Brad Shelton

KOSZUL ALGEBRAS FROM GRAPHS AND HYPERPLANE ARRANGEMENTS

This work was started as an attempt to apply theory from noncommutative graded algebra to questions about the holonomy algebra of a hyperplane arrangement. We soon realized that these algebras and their deformations, which form a class of quadratic graded algebras, have not been studied much and yet are interesting to algebra, arrangement theory and combinatorics.

Embedding a quantum rank three quadric in a quantum P3

We continue the classification, begun in [11], [14] and [12], of quadratic Artin-Schelter regular algebras of global dimension 4 which map onto a twisted homogeneous coordinate ring of a quadric hypersurfcice in P3. In this paper, we consider those cases where the quadric has rank 3. We also give sufficient conditions for the point scheme of any quadratic regular algebra of global dimension 4 to be the graph of an automorphism.

Schemes of line modules I

It is proved that there exists a scheme that represents the functor of line modules over a graded algebra, and it is called the line scheme of the algebra. Its properties and its relationship to the point scheme are studied. If the line scheme of a quadratic, Auslander-regular algebra of global dimension 4 has dimension 1, then it determines the defining relations of the algebra. Moreover, the following counter-intuitive result is proved. If the zero locus of the defining relations of a quadratic (not necessarily regular) algebra on four generators with six defining relations is finite, then it determines the defining relations of the algebra. Although this result is non-commutative in nature, its proof uses only commutative theory. The structure of the line scheme and the point scheme of a 4-dimensional regular algebra is also used to determine basic incidence relations between line modules and point modules.

PBW-deformation theory and regular central extensions

Abstract A deformation U, of a graded K-algebra A is said to be of PBW type if gr U is A. It has been shown for Koszul and N-Koszul algebras that the deformation is PBW if and only if the relations of U satisfy a Jacobi type condition. In particular, for these algebras the determination of the PBW property is a finite and explicitly determined linear algebra problem. We extend these results to an arbitrary graded K-algebra, using the notion of central extensions of algebras and a homological constant attached to A which we call the complexity of A.

Some quantum P3s with one point

We study a certain 1-parameter family of non-commutative graded regular algebras of global dimension four which were introduced by Vancliff, Van Rompay and Willaert in [12]. Most members of the family have a singleton point scheme of multiplicity twenty. Our objective is to analyse the point scheme of these algebras and the coordinate ring of the point scheme. In particular, we prove that the point scheme determines the defining relations and that its coordinate ring is a Frobenius algebra.

Schemes of line modules. II

ABSTRACT In this sequel to[1], we study the scheme of line modules for several classes of quantum s, including Clifford algebras, homogenized and algebras associated to smooth quadrics in . We also prove that a quantum with enough symmetry in its defining relations has a line scheme of dimension at least two, with infinitely many line modules incident to any point module.

NONCOMMUTATIVE KOSZUL ALGEBRAS FROM COMBINATORIAL TOPOLOGY

Abstract Associated to any uniform finite layered graph Γ there is a noncommutative graded quadratic algebra A(Γ) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW-complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology groups HX (n, k), generalizing the usual cohomology groups Hn (X). Along with several other results, our methods give a new and primarily topological proof of the main result of [Serconek and Wilson, J. Algebra 278: 473–493, 2004] and [Piontkovski, J. Alg. Comput. 15, 643–648, 2005].

Extensions between generalized Verma modules: The Hermitian symmetric cases

1.1. Let g be a semisimple complex Lie algebra with Cartan subalgebra _h and let b be a Borel subalgebra of g with nilradical n and Levi component _h. Let p(p~_b) be a parabolic subalgebra of g with nilradical _u(_u_~n) and Levi component _m(_m_ 0). The problem of computing the u-cohomology of irreducible highest weight (g, p)-modules is completely solved by the Kazhdan-Lusztig conjectures. On the other hand, the u-cohomology of the (g, p)-generalized Verma modules is a completely open question, even in the case when _/2 =-b. In this paper we give a solution to this problem for regular integral character in the cases when (g, p) corresponds to an indecomposable Hermitian symmetric pair. Our solution is given as a set of recursion relations for the dimensions of the spaces of extensions between generalized Verma modules.

The Yoneda Algebra of a 𝒦2 Algebra Need not be Another 𝒦2 Algebra

The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. 𝒦2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a 𝒦2 algebra would be another 𝒦2 algebra. We show that this is not necessarily the case by constructing a monomial 𝒦2 algebra for which the corresponding Yoneda algebra is not 𝒦2.