical of KG by JJ. Recall that anideal I of KG is said to be admissible if there exists some integer m � 1 suchthat JJmþ1 � I � JJ2, and that an admissible ideal I of KG is called an ade-quate ideal for A if KG=I ffi A. Gabriel has shown that there exists at leastone adequate ideal for A (see [6]). Now let I be an adequate ideal for A, andlet p : KG �! A be an epimorphism with kernel I. Since pðP�Þ spans A overK, there exists a K-basis B for A with B � pðP�Þ. We say that such a K-basisfor A is standard with respect to I. Notice that if I is an adequate ideal for A

BINOMIAL ALGEBRAS 1965

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

and if B is a K-basis for A that is standard with respect to I, then, sinceI � JJ2, we must have both pðEÞ and pðAÞ in B; moreover, we may identifypðEÞ and pðAÞ with their respective pre-images E and A in KG.

2. CHARACTERIZATION OF BINOMIAL ALGEBRAS

Let K be a field (of arbitrary characteristic), and let G be any finitequiver. We say that an ideal I of the path algebra KG is binomial if it isgenerated (as an ideal) by elements of the form p� lq, where l 2 K, andwhere p and q are paths in G. Let A be a split finite-dimensional K-algebrawith left quiver G. Recall that A is Morita equivalent to its basic algebra; wetherefore may assume that A itself is basic. We then say that A is a binomialalgebra if there exists an adequate binomial ideal I � KG for A.

Example 2.1. Let G be the quiver

and let A be the algebra KG=I, where I is the ideal of KG generated by theelement ad� ge. Then A is a binomial algebra, which is neither monomialnor square-free. j

As noted in [5], if an adequate ideal I for A is binomial, then any K-basis B for A that is standard with respect to I is nearly multiplicativelyclosed: that is, for every c and d in B, there exists b ¼ bc;d in B such thatcd 2 Kb. Thus, S ¼ B [ fyg is a multiplicative semigroup, where y acts aszero, and where, for each pair s; t 2 S�,

s � t ¼ bs;t; if st 6¼ 0;y; otherwise.

�

Clearly, S is an algebra semigroup, with set E of nonzero idempotents. Wecall S an associated semigroup for A. (We show in Example 2.5, adaptedfrom Example 6b in [10], that S need not be unique to up to isomorphism.)Now, for each pair s; t 2 S� with s � t 6¼ y there exists a unique elementxðs; tÞ 2 K� with st ¼ xðs; tÞs � t. Since multiplication in A is associative,it follows that the map x : Sh2i �! K� is a two-dimensional cocycle; thus,

1966 SKLAR

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

A ffi KxS as K-algebras. This proves half of the following characterization ofbinomial algebras as twisted semigroup algebras of algebra semigroups.

Theorem 2.2. Let A be a K-algebra. Then A is a binomial algebra if and onlyif A ffi KxS for some algebra semigroup S and some x 2 Z2ðS;K�Þ.

Proof. Let S be an algebra semigroup, and let A ffi KxS for somex 2 Z2ðS;K�Þ. It suffices to prove that A is a binomial algebra. Clearly wemay assume that A ¼ KxS and, by Lemma 1.1, that x is normal.

As before, let E be the set of nonzero idempotents of S, and letJ ¼ SnE, so that A is a split basic finite-dimensional K-algebra with radicalJ ¼ KJ . Denoting J � J by J 2, it is easy to see that J2 is the K-subspace ofA spanned by the nonzero elements of J 2; so if we let

A ¼ J nðJ 2 [ fygÞ;

then faþ J2 : a 2 Ag is a K-basis for J=J2.Next, let G be the left quiver of A with path semigroup P, and, for each

i � 0, let Pi be the set of paths in G of length i. Then there exists a K-algebraepimorphism p : KG �! A sending P0 and P1 bijectively onto E and A,respectively. It suffices to show that Ker p is a binomial ideal of KG. SincepðP0 [ P1Þ � S� [ f0g, and since S� is a nearly multiplicatively closedK-basis of A, there exists a surjection s : P�nKer p �! S� with pðpÞ 2K�sðpÞ for every p 2 P�nKer p. Now, if p; q 2 s ðsÞ for some s 2 S�, thenpðpÞ ¼ lp;qpðqÞ for unique lp;q 2 K�. Let I be the (necessarily binomial) idealof KG generated by the set

ðP� \Ker pÞ [ fp� lp;qq : p; q 2 P�nKer p with sðpÞ ¼ sðqÞg:

Clearly, I � Ker p. Now, we can decompose any x 2 Ker p uniquely as

x ¼ x0 þX

s2S�xs;

where x0 is a linear combination of elements of P� \Ker p, and where eachxs is a linear combination of elements s ðsÞ. Now,

Ps2S� pðxsÞ ¼

pðx� x0Þ ¼ 0. But for every s 2 S�, pðxsÞ 2 Ks; so, for each s, we must havepðxsÞ ¼ 0. Fix s 2 S�, and write xs as

xs ¼Xr

i¼1

mipi;

where the pi are distinct elements of s ðsÞ. Let

BINOMIAL ALGEBRAS 1967

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

qi ¼ pi � lpi;piþ1piþ1 for 1 � i � r� 1; and qr ¼ pr:

Since fqigri¼1 is then a K-basis for the subspace Kp1 þ Kp2 þ � � � þ Kpr ofKG, there exist fnigri¼1 � K with

xs ¼Xr

i¼1

niqi:

But 0 ¼ pðxsÞ ¼Pr

i¼1 nipðqiÞ, and for each 1 � i � r� 1, we haveqi 2 I � Ker p; thus, 0 ¼ nrpðqrÞ ¼ nrpðprÞ 2 nrK�s; and so nr ¼ 0. It followsthat xs 2 I for each s 2 S�; as x0 is also in I, we have x 2 I. Thus, Ker p ¼ I,so that A is binomial. j

Recall that a finite-dimensional K-algebra A is said to be square-free iffor every pair of primitive idempotents e; f in A we have dimKðeAfÞ � 1 (see[1]). The above theorem and Theorem 1.14 in [1] then yield the following:

Corollary 2.3. Let K be a field. Then every basic square-free K-algebra isbinomial. j

In [4], Fuller introduces the notion of diagram algebras: that is, K-semigroup algebras KS, where the semigroup S is the node semigroup of analgebra diagram. It follows from Lemma 2.2, Lemma 2.3 and Proposition2.4 in [4] that a semigroup S is the node semigroup of an algebra diagram ifand only if it is an algebra semigroup; thus, given a field K, it follows fromTheorem 2.2 that the diagram K-algebras are exactly the untwisted binomialK-algebras. We summarize this in the following theorem.

Theorem 2.4. Let K be a field, and A a K-algebra. Then A is an untwistedbinomial algebra if and only if it is a diagram algebra. j

The following example shows that an associated semigroup of abinomial algebra need not be unique to within isomorphism.

Example 2.5. Let G be the quiver

with path semigroup P and path algebra KG. Let I be the ideal of KGgenerated by fbd; gad; adb� adgag, and let A be the K-algebra

A ¼ KG=I:

1968 SKLAR

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

Now, I is an adequate ideal for A; as Saorın notes in [10], the ideal L of KGthat is generated by the set fbd; gad; adbg is also adequate for A. So, with theobvious induced semigroup structures, the sets

S ¼ P� þ I ¼ fpþ I : p 2 P�g and T ¼ P� þ L ¼ fpþ L : p 2 P�g

are both associated semigroups for A, with respective zero elements

y1 ¼ I and y2 ¼ L:

But S and T are not isomorphic. Indeed, if j is any semigroup isomorphismfrom S to T then it must induce a digraph automorphism of G, so thatjðpþ IÞ ¼ pþ L for every p 2 P�. In particular, adbþ I 6¼ y1, butjðadbþ IÞ ¼ adbþ L ¼ y2, a contradiction. j

Let S be an algebra semigroup, and let x; z 2 Z2ðS;K�Þ. We wish toknow under which circumstances we have KxS ffi KzS. We have alreadyshown in Lemma 1.1 that if there exists an automorphism j 2 AutS suchthat ½xj� ¼ ½z�, then KxS ffi KzS as K-algebras. We now make two addi-tional assumptions about S. Let S be an algebra semigroup, with E, Jand A as before. We say that S is almost strongly acyclic if for everye; f 2 E,

e � A � f 6¼ fyg¼) e � J 2 � f ¼ fyg;

and that S is locally square-free if for every e; f 2 E we have

je � A � f nfygj � 1:

Notice that if we let S be an algebra semigroup and x 2 Z2ðS;K�Þ, then S islocally square-free if and only if the left quiver of KxS has no multiplearrows. Moreover, S is almost strongly acyclic if and only if, lettingJ ¼ RadKxS, we have eJ2f ¼ 0 whenever e; f 2 E with eJf=eJ2f 6¼ 0; in otherwords, S is almost strongly acyclic if and only if KxS is an almost stronglyacyclic K-algebra in the sense of [8].

Now let S be an algebra semigroup, and let x; z be normal cocycles inZ2ðS;K�Þ. We say that a K-algebra isomorphism g : KxS �! KzS is normalif gðEÞ ¼ E. The next lemma follows from Theorem 3 in [10]; we hereprovide a short direct proof.

BINOMIAL ALGEBRAS 1969

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

Lemma 2.6. Let S be an almost strongly acyclic locally square-free algebrasemigroup, let x and z be normal cocycles in Z2ðS;K�Þ, and let g be a normalK-algebra isomorphism from KxS to KzS. Then there exists an automorphismg0 of S and a map lg 2 F1ðS;K�Þ such that

gðsÞ ¼ lgðsÞg0ðsÞ

for every s 2 S�.

Proof. We claim that for every s 2 S�, there exists a unique element g0ðsÞ 2S� with gðsÞ 2 K�g0ðsÞ. First, we define g0ðeÞ ¼ gðeÞ for every e 2 E. Next, letJ ¼ RadKzS, let e; f 2 E, and let 0 6¼ a 2 eAf � KxS. Then

0 6¼ gðaÞ ¼ gðeÞgðaÞgð f Þ 2 gðeÞJgð f ÞngðeÞJ2gð f Þ:

But S almost strongly acyclic implies gðeÞJgð f Þ ¼ KgðeÞAgð f Þ, which in turnmust equal Kg0ðaÞ for some unique g0ðaÞ 2 A, since the left quiver of KzShas no multiple arrows. Thus, we have 0 6¼ gðaÞ 2 K�g0ðaÞ for each a 2 A.Since S� is nearly multiplicatively closed in KzS, our claim follows. Definingg0ðyÞ ¼ y, the map g0 : S �! S is easily then seen to be an automorphismof S. j

We now have the following theorem:

Theorem 2.7. Let S be an almost strongly acyclic locally square-free algebrasemigroup, and let x; z 2 Z2ðS;K�Þ. Then

KxS ffi KzS

if and only if there exists an automorphism j 2 AutS with

½xj� ¼ ½z�:

Proof. We showed in Lemma 1.1 that if ½xj� ¼ ½z�, then KxS ffi KzS as K-algebras. So suppose now that KxS ffi KzS via K-algebra isomorphism g, andassume, without loss of generality, that x and z are normal. Since g is anisomorphism, we have that E and gðEÞ are both complete sets of idempo-tents for KzS; therefore, there exists an invertible element u 2 KzS such that

augðEÞ ¼ E;

1970 SKLAR

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

where au 2 AutKzS is conjugation by u (see Lemma 2.4 in [1]). Then aug is anormal isomorphism from KxS to KzS; thus, without loss of generality, wemay assume that g is normal. Then by Lemma 2.6, there exists map lg 2F1ðS;K�Þ and automorphism g0 2 AutS such that

gðsÞ ¼ lgðsÞg0ðsÞ

for every s 2 S�. Let j ¼ g�10 2 AutS, and define Z 2 F1ðS;K�Þ by

ZðsÞ ¼ lgðjðsÞÞ

for every s 2 S�. Then xjz�1 ¼ @Z 2 B2ðS;K�Þ; indeed, for every s 2 S�, wehave

gðsÞ ¼ Zðj�1ðsÞÞj�1ðsÞ:

Therefore, for every s; t 2 S with s � t 6¼ y (hence jðsÞ � jðtÞ 6¼ y), we have

xjðs; tÞZðs � tÞs � t ¼ xjðs; tÞgðjðs � tÞÞ¼ gðxjðs; tÞjðsÞ � jðtÞÞ¼ gðjðsÞjðtÞÞ¼ gðjðsÞÞgðjðtÞÞ¼ ZðsÞsZðtÞt¼ ZðsÞZðtÞzðs; tÞs � t:

Therefore, xjz�1 ¼ @Z, and hence ½xj� ¼ ½z� 2 H2ðS;K�Þ, as desired. j

Consider now our action of AutS on H2ðS;K�Þ. For each ½x� 2H2ðS;K�Þ, let

½x�AutS ¼ f½xj� : j 2 AutSg

denote the orbit of ½x� under the action of AutS. Then from Theorem 2.7 wehave:

Theorem 2.8. Let S be an almost strongly acyclic locally square-free algebrasemigroup, and K a field. Then

½x�AutS 7! ½KxS�

is a bijection from the set of AutS orbits in H2ðS;K�Þ onto the set of iso-morphism classes of binomial K-algebras with associated semigroup S. j

BINOMIAL ALGEBRAS 1971

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

3. THE AUTOMORPHISM GROUP OF A BINOMIAL

ALGEBRA

Let A be a binomial K-algebra. By Theorem 2.2 and Lemma 1.1, wemay assume that A ¼ KxS, where S is an algebra semigroup and x is anormal cocycle in Z2ðS;K�Þ. Let E be the set of nonzero idempotents of S,let J ¼ SnE, and let J be denote the radical of A. In this section weinvestigate the outer automorphism group

OutA ¼ AutA=InnA

of A, where AutA is the group of algebra automorphisms of A, and InnA isthe subgroup of AutA consisting of inner automorphisms. For every unitu 2 A, we denote by au the inner automorphism of A determined by u; thatis, we set

auðxÞ ¼ u�1xu

for each x 2 A. Finally, we let 1S be the identity element of the group AutSof semigroup automorphisms of S.

For every Z 2 Z1ðS;K�Þ, define sZ : A �! A to be the unique K-linearmap with

sZðsÞ ¼ ZðsÞs

for every s 2 S�. Since Z is a one-dimensional cocycle, it easily follows thatsZ 2 AutA. Now define a map

~LL : Z1ðS;K�Þ �! AutA

by~LL : Z 7�! sZ:

It is straightforward to show that ~LL is a group homomorphism. We thenhave:

Theorem 3.1. There exists a monomorphism

L : H1ðS;K�Þ �! OutA

defined by

1972 SKLAR

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

L : ðB1ðS;K�ÞÞZ 7�! ðInnAÞsZ;

for every Z 2 Z1ðS;K�Þ.

Proof. It suffices to show that for Z 2 Z1ðS;K�Þ, then Z 2 B1ðS;K�Þ if andonly if sZ 2 InnA. The arguments of Lemma 2.2 in [1] show that ifZ 2 B1ðS;K�Þ, then sZ 2 InnA. Next, let sZ 2 InnA, with invertible elementu 2 A such that sZ ¼ au. We have

u ¼X

r2S�krr

and

u�1 ¼X

t2S�ltt

for some kr and lt 2 K. Let e; f 2 E, and let y 6¼ s 2 e � ðSnJ 2Þ � f. Then

ZðsÞs ¼ auðsÞ ¼X

t;r2S�ltkrtsr:

Now, for each pair t; r 2 S� with t 6¼ e or r 6¼ f, we have that tsr is in theK-linear subspace of A spanned by S� n fsg; hence, ZðsÞ ¼ lekf. In parti-cular, for every e 2 E we have 1 ¼ ZðeÞ ¼ leke, so that le ¼ k�1

e ; since Z is acocycle, it follows that ZðsÞ ¼ k�1

e kf for every y 6¼ s 2 e � S � f. Defining m :E �! K� by mðeÞ ¼ ke for each e 2 E, we have ZðsÞ ¼ mð f ÞmðeÞ�1 for everyy 6¼ s 2 e � S � f; hence, Z ¼ @m 2 B1ðS;K�Þ, as desired. j

We now assume that S is both almost strongly acyclic and locallysquare-free. Let Aut0 A be the subgroup of normal automorphisms of A:that is,

Aut0 A ¼ fg 2 AutAjgðEÞ ¼ Eg:

By Lemma 2.6, for every g 2 Aut0 A, there exists an automorphism g0 2AutS such that

gðsÞ 2 K�g0ðsÞ

for every s 2 S�.Now, it’s easy to see (using Lemma 2.4 of [1]) that given any r 2 AutA

there exists a normal automorphism g of A with ðInnAÞr ¼ ðInnAÞg.

BINOMIAL ALGEBRAS 1973

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

Moreover, if u is an invertible element of A with au 2 InnA \Aut0 A, thenauðEÞ ¼ E; but for every e 2 E, Ae ffi AauðeÞ, so, as A is basic, we must haveauðeÞ ¼ e for every e 2 E. Thus, if g; g0 2 Aut0 A with ðInnAÞg ¼ ðInnAÞg0,then we have gðeÞ ¼ g0ðeÞ for each e 2 E, and so g0 ¼ g00 on E. Since S islocally square-free, it follows that g0 and g00 are equal on A, and hence areequal on all of S. Therefore, we have a well-defined map

F : OutA �! AutS

sending

ðInnAÞg 7�! g0;

where each g is in Aut0 A. It is straightforward to verify that F is a grouphomomorphism.

Lemma 3.2. ImL ¼ KerF.

Proof. First, let B1ðS;K�ÞZ 2 H1ðS;K�Þ (where Z 2 Z1ðS;K�Þ). Then

FLðB1ðS;K�ÞZÞ ¼ FððInnAÞsZÞ ¼ ðsZÞ0;

which clearly equals 1S; therefore, we have ImL � KerF. Next, let g 2Aut0 A with ðInnAÞg 2 KerF. Then g0 ¼ 1S, so for each s 2 S�, there existsa unique element ZðsÞ 2 K� such that gðsÞ ¼ ZðsÞs. It is straightforward toshow that the resulting map Z : S� �! K� is a cocycle, so that ðInnAÞG ¼ðInnAÞsZ ¼ LððB1ðS;K�ÞZÞ 2 ImL. j

We now turn to the question of finding ImF in AutS. Recall ourpreviously defined action of AutS on H2ðS;K�Þ. If ½o� 2 H2ðS;K�Þ, wedenote its stabilizer in AutS under this action by

StaboðAutSÞ ¼ fj 2 AutSj½oj� ¼ ½o�g:

Further, given any Z : S� �! K� and j 2 AutS, let

sZj : A �! A

be the (necessarily bijective) K-linear map with sZjðsÞ ¼ ZðsÞjðsÞ for everys 2 S�.

Then we have:

1974 SKLAR

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

Lemma 3.3. ImF ¼ StabxðAutSÞ.

Proof. Given j 2 AutS and Z : S� �! K�, straightforward computationyields that xðxjÞ�1 ¼ @Z if and only if sZj 2 AutA. If j 2 StabxðAutSÞ,then ½x� ¼ ½xj� 2 H2ðS;K�Þ, so there exists some map Z : S� �! K� withxðxjÞ�1 ¼ @Z, and hence with sZj 2 AutA. Since for every e 2 E

ZðeÞ ¼ ZðeÞZðe � eÞ�1ZðeÞ ¼ ð@ZÞðe; eÞ ¼ xðe; eÞxjðe; eÞ�1 ¼ 1;

we have sZj 2 Aut0A, and so j ¼ FððInnAÞsZjÞ. Conversely, if g 2 Aut0 Awith j ¼ FððInnAÞgÞ, then there exists some Z : S� �! K� withsZj ¼ g 2 Aut0 A, so that ½x� ¼ ½xj� 2 H2ðS;K�Þ. j

Summarizing the contents of this section, we have our main theorem.

Theorem 3.4. Let A be an almost strongly acyclic binomial K-algebra withassociated semigroup S, and let x 2 Z2ðS;K�Þ so that A ffi KxS. If the leftquiver of A has no multiple arrows, then the following sequence is exact:

1 �! H1ðS;K�Þ �!L OutA �!F StabxðAutSÞ �! 1:

If ½x� ¼ 1, then StabxðAutSÞ ¼ AutS, and so the following sequence is exact:

1 �! H1ðS;K�Þ �!L OutA �!F AutS �! 1:

Moreover, this last sequence splits, and so in this case OutA is isomorphic to asemidirect product of H1ðS;K�Þ by AutS.

Proof. By Lemma 1.1, we may assume that x is normal; all but the laststatement of the theorem then follows from the preceding remarks. Next,suppose that we have ½x� ¼ 1; without loss of generality, we may assume thatx ¼ 1. For each j 2 AutS, let s1j be the unique K-linear map from A to Asending s to jðsÞ for every s 2 S�. Easily, each s1j is a normal auto-morphism of A. Now define

C : AutS! OutA

by

C : j 7! ðInnAÞs1j

for every j 2 AutS. As in the proof of Theorem 2.7 in [1], C is a homo-morphism of groups with FC ¼ 1AutS; so the sequence splits, and henceOutA ffi H1ðS;K�Þ �j AutS: j

BINOMIAL ALGEBRAS 1975

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

We note that in the case of a locally square-free binomial algebra A,the shorter exact sequence

1 �! H1ðS;K�Þ �!L OutA �!F AutS

is (essentially) a cohomological version of an exact sequence derived bySaorın (see Theorem 7 in [10]) using the notion of changes of variables. Wefurther note that several questions about a possible generalization of theabove theorem, or of Theorem 7 in [10], remain. In particular, it is unknownwhether one can drop the hypotheses of these theorems that A is almoststrongly acyclic. The question of whether the short exact sequence of The-orem 3.4 splits for arbitrary x 2 Z2ðS;K�Þ is also unresolved.

The following two corollaries of Theorem 3.4 are immediate.

Corollary 3.5. Let A ffi KxS be an almost strongly acyclic binomial K-algebra whose left quiver has no multiple arrows. If H1ðS;K�Þ ¼ 1, then

OutA ffi StabxðAutSÞ:

If we further have that A ffi KS is untwisted, then

OutA ffi AutS: j

Corollary 3.6. Let A ffi KxS be an almost strongly acyclic binomial K-algebra whose left quiver has no multiple arrows. Then every automorphism ofA is inner if and only if

H1ðS;K�Þ ¼ StabxðAutSÞ ¼ 1: j

We conclude with three examples of binomial algebras and their outerautomorphism groups.

Example 3.7 Let K be a field, and let G be the left quiver

with path semigroup S. Let A be any binomial algebra with associatedsemigroup S. Straightforward calculation yields

1976 SKLAR

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

H1ðS;K�Þ ffi K� � K� and H2ðS;K�Þ ¼ 1;

therefore, A is of necessity isomorphic to the untwisted K-algebra KS. Since,moreover, we clearly have

AutS ffi Z2 �Z2;

we conclude that

OutA ffi ðK� � K�Þ �j ðZ2 �Z2Þ: j

Example 3.8 Let K be a field, and let G be the quiver in Example 3.7, withpath semigroup P. Let S be the semigroup induced from P by identifying allpaths sharing common initial and terminal vertices. (In other words, let S bethe natural semigroup of the poset X ¼ fe1; e2; . . . ; e5g with Hassediagram G.) In this case, we have

H1ðS;K�Þ ¼ 1 ¼ H2ðS;K�Þ;

therefore, for any binomial algebra A with associated semigroup S we have

OutA ffi AutS ffi Z2 �Z2: j

Example 3.9. Let K be a field with char K 6¼ 2, and let G be the left quiver

with path semigroup P. Let S be the semigroup induced from P by identi-fying all paths sharing common initial and terminal vertices. Then, as in [1],we have

H1ðS;K�Þ ¼ 1 and H2ðS;K�Þ ffi K�:

(See also [2].)Now let A ffi KxS for some x 2 Z2ðS;K�Þ. If ½x� 6¼ 1 in H2ðS;K�Þ, then,

as Anderson and D’Ambrosia show in [1], we have

BINOMIAL ALGEBRAS 1977

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014

StabxðAutSÞ ffi Z2 �Z2;

if, on the other hand, ½x� ¼ 1, then we have

StabxðAutSÞ ¼ AutS ffi Z2 �Z2 �Z2:

Since H1ðS;K�Þ ¼ 1, we therefore conclude that

OutA ffi Z2 �Z2 �Z2; if ½x� ¼ 1;Z2 �Z2; otherwise:

�j

REFERENCES

1. Anderson, F.W.; D’Ambrosia, B.K. Square-free Algebras and theirAutomorphism Groups. Comm. Alg. 1996, 24 (10), 316373191.

2. Clark, W.E. Cohomology of Semigroups via Topology 7 with anApplication to Semigroup Algebras. Comm. Alg. 1976, 4, 9797997.

3. Coelho, S.P. The Automorphism Group of a Structural MatrixAlgebra. Linear Algebra Appl. 1993, 195, 35758.

4. Fuller, K.R. Algebras from Diagrams. J. Pure Appl. Algebra 1987, 48,23737.

5. Fuller, K.R.; Nicholson, W.K.; Watters, J.F. Algebras whose projec-tive modules are reflexive. J. Pure Appl. Algebra 1995, 98, 1357150.

6. Gabriel, P. Auslander-Reiten sequences and representation-finitealgebras. Springer-Verlag LNM 1979, 831, 1771.

7. Guil-Asensio, F.; Saorın, M. The automorphism group and the Picardgroup of a monomial algebra. Comm. Alg. 1999, 27 (2), 8577887.

8. Guil-Asensio, F.; Saorın, M. The group of outer automorphisms andthe Picard group of an algebra. Preprint.

9. Jianbo, D. On finite dimensional algebras isomorphic to monomialalgebras. Ph.D. Thesis, U. of Ottawa, 1997.

10. Saorın, M. Isomorphisms between representations of algebras. Publ.Mat. 1992, 36, 9557964.

11. Stanley, R.P. Structure of incidence algebras and their automorphismgroups. Bull. Amer. Math. Soc. 1970, 76, 123671239.

Received October 2000Revised May 2001

1978 SKLAR

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ew M

exic

o] a

t 07:

06 0

3 O

ctob

er 2

014