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March 3, 2016
REPETITIVE ALGEBRAS AND TRIVIAL EXTENSION
ALGEBRAS
JAN SCHRÖER
Contents
1. Repetitive algebras
2. Repetitive algebras and derived categories
3. Trivial extension algebras
References
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2
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Let K be a field, and let A be a finite-dimensional K-algebra.
1. Repetitive algebras
b of A is defined as
The underlying vector space of the repetitive algebra A


.. ..
.
.




A A∗


∗


A A


∗
b


A A
A := 

A A∗




.

. .


..
.
b are infinite matrices
where A∗ := D(A) := HomK (A, K). Thus the elements in A
M = (mij )ij with rows and columns indexed by Z with only finitely many non-zero
entries. The entries on the diagonal are in A, the entries on the upper off diagonal
are in A∗ , and all other entries are 0. We can identify such an element (mij )ij with
the tuple ((ai , bi ))i where ai = mii and bi := mi,i+1 .
Note that A∗ is an A-A-bimodule in the obvious way, thus for a, a0 and b ∈ A∗ the
b is induced by the
expressions ab and ba0 are elements in A∗ . The multiplication in A
0
usual matrix multiplication with the additional rule that bb := 0 for all b, b0 ∈ A∗ .
b we define
More explicitely, for ((ai , bi ))i and ((a0i , b0i ))i in A
((ai , bi ))i · ((a0i , b0i ))i := ((ci , di ))i .
with ci := ai a0i and di := ai b0i + bi a0i+1 .
Suppose A is defined by a quiver with relations. Then one can often also explicitely
b by a quiver with relations, see for example [Sch] which deals with path
describe A
algebras modulo admissible ideals generated by zero relations and commutativity
relations.
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JAN SCHRÖER
2. Repetitive algebras and derived categories
b is infinite-dimensional provided A 6= 0. The repetitive algebra A
b
Obviously, A
is a selfinjective algebra. The indecomposable projective-injective A-modules are
b is a triangulated category, see [H].
finite-dimensional. The stable category mod(A)
Happel [H] constructed a functor
b
F : Db (mod(A)) → mod(A)
of triangulated categories. We refer to [BM] for a detailed explanantion of the
construction of F .
Theorem 2.1 (Happel [H]). The functor F is full and faithful. It is an equivalence
if and only if gl. dim(A) < ∞.
3. Trivial extension algebras
The trivial extension algebra T (A) of A has as an underyling vector space
T (A) := A ⊕ A∗ .
The multiplication is defined by
(a, b) · (a0 , b0 ) := (aa0 , ab0 + ba0 ).
Trivial extension algebras are symmetric algebras.
The algebra T (A) is Z-graded with deg(A) := 0 and deg(A∗ ) := 1. Let modZ (T (A))
be the category of finite-dimensional Z-graded T (A)-modules.
b and modZ (T (A)) are equivProposition 3.1 (Happel [H]). The categories mod(A)
alent.
References
[BM] M. Barot, O. Mendoza, An explicit construction for the Happel functor, Colloq. Math. 104
(2006), no. 1, 141–149.
[H] D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, 119. Cambridge University Press,
Cambridge, 1988. x+208pp.
[Sch] J. Schröer, On the quiver with relations of a repetitive algebra, Arch. Math. (Basel) 72 (1999),
no. 6, 426–432.
Jan Schröer
Mathematisches Institut
Universität Bonn
Endenicher Allee 60
53115 Bonn
Germany
E-mail address: [email protected]