Related papers: On the classification of graded twisted planes
This thesis is concerned with studying the properties of gradings on several examples of cluster algebras, primarily of infinite type. We first consider two finite type cases: $B_n$ and $C_n$, completing a classification by Grabowski for…
Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist family of knots $\{K_n\}$ indexed by the integers. Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that…
We categorify a quantized Heisenberg algebra associated to a finite subgroup of SL(2,C).
We calculate the Cuntz semigroup of the tensor product A with A. We restrict our attention to C*-algebras A which are unital, simple, nuclear, stably finite, have stable rank one, absorbs the Jiang-Su algebra tensorially and satisfy the…
We study a class of quantized enveloping algebras, called twisted Yangians, associated with the symmetric pairs of types B, C, D in Cartan's classification. These algebras can be regarded as coideal subalgebras of the extended Yangian for…
It is well-known that the tensor product of two bialgebras constitutes the binary product in the category of cocommutative bialgebras and morphisms of bialgebras between them. In this paper, we extend this result to triangular bialgebras…
We use product systems of $C^*$-correspondences to introduce twisted $C^*$-algebras of topological higher-rank graphs. We define the notion of a continuous $\mathbb{T}$-valued $2$-cocycle on a topological higher-rank graph, and present…
Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi-Yau triangulated categories with a choice of cluster-tilting object. The grading on the category corresponds to a grading on the…
On the set H_n(K) of symmetric n by n matrices over the field K we can define various binary and ternary products which endow it with the structure of a Jordan algebra or a Lie or Jordan triple system. All these non-associative structures…
In a projective plane PG(2,K) defined over an algebraically closed field K of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky, arises from plane cubics and…
We study deformation of tube algebra under twisting of graded monoidal categories. When a tensor category $\mathcal{C}$ is graded over a group $\Gamma$, a torus-valued 3-cocycle on $\Gamma$ can be used to deform the associator of…
We study the congeniality property of algebras, as defined by Bao, He, and Zhang, in order to establish a version of Auslander's theorem for various families of filtered algebras. It is shown that the property is preserved under homomorphic…
The paper studies the $K$-theoretic invariants of the crossed product $C^{*}$-algebras associated with an important family of homeomorphisms of the tori $\Bbb{T}^{n}$ called {\em Furstenberg transformations}. Using the Pimsner-Voiculescu…
An algebraic representation of the Turing machines is given, where the configurations of Turing machines are represented by 4 order tensors, and the transition functions by 8 order tensors. Two types of tensor product are defined, one is to…
This paper explores formalizing Geometric (or Clifford) algebras into the Lean 3 theorem prover, building upon the substantial body of work that is the Lean mathematics library, mathlib. As we use Lean source code to demonstrate many of our…
In this paper, we study homological properties of twisted tensor products of connected graded algebras. We focus on the Ext-algebras of twisted tensor products with a certain form of twisting maps firstly. We show those Ext-algebras are…
We define a general notion of centrally $\Gamma$-graded sets and groups and of their graded products, and prove some basic results about the corresponding categories: most importantly, they form braided monoidal categories. Here, $\Gamma$…
We endow twisted tensor products with a natural notion of counit and comultiplication, and we provide sufficient and necessary conditions making the twisted tensor product a counital coassociative coalgebra. We then characterize when the…
The loop algebra construction by Allison, Berman, Faulkner, and Pianzola, describes graded-central-simple algebras with split centroid in terms of central simple algebras graded by a quotient of the original grading group. Here the…
One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we continue the work of [7] to adapt the machinery of globular operads [4] to…