3. The Radical
3.6 Diagram of an Algebra
60 3. The Radical
(up to an isomorphism) uniquely and is isomorphic to the basic algebra of the algebra A.
Let us remark that the module P is, in general, not uniquely determined (see Exercise 16 of this chapter).
3.6 Diagram of an Algebra 61 In general, by a diagram V we shall understand an arbitrary finite set of points together with arrows between them. Usually, the points will be denoted by the numbers 1,2, ... ,s. Then, the diagram is given by its incidence matrix
where tij is the number of arrows from the point i to the point j. If an arrow CT of the diagram V joins the point i with the point j, then i is called the tail ( origin) and j the head (top) of the arrow CT. This fact will be recorded as follows: CT : i ~ j.
Two diagrams VI and V2 are called isomorphic if there is a bijective correspondence between their points and arrows such that the tails and the heads of the corresponding arrows map one to the other. It is not difficult to see that VI ~ V 2 if and only if the incidence matrix [VI] can be transformed into the incidence matrix [V2 ] by simultaneous permutations of the rows and columns. In particular, the diagram of an algebra is determined uniquely up to an isomorphism.
A path of a diagram V is an ordered sequence of arrows {CTl' CT2, ... ,CTk}
such that the head of the arrow CT/ coincides with the tail of the arrow CT£+1 (£
=
1,2, ... , k - 1). The number of the arrows k is called the length of the path. The tail of the arrow CTI is called the tail of the path and the head of the arrow CTk the head of the path. We shall say that the path connects the point i with the point j and write CTI CT2 ... CTk : i ~ j.We shall assume that the algebra A is basic. Then A ~ Dl X D2 x ... x Ds ,
where Di = EA(Ui) and Ui can be considered as the regular Di-module. Let 1 =
el + e2 + ... + e
s be the decomposition of the identity of the algebraA
such that eiA ~ Di and 1
=
el+
e2+ ... +
es the corresponding decomposition of the identity of the algebra A (see Corollary 3.3.9). In this case, Pi=
eiA, Ri=
eiR and Vi=
ei V, where V=
R/ R2. Write Vij=
ei Vej . Now, Vij is a right Dj-module and as such Vij ~ iijUj. Thus, tLDj(Vij)=
tij. Let us choose in Vij a generating set of tij elements and index them by the arrows of the diagram V(A) which point from i to j (their number is also tij). Let v" be the generator corresponding to the arrow CT : i ~ j and r" its preimage in Rij = eiRej . The set of all elements {v,,} (over all arrows of the diagram V) is a generating set of the module V. By Nakayama's lemma (Corollary 3.1.5), {r,,} is a generating set of R (as a right module). Note that if CT: i ~ j, then eir"=
r"ej=
r" and 1',,1'r=
0 if the head of the arrow CT does not coincide with the tail of ToLemma 3.6.1. If the1'e is no path in the diagram V(A) which connects the point i with the point j (i
i=
j), then HomA(Pj,Pi)=
O. If the al- gebra A is basic, then every element r E Rij can be represented in the form r =L
r",1' "2 ••. 1'''k a", "2 ... "k , where the summation runs over all paths CTICT2 ... CTk : i ~ j and a"'''2'''''k E Ajj with Ajj=
ejAej.62 3. The R ... dical
Proof. By Lemma 3.5.5, the algebra A can be assumed to be basic and there- fore, by Theorem 3.5.3, HomA(Pj, Pi) ~ eiAej = Rij (for i =1= j). Therefore, it suffices to prove only the second assertion. The considerations introduced above show that if l' E Rij, then l' ==
L:
rtTatT (modR2), where atT E Ajj and the summation runs over all arrows (J" : i ~ j. Then the element 1" : l' -L:
rtTatT belongs to eiR2ej. However, R= L:
Rij and therefore ei R2ej= L:
RikRkj,i j k
i.e. 1"
=
L:XkYk, where Xk ERik, Yk E Rkj. Again, Xk == L:rrar (modR2),k r
where T : i ~ k, ar E Au and arYk == L:rparp (modR2 ), where p: k ~ j,
p
arp E Ajj . Therefore, 1" ==
L:
rrrparp (modR3), where TP: i ~ j. Continuing this process and taking into account that the radical is nilpotent, we obtain the required expression for r.Let us remark here that HomA(Pj, Pi) = 0 is possible even if there is a
path from i to j (see Exercise 12). 0
A diagram D is called connected if it cannot be divided into two non-empty disjoint subsets which are not connected by any arrows.
Theorem 3.6.2. An algebra A is a non-trivial direct product if an only if the diagram D(A) is disconnected.
Proof. Let the diagram D = D( A) be disconnected: D
=
DI U D2 , DIn
D2=
0,
DI =1=0,
D2 =1=0
and there are no arrows between the points of DI and D2. Thus, by Lemma 3.6.1, if i E DI , j E D2, then HomA(Pi, Pj) = 0 and HomA(Pj,Pi)=
O. By Corollary 1.7.9, the algebra A is d~composable. Con- versely, if A decomposes, A = Al X A2 , then, for any principal AI-module Pi and any principal A2-module Pj, HomA(Pi,Pj)=
0 and HomA(Pj,Pi)=
0;it follows that the points i and j are not connected and the diagram D( A) is
disconnected. 0
Corollary 3.6.3. Algebras A and AI R2 are either both decomposable or both indecomposable.
In addition to the diagram D(A), an algebra A has a number of important invariants. First of all, such are the division algebras Di
=
EA(Ui) and the multiplicities ni of the Pi in the decomposition of the regular module. For a basic algebra, all ni=
1, but the division algebras can be arbitrary. If the field K is algebraically closed then the situation simplifies significantly: All Di coincide with the ground field K.An algebra A over the field K is called split if AI R ~ Mn1 (K) x Mn2 (K) x ... x Mn , (K). All algebras over algebraically closed fields are split.
For split algebras, Lemma 3.6.1 can be strengthened.
Lemma 3.6.4. Let A be a basic split algebra. Then every element l' E Rij can be rep res ented in the form l'
= L:
r <71 1'<72 ... r <7k C<71 <72 ••• <7k , where C<71 <72 ••• tTk E K3.6 Diagram of an Algebra 63 and the summation runs through all paths O"} 0"2 ••• O"k : i -t j (here, possibly, i =j).
Proof. The proof of Lemma 3.6.1 can be repeated word by word. Observe that in this case Ajj j Rjj = K, i. e. every element of the algebra Ajj is of the form
C
+
x, where C E K, x E Rjj. 0A cycle of the diagram D is a path whose tail coincides with its head.
Corollary 3.6.5. If there is no cycle in the diagram of a basic split algebra A, then EA(P) = K for each principal A-module P.
Lemma 3.6.4 enables us to construct for every diagram 1J a K -algebra K(1J), in general infinite dimensional, such that every basic split algebra with the given diagram D is its quotient algebra.
A basis of the space K(D) is formed by all possible paths of the di- agram and by the symbols {cd (indexed by the points of D). In this way, every element of K(D) can be uniquely written in the form
L:
8 Cic;+
L:
CO"l0"2 ... O"k O"} 0"2 ••• O"k (the second sum runs over all paths of the diagram D), ;=}where Ci E K, CO"l0"2 ••• O"k E K. It will be convenient to interpret the symbol C;
as the path of length 0 with its head and tail at the point i.
Define the product of the paths a and
/3
as the path0/3
if the head of a coincides with the tail of/3,
and as 0 otherwise. In other words,if the head of O"k coin- cides with the tail of 7},
otherwise;
if i is the tail of O"},
otherwise;
if i is the head of O"k, otherwise;
if i
=
j,if i '" j.
Extend this definition to the whole space K(D) "by linearity" putting
(L: c",a) (L: c'p/3) = L: c",c'p( 0/3),
where0, /3
are paths of the diagram D and '" p "',pC"" c'p elements of the field K. A trivial verification shows that in this way K(D) becomes an algebra over the field K with the identity 1
=
C}+
C2+
... +cs·
Denote by J the set of those elements of the algebra K(D) whose coeffi- cients of ci are equal to 0 for all i. Evidently, J is an ideal of K(D). An ideal I C K(D) is called admissible if J2 :> I :> In for some n ? 2.
Theorem 3.6.6. For any admissible ideal I C K(D), the quotient algebra K(1J)j I is a split basic algebra with the diagram D. Conversely, every split
64 3. The Radical
basic algebra with a diagram D is isomorphic to a quotient algebra of the algebra K(D) by an admissible ideal I.
Proof. Let An = K(D)I In+I (n ~ 1). The classes Q = ll'
+
In+I, wherell' is an arbitrary path of the diagram D of length smaller or equal than n, form a basis of the algebra An. The ideal
J
= J I In+I of .4.n is nilpotent and AniJ ::::
K(D)I J :::: Ie (a basis of this algebra is formed by the classes ti = ei+
J; moreover eiej = Oije;). By Proposition 3.1.13,J
= radAn. In this way, An is a basic split algebra and AnlJ2 :::: AI, i.e. D(An) = D(Ad.In the algebra AI, tdtj is a vector space over K with a basis {a}, where (7 are the arrows from i to j. Therefore, if [D]
=
(tij) is the incidence matrix of the diagram D, then tdtj :::: tijUj, where Uj = tjAI/tjJ and D(Ad = D.From here we get the first statement of the theorem (taking into account Corollary 3.1.14).
Now, let A be an arbitrary basic split algebra with the diagram D, 1 = el
+
e2
+ ... +
es be a decomposition of the identity into minimal idempotents and {r u} a generating set of the radical constructed earlier (before Lemma 3.6.1).For every path ll' = (71(72 ••• (7k of the diagram D, we write ra =
rU,rU2 ••• rUk ' re; = ei and put I(Lcall') = LCaTa. The relations between
a a
Tu and ei imply that
I
is a homomorphism of the algebra K(D) into the algebra A, and that, in view of Lemma 3.6.4, it is an epimorphism. There- fore, A :::: K(D)I I with I = KerI.
One can see easily thatI(
J) = R, where R = rad A. Since Rn = 0 for some n, Inc
I. Finally, the elementsVu
=
Tu+
R2 are linearly independent in RI R2 and therefore the homomor- phism Al ---+ AI R2 is an isomorphism; hence I C J2. The proof of the theoremis completed. 0
The algebra K(D) is called the path algebra or the universal algebra of the diagram D.
Of course, a similar construction can be performed for the left diagrams.
However, it turns out that the following proposition holds.
Proposition 3.6.7. If A is a split algebra, then the (left) diagram D'(A) can be obtained from the diagram D(A) by reversing all arrows, OT by transposition of the incidence matrix.
Proof. The algebra A can be assumed to be basic. Then, as we have already seen, [D(A)] = (tij), where iij is the dimension of the space eiVej (here, V