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 -