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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 1 2 2 2 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. 1 2 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]