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- W2094089382 abstract "A quiver Q is a finite oriented graph which can contain more than one arrow between two vertices, as well as loops and oriented cycles. For n a positive integer, Q, is the set of oriented paths of length n of Q, where the length is the number of arrows of the oriented path. Notice that Q, is the set of vertices and Q, the set of arrows. Let k be a field and kQo = x St po ks be the commutative semi-simple algebra with Q0 as a k-basis of idempotents. For each arrow a E Q, with source vertex s(a) = s and end vertex r(u) = t, the one dimensional vector space k, has an evident kQ,-bimodule structure: for u E Q,, we have au=6,,sa and ua=d,.,a, where 6 is the Kronecker symbol. In this way kQ, = @,, o, ka is a kQ,-bimodule and the quiver algebra kQ is the tensor algebra over kQo of the kQ,-bimodule kQt. Of course, kQ can also be described as the vector space kQo@ kQl 0 kQ, 0 ... where the multiplication of /?E Qj and CI E Qi is /3~ E Qj+, if t(a) = s(p) and 0 otherwise. Let now n be a finite dimensional k-algebra. We suppose that /i is Morita reduced and that the endomorphism ring of each simple n-module is k. This is equivalent to A/r = k x . . . x k, were r is the Jacobson radical of /i. By definition, the set of vertices of the Gabriel’s quiver Q of /i is the set of isomorphism classes of simple /i-modules. If S and T are simple n-modules, the number of arrows from S to T is dim,Exti(S, T). By an observation of Gabriel [6; 7,4.3] every k-algebra n such that A/r = k x . .. x k admits a presentation, that is, an algebra surjection cp: kQ,, -+ A whose kernel Z verifies F” c Zc F2, where F is the two-sided ideal of kQ,, generated by Ql and m is some positive integer. Such ideals Z of a quiver algebra are called admissible. In general an algebra n has not a unique presentation, that is, two different admissible two-sided ideals Z and J of a quiver algebra kQ can give isomorphic k-algebras kQ/Z and kQ/J By definition, we say that a k-algebra /i is a truncated quiver algebra if" @default.
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- W2094089382 date "1990-01-01" @default.
- W2094089382 modified "2023-09-25" @default.
- W2094089382 title "Rigidity of truncated quiver algebras" @default.
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- W2094089382 doi "https://doi.org/10.1016/0001-8708(90)90057-t" @default.
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