Related papers: Projective modules and Gr\"obner bases for skew PB…
From symplectic reflection algebras, some algebras are naturally introduced. We show that these algebras are non-homogeneous N-Koszul algebras, through a PBW theorem.
We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensaetze) from real algebraic geometry can be generalized to noncommutative $*$-algebras. A…
This paper is lead by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, R of a p-adic analytic group G. For G without any p-torsion element we…
Lie conformal algebras $\mathcal{W}(a,b)$ are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we give a complete classification of extensions of finite irreducible…
Let $U$ be either classical or quantized universal enveloping algebra of $\s\l(n+1)$ extended over the field of fractions of the Cartan subalgebra. We suggest a PBW basis in $U$ over the extended Cartan subalgebra diagonalizing the…
In this paper, we define and study a notion of Ding projective dimension for complexes of left modules over associative rings. In particular, we consider the class of homologically bounded below complexes of left R-modules, and show that…
The present survey results from the will to reconcile two approaches to quantum probabilities: one rather physical and coming directly from quantum mechanics, the other more algebraic. The second leading idea is to provide a unified picture…
Persistence modules serve as the algebraic foundation for topological data analysis, typically studied as representations of posets over a field. This article extends the structural and decomposition theory of persistence modules to the…
Let R be a commutative algebra. In this paper we show that constant skew PBW extensions of a generalized Koszul algebra R are also generalized Koszul. Let A be a semi-commutative skew PBW extension of R such that A is R-augmented. We show…
We consider algebras that can be realized as PBW deformations of (Artin-Schelter) regular algebras. This is equivalent to the homogenization of the algebra being regular. It is shown that the homogenization, when it is a geometric algebra,…
These are expanded lecture notes of a mini-course whose objectives were to introduce the basic concepts, constructions and techniques of noncommutative geometry, as well as their uses as a framework for modelling quantum spacetime. Key…
In recent work of T. Cassidy and the author, a notion of complete intersection was defined for (non-commutative) regular skew polynomial rings, defining it using both algebraic and geometric tools, where the commutative definition is a…
In this paper, we introduce a new homological invariant called quasi-projective dimension, which is a generalization of projective dimension. We discuss various properties of quasi-projective dimension. Among other things, we prove the…
We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for…
Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a connected reductive algebraic group over $k$. Under some standard hypothesis on $G$, we give a direct approach to the finite $W$-algebra $U(\mathfrak g,e)$…
The construction of HNN-extensions of involutive Hom-associative algebras and involutive Hom-Lie algebras is described. Then, as an application of HNN-extension, by using the validity of Poincar\'e-Birkhoff-Witt theorem for involutive…
We introduce and study action of quantum groups on skew polynomial rings and related rings of quotients. This leads to a ``q-deformation'' of the Gel'fand-Kirillov conjecture which we partially prove. We propose a construction of…
The classical Gelfand-Kirillov dimension for algebras over fields has been extended recently by J. Bell and J.J Zhang to algebras over commutative domains. However, the behavior of this new notion has not been enough investigated for the…
Using the fact that $\Pi$-invertible sheaves can be interpreted as locally free sheaves of modules for the super skew field $\mathbb{D}$, we give a new construction of the $\Pi$-projective superspace $\mathbb{P}^n_{\Pi, B}$ over affine $k$…
We describe weighted projective lines in the sense of Geigle and Lenzing by a moduli problem on the canonical algebra of Ringel. We then go on to study generators of the derived categories of coherent sheaves on the total spaces of their…