Semisimple representations of quivers

Abstract

Abstract.We discuss the invariant theory of the variety of representations of a quiver and present generators and relations.We connect this theory of algebras with a trace satisfying a formal CayleyHamilton identity

Cohomology of Noncommutative Hilbert Schemes

“…(see [16]) and the theory of nullcones of quiver representations [15]. This is worked out in [18] and allows to derive, for example, a determination of the irreducible components of the fibres and their dimensions.…”

Isolated Singularities, Smooth Orders, and Auslander Regularity

“…Assume v is a vertex having maximal α v ≥ 2. Because (Q, α) is reduced it follows from the definition of reduction step R w I that for all vertices w we have χ Q (ǫ w , α) < 0 and χ Q (α, ǫ w ) < 0 Therefore, by [14] we have that α − ǫ v is the dimension vector of a simple representation of Q and we look at the local quiver setting (Q τ , α τ ) for the representation type τ = (1, ǫ v ; 1, α − ǫ v ). This is of the form…”

“…To start, ǫ is the dimension vector of a simple representation of Q. By the results of [14] this implies that Q is a strongly connected quiver (any pair of vertices v i , v j is connected by an oriented path in Q starting at v i and ending in v j ) and that the dimension vector ǫ satisfies the numerical conditions…”

“…where the sum is taken over all vertices w j having loops in Q τ , see [14]. If we apply this to the representation type (α 1 , ǫ 1 ; .…”

“…We give two proofs of this result. By [14] the coordinate ring of a quotient variety iss ǫ Q is generated by traces along oriented cycles in the quiver Q. In the case of the left hand quiver settings, these invariants are easy to determine : the dimension of the power m i of the maximal graded ideal m i /m i+1 is equal to…”