Natalia Iyudu

Representation Spaces of the Jordan Plane

We investigate relations between the properties of an algebra and its varieties of finite-dimensional module structures, on the example of the Jordan plane R = k ⟨ x, y ⟩ /(xy − yx − y 2). A complete description of irreducible components of the representation variety mod(R, n) is obtained for any dimension n, it is shown that the representation variety is equidimensional. We investigate the influence of the property of the noncommutative Koszul (or Golod–Shafarevich) complex to be a DG-algebra resolution of an algebra, on the structure of representation spaces. It is shown that the Jordan plane provides a new example of representational complete intersection (RCI), which is not a preprojective algebra.

The Golod-Shafarevich inequality for Hilbert series of quadratic algebras and the Anick conjecture

We study the question on whether the famous Golod-Shafarevich estimate, which gives a lower bound for the Hilbert series of a (noncommutative) algebra, is attained. This question was considered by Anick in his 1983 paper ’Generic algebras and CWcomplexes’, Princeton Univ. Press., where he proved that the estimate is attained for the number of quadratic relations d 6 n 2 4 and d > n 2 2 , and conjectured that it is the case for any number of quadratic relations. The particular point where the number of relations is equal to n(n 1) 2 was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional. We prove that over any infinite field, the Anick conjecture holds for d > 4(n 2 +n) 9 and arbitrary number of generators, and confirm the Vershik conjecture over any field of characteristic 0. We give also a series of related asymptotic results.

Non-trivial stably free modules over crossed products

We consider the class of crossed products of Noetherian domains with universal enveloping algebras of Lie algebras. For algebras from this class we give a sufficient condition for the existence of projective non-free modules. This class includes Weyl algebras and universal envelopings of Lie algebras, for which this question, known as the non-commutative Serres problem, has been extensively studied previously. It turns out that the method of lifting of non-trivial stably free modules from simple Ore extensions can be applied to crossed products after an appropriate choice of filtration. The motivating examples of crossed products are provided by the class of relativistic internal time algebras, originating in non-equilibrium physics.

Asymptotically optimal k-step nilpotency of quadratic algebras and the Fibonacci numbers

It follows from the Golod-Shafarevich theorem that if k ∈ N and R is an associative algebra given by n generators and

Three dimensional Sklyanin algebras and Gröbner bases

d< \frac{{{n^2}}}{4}{\cos ^{ - 2}}\left( {\frac{\pi }{{k + 1}}} \right)

On Koszulity for operads of Conformal Field Theory

d<n24cos−2(πk+1) quadratic relations, then R is not k-step nilpotent. We show that the above estimate is asymptotically optimal. Namely, for every k ∈ N, there is a sequence of algebras Rn given by n generators and dn quadratic relations such that Rn is k-step nilpotent and

Minimal Hilbert series for quadratic algebras and the Anick conjecture

\mathop {\lim }\limits_{n \to \infty } \frac{{{d_n}}}{{{n^2}}} = \frac{1}{4}{\cos ^{ - 2}}\left( {\frac{\pi }{{k + 1}}} \right)