Related papers: Noncommutative tensor triangular geometry
We investigate non-semisimple modular categories with an eye towards a structure theory, low-rank classification, and applications to low dimensional topology and topological physics. We aim to extend the well-understood theory of…
This paper reviews several analytic tools for the field of topological insulators, developed with the aid of non-commutative calculus and geometry. The set of tools includes bulk topological invariants defined directly in the thermodynamic…
This paper begins by noting that, in a 1969 paper in the Transactions, M.C.McCord introduced a construction that can be interpreted as a model for the categorical tensor product of a based space and a topological abelian group. This can be…
This paper is concerned with the construction of a small, but non-trivial, example of a polynomial identity algebra, which we call the \emph{Jackson algebra}, that will be used in sequels to this paper to study non-commutative arithmetic…
We study ideals in Hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations. These conditions include the property that the monoid act via partial permutations, that the…
For each endotrivial complex arising from Bredon homology of a representation sphere, we construct $p$-local quasi-isomorphisms, called forerunners, enabling us to extend Balmer--Gallauer's results in arXiv:2307.04398 Part II concerning the…
There are theories of coverings of $C^*$-algebras which can be included into a following list: coverings of commutative $C^*$-algebras, coverings of $C^*$-algebras of groupoids and foliations, coverings of noncommutative tori, the double…
We introduce Hopf algebroid covariance on Woronowicz's differential calculus. Using it, we develop quite a general framework of noncommutative complex geometry that subsumes the one in [2]. We present transverse complex and K\"ahler…
In this paper, our objects of interest are Hopf Galois extensions (e.g., Hopf algebras, Galois field extensions, strongly graded algebras, crossed products, principal bundles, etc.) and families of noncommutative rings (e.g., skew…
Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of…
This is a review of concepts of noncommutative supergeometry - namely Hilbert superspace, C*-superalgebra, quantum supergroup - and corresponding results. In particular, we present applications of noncommutative supergeometry in harmonic…
In this paper, we study geometric points in tensor triangular geometry. In doing so, we construct a counter-example to Balmer's Nerves of Steel conjecture using free constructions in higher Zariski geometry. We then go on to introduce and…
For a braided finite tensor category $\mathcal{C}$ with unit object $1 \in \mathcal{C}$, Lyubashenko considered a certain Hopf algebra $\mathbb{F} \in \mathcal{C}$ endowed with a Hopf pairing $\omega: \mathbb{F} \otimes \mathbb{F} \to 1$ to…
In Kapranov, M. {\it Noncommutative geometry based on commutator expansions,} J. reine angew. Math {\bf 505} (1998) 73-118, a theory of noncommutative algebraic varieties was proposed. Here we prove a structure theorem for the…
From a geometrical point of view it is, so far, not sufficiently well understood what should be a "noncommutative principal bundle". Still, there is a well-developed abstract algebraic approach using the theory of Hopf algebras. An…
Tensors are multiway arrays of data, and transverse operators are the operators that change the frame of reference. We develop the spectral theory of transverse tensor operators and apply it to problems closely related to classifying…
We describe an algebraic approach to the time-dependent noncommutative geometry of a six-dimensional Cahen-Wallach pp-wave string background supported by a constant Neveu-Schwarz flux, and develop a general formalism to construct and…
The following article is an application of commutative algebra to the study of multiparameter persistent homology in topological data analysis. In particular, the theory of finite free resolutions of modules over polynomial rings is applied…
Given a rigidly-compactly generated tensor-triangulated category whose Balmer spectrum is finite dimensional and Noetherian, we construct a torsion model for it, which is equivalent to the original tensor-triangulated category. The torsion…
We look at various forms of spectrum and associated pseudospectrum that can be defined for noncommuting $d$-tuples of Hermitian elements of a $C^*$-algebra. The emphasis is on theoretical calculations of examples, in particular for…