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Structure Theorems for Basic Algebras
A basic finite dimensional algebra over an algebraically closed field is
isomorphic to a quotient of a tensor algebra by an admissible ideal. The
category of left modules over the algebra is isomorphic to the category of
representations of a finite quiver with relations. In this article we will
remove the assumption that is algebraically closed to look at both perfect
and non-perfect fields. We will introduce the notion of species with relations
to describe the category of left modules over such algebras. If the field is
not perfect, then the algebra is isomorphic to a quotient of a tensor algebra
by an ideal that is no longer admissible in general. This gives hereditary
algebras isomorphic to a quotient of a tensor algebra by a non-zero ideal. We
will show that these non-zero ideals correspond to cyclic subgraphs of the
graph associated to the species of the algebra. This will lead to the ideal
being zero in the case when the underlying graph of the algebra is a tree
On the lower central series of an associative algebra
This paper continues the study of the lower central series quotients of an
associative algebra A, regarded as a Lie algebra, which was started in
math/0610410 by Feigin and Shoikhet. Namely, it provides a basis for the second
quotient in the case when A is the free algebra in n generators (note that the
Hilbert series of this quotient was determined earlier in math/0610410).
Further, it uses this basis to determine the structure of the second quotient
in the case when A is the free algebra modulo the relations saying that the
generators have given nilpotency orders. Finally, it determines the structure
of the third and fourth quotient in the case of 2 generators, confirming an
answer conjectured in math/0610410. Finally, in the appendix, the results of
math/0610410 are generalized to the case when A is an arbitrary associative
algebra (under certain conditions on ).Comment: 23 pages, latex, corrected some more typo
PBW-deformations of N-Koszul algebras
For a quotient algebra of the tensor algebra we give explicit conditions
on its relations for being a PBW-deformation of an -Koszul algebra .
We show there is a one-one correspondence between such deformations and a class
of -structures on the Yoneda algebra of . We
compute the PBW-deformations of the algebra whose relations are the
anti-symmetrizers of degree and also of cubic Artin-Schelter algebras.Comment: 35 pages. Some minor correction
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Free resolutions of algebras
Given an algebra A, presented by generators and relations, i.e. as a quotient of a tensor algebra by an ideal, we construct a free algebra resolution of A, i.e. a differential graded algebra which is quasi-isomorphic to A and which is itself a tensor algebra. The construction rests combinatorially on the set of bracketings that arise naturally in the description of a free contractible differential graded algebra with given generators
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