TY - JOUR
T1 - Full quivers of representations of algebras
AU - Belov-Kanel, Alexei
AU - Rowen, Louis H.
AU - Vishne, Uzi
PY - 2012
Y1 - 2012
N2 - We introduce the notion of the full quiver of a representation of an algebra, which is a cover of the (classical) quiver, but which captures properties of the representation itself. Gluing of vertices and of arrows enables one to study subtle combinatorial aspects of algebras which are lost in the classical quiver. Full quivers of representations apply especially well to Zariski closed algebras, which have properties very like those of finite dimensional algebras over fields. By choosing the representation appropriately, one can restrict the gluing to two main types: Frobenius (along the diagonal) and, more generally, proportional Frobenius gluing (above the diagonal), and our main result is that any representable algebra has a faithful representation described completely by such a full quiver. Further reductions are considered, which bear on the polynomial identities.
AB - We introduce the notion of the full quiver of a representation of an algebra, which is a cover of the (classical) quiver, but which captures properties of the representation itself. Gluing of vertices and of arrows enables one to study subtle combinatorial aspects of algebras which are lost in the classical quiver. Full quivers of representations apply especially well to Zariski closed algebras, which have properties very like those of finite dimensional algebras over fields. By choosing the representation appropriately, one can restrict the gluing to two main types: Frobenius (along the diagonal) and, more generally, proportional Frobenius gluing (above the diagonal), and our main result is that any representable algebra has a faithful representation described completely by such a full quiver. Further reductions are considered, which bear on the polynomial identities.
UR - http://www.scopus.com/inward/record.url?scp=84862889415&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-2012-05565-6
DO - 10.1090/S0002-9947-2012-05565-6
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AN - SCOPUS:84862889415
SN - 0002-9947
VL - 364
SP - 5525
EP - 5569
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 10
ER -