A Minimalist Construction of the Geometric Algebra

转置 Poisson 代数的代数分类与几何分类

The algebraic and geometric classification of transposed Poisson algebras

该研究对复数域上的全部三维转置 Poisson 代数进行代数与几何分类，并进一步探讨强特殊三维转置 Poisson 代数。

幂等二元与一元 Leibniz 代数的代数与几何分类

The algebraic and geometric classification of nilpotent binary and mono Leibniz algebras

研究给出复5维幂零二元Leibniz代数和4维幂零一元Leibniz代数的完整代数与几何分类，并由此推出幂零指数3的复4维幂零代数分类。

The algebraic and geometric classification of nilpotent anticommutative algebras

We give algebraic and geometric classifications of $6$-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are $14$ one-parameter families of $6$-dimensional nilpotent anticommutative algebras, complemented by $130$ additional isomorphism classes. The corresponding geometric variety is irreducible and determined by the Zariski closure of a one-parameter family of algebras. In particular, there are no rigid $6$-dimensional complex nilpotent anticommutative algebras.

The geometric classification of nilpotent Tortkara algebras

We give a geometric classification of all $6$-dimensional nilpotent Tortkara algebras over $\mathbb C$

The algebraic and geometric classification of nilpotent left-symmetric algebras

This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent left-symmetric algebras. The corresponding geometric variety has dimension $15$ and decomposes into $3$ irreducible components determined by the Zariski closures of two one-parameter families of algebras and a two-parameter family of algebras (see Theorem B). In particular, there are no rigid $4$-dimensional complex nilpotent left symmetric algebras.

The algebraic and geometric classification of Zinbiel algebras

This paper is devoted to the complete algebraic and geometric classification of complex $5$-dimensional Zinbiel algebras. In particular, we proved that the variety of complex $5$-dimensional Zinbiel algebras has dimension $24$, it is defined by $16$ irreducible components and it has $11$ rigid algebras.

The algebraic and geometric classification of nilpotent terminal algebras

We give algebraic and geometric classifications of $4$-dimensional complex nilpotent terminal algebras. Specifically, we find that, up to isomorphism, there are $41$ one-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, $18$ two-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, $2$ three-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, complemented by $21$ additional isomorphism classes. The corresponding geometric variety has dimension 17 and decomposes into 3 irreducible components determined by the…

The geometric classification of nilpotent commutative $\mathfrak{CD}$-algebras

We give a geometric classification of complex $5$-dimensional nilpotent commutative $\mathfrak{CD}$-algebras. The corresponding geometric variety has dimension $24$ and decomposes into $10$ irreducible components determined by the Zariski closures of a two-parameter family of algebras, three one-parameter families of algebras, and $6$ rigid algebras.

Introduction to Clifford's Geometric Algebra

Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This tutorial explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, and the popular conformal model. Geometric algebras are ideal to represent geometric transformations in the general framework of Clifford groups (also called versor or Lipschit…

The geometric classification of nilpotent $\mathfrak{CD}$-algebras

We give a geometric classification of complex $4$-dimensional nilpotent $\mathfrak{CD}$-algebras. The corresponding geometric variety has dimension 18 and decomposes into $2$ irreducible components determined by the Zariski closures of a two-parameter family of algebras and a four-parameter family of algebras (see Theorem 2). In particular, there are no rigid $4$-dimensional complex nilpotent $\mathfrak{CD}$-algebras.

幂零代数的几何分类

The geometric classification of nilpotent algebras

本文对 n 维幂零代数及其交换与反交换子类进行几何分类，证明相应簇不可约并给出维数与显式泛型族，同时应用于反交换代数长度研究。

幂零Leibniz代数的代数与几何分类

The algebraic and geometric classification of nilpotent Leibniz algebras

本文对复5维幂零Leibniz代数进行了完整的代数与几何分类，并确定其代数簇维数为24，包含10个不可约分支及一个刚性代数。

The algebraic and geometric classification of nilpotent weakly associative and symmetric Leibniz algebras

This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent weakly associative, complex $4$-dimensional symmetric Leibniz algebras, and complex $5$-dimensional nilpotent symmetric Leibniz algebras. In particular, we proved that the variety of complex $4$-dimensional symmetric Leibniz algebras has no Vergne--Grunewald--O'Halloran Property (there is an irreducible component formed by only nilpotent algebras), but on the other hand, it has Vergne Property (there are no rigid nilpotent algebras).

几何型广义簇代数的若干性质

Some properties of generalized cluster algebras of geometric types

本文研究几何型无环广义簇代数，证明互素条件下下界代数等于广义簇代数，由此获得对偶 PBW 基，并建立标准单项式线性无关与初始种子图无 3-环之间的关系。

导出约尔旦和双交换代数的代数与几何分类

The algebraic and geometric classification of derived Jordan and bicommutative algebras

本文提出新方法分类n维导出约尔旦代数，并应用于3维情况，同时得到3维metabelian交换代数和导出交换结合代数的代数分类，进而分类双交换代数，并探讨3维metabelian交换、导出交换结合、导出约尔旦和双交换代数的几何分类。

Geometric (Clifford) algebra and its applications

In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to define Clifford algebras with scalars in arbitrary rings and provides new suggestions for an infinite-dimensional approach. Furthermore, I give a quick review of classic results regarding geometric algebras, such as their classification in terms of matrix algebras, the connection to orthogonal and Spin groups, and their representation theory. A number of lowe…

Euclidean Geometric Objects in the Clifford Geometric Algebra of {Origin, 3-Space, Infinity}

This paper concentrates on the homogeneous (conformal) model of Euclidean space (Horosphere) with subspaces that intuitively correspond to Euclidean geometric objects in three dimensions. Mathematical details of the construction and (useful) parametrizations of the 3D Euclidean object models are explicitly demonstrated in order to show how 3D Euclidean information on positions, orientations and radii can be extracted.

反结合代数的代数与几何分类

The algebraic and geometric classification of antiassociative algebras

本文对复数域上4维与5维反结合代数给出完整的代数与几何分类，确定了两类簇的维数、不可约分支数及刚性代数个数。

幂零 Lie-Yamaguti 代数、Bol 代数与相容 Lie 代数的几何分类

The geometric classification of nilpotent Lie-Yamaguti, Bol and compatible Lie algebras

本文对复 4 维幂零 Lie-Yamaguti 代数、Bol 代数及相容 Lie 代数进行几何分类，系统给出其完整分类结果。

诺维科夫-泊松代数与盖尔范德-多夫曼代数的几何分类

The geometric classification of Novikov-Poisson and Gelfand-Dorfman algebras

本文对复三维诺维科夫-泊松代数、盖尔范德-多夫曼代数及诺维科夫代数开展几何分类，明确了各类代数对应簇的维数、不可约分支数量与刚性代数数量。

Geometric classification of nilpotent Jordan algebras of dimension five

The variety $JorN_{5}$ of five-dimensional nilpotent Jordan algebras structures over an algebraically closed field is investigated. We show that $JorN_{5}$ is the union of five irreducible components, four of them correspond to the Zariski closure of the $GL_5$-orbits of four rigid algebras and the other one is the Zariski closure of an union of orbits of infinite family of algebras, none of them being rigid.

On the Generalized Cluster Algebras of Geometric Type

We develop and prove the analogs of some results shown in [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52] concerning lower and upper bounds of cluster algebras to the generalized cluster algebras of geometric type. We show that lower bounds coincide with upper bounds under the conditions of acyclicity and coprimality. Consequently, we obtain the standard monomial bases of these generalized cluster algebras. Moreover, in the appendix, we prove that an acyclic generalized cluster algebra is equal to the corresponding generalized upper cluster algebra without the assumpti…

Geometric operations implemented by conformal geometric algebra neural nodes

Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this approach to elementary representations of arbitrary points, point pairs, lines, circles, planes and spheres. Apart from including curved objects, conformal geometric algebra has an elegant unified quaternion like representation for all proper and improper Euclidean transformations, including reflections at spheres, general screw transformations and scaling. Expandi…

The geometric classification of $2$-step nilpotent algebras and applications

We give a geometric classification of complex $n$-dimensional $2$-step nilpotent (all, commutative and anticommutative) algebras. Namely, has been found the number of irreducible components and their dimensions. As a corollary, we have a geometric classification of complex $5$-dimensional nilpotent associative algebras. In particular, it has been proven that this variety has $14$ irreducible components and $9$ rigid algebras.

Geometric Roots of -1 in Clifford Algebras $\G_{p,q}$ with $p+q \leq 4$

It is known that Clifford (geometric) algebra offers a geometric interpretation for square roots of -1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Research has been done [S. J. Sangwine, Biquaternion (Complexified Quaternion) Roots of -1, Adv. Appl. Cliford Alg. 16(1), pp. 63-68, 2006.] on the biquaternion roots of -1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra $Cl_{3}$ of $\R^3$. All these roots of -1 find immediate applications in …

Representation of Crystallographic Subperiodic Groups in Clifford's Geometric Algebra

This paper explains how, following the representation of 3D crystallographic space groups in Clifford's geometric algebra, it is further possible to similarly represent the 162 so called subperiodic groups of crystallography in Clifford's geometric algebra. A new compact geometric algebra group representation symbol is constructed, which allows to read off the complete set of geometric algebra generators. For clarity moreover the chosen generators are stated explicitly. The group symbols are based on the representation of point groups in geometric algebra by versors (Clifford monomials, Lipsch…

几何代数中多重向量的秩研究

On Rank of Multivectors in Geometric Algebras

该研究在任意维度Clifford几何代数中，仅用几何代数运算、结合特征多项式概念与SVD方法定义了多重向量的秩，成果可应用于计算机科学、工程、物理等多领域的几何代数相关场景。

Pre- and Post-Lie Algebras: The Algebro-Geometric View

We relate composition and substitution in pre- and post-Lie algebras to algebraic geometry. The Connes-Kreimer Hopf algebras, and MKW Hopf algebras are then coordinate rings of the infinite-dimensional affine varieties consisting of series of trees, resp.\ Lie series of ordered trees. Furthermore we describe the Hopf algebras which are coordinate rings of the automorphism groups of these varieties, which govern the substitution law in pre- and post-Lie algebras.

Clifford algebra, geometric algebra, and applications

These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of the algebra. The various applications presented include vector space and projective geometry, orthogonal maps and spinors…

Hochschild cohomology of Jacobian algebras from unpunctured surfaces: A geometric computation

There are several examples in which algebraic properties of Jacobian algebras from (unpunctured) Riemann surfaces can be computed from the geometry of the Riemann surface. In this work, we compute the dimension of the Hochschild cohomology groups of any Jacobian algebra from unpunctured Riemann surfaces. In those expressions appear geometric objects of the triangulated surface, namely: the number of internal triangles and certain types of boundaries. Moreover, we give geometric conditions on the triangulated surface $(S,M,\mathbb T)$ such that the Gerstenhaber algebra $\operatorname{HH}^*(A_{\…

A geometric realisation of 0-Schur and 0-Hecke algebras

We define a new product on orbits of pairs of flags in a vector space, using open orbits in certain varieties of pairs of flags. This new product defines an associative $\mathbb{Z}$-algebra, denoted by $G(n,r)$. We show that $G(n,r)$ is a geometric realisation of the 0-Schur algebra $S_0(n, r)$ over $\mathbb{Z}$, which is the $q$-Schur algebra $S_q(n,r)$ at q=0. We view a pair of flags as a pair of projective resolutions for a quiver of type $\mathbb{A}$ with linear orientation, and study $q$-Schur algebras from this point of view. This allows us to understand the relation between $q$-Schur al…

Representation of Crystallographic Subperiodic Groups by Geometric Algebra

We explain how following the representation of 3D crystallographic space groups in geometric algebra it is further possible to similarly represent the 162 socalled subperiodic groups of crystallography in geometric algebra. We construct a new compact geometric algebra group representation symbol, which allows to read off the complete set of geometric algebra generators. For clarity we moreover state explicitly what generators are chosen. The group symbols are based on the representation of point groups in geometric algebra by versors (Clifford group, Lipschitz elements).

从三维几何代数恢复合成代数

Recovering Composition Algebras from 3D Geometric Algebras

本文基于广义Hurwitz定理，研究从三维几何代数中恢复十五种合成代数的方法，通过引入不同对合恢复范数，并特别处理八维代数如八元数等，需引入新乘积。

Clifford几何代数中的非交换Vieta定理

Noncommutative Vieta Theorem in Clifford Geometric Algebras

本文将Vieta定理推广至Clifford几何代数，借助Gelfand--Retakh非交换Vieta定理与行列式无基公式，证明任意维几何代数中的广义Vieta定理，可应用于符号计算及计算机科学等领域。

分次gentle代数的Koszul对偶的几何实现

A geometric realization of Koszul duality for graded gentle algebras

本文证明了同调光滑分次gentle代数的Koszul函子可由几何模型中的半旋转实现，并以此证明了涉及该函子的交维数公式。

On Structure of cluster algebras of geometric type I: In view of sub-seeds and seed homomorphisms

Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point of gluing of seeds. As an application, for (rooted) cluster algebras, we completely classify rooted cluster subalgebras and characterize rooted cluster quotient algebras in detail. Also, we build the relationship between the categorification of a rooted cluster algebra and that of its rooted cluster subalgebras. Note that cluster algebras of geometric type …

Global geometric deformations of current algebras as Krichever-Novikov type algebras

We construct algebraic-geometric families of genus one (i.e. elliptic) current and affine Lie algebras of Krichever-Novikov type. These families deform the classical current, respectively affine Kac-Moody Lie algebras. The construction is induced by the geometric process of degenerating the elliptic curve to singular cubics. If the finite-dimensional Lie algebra defining the infinite dimensional current algebra is simple then, even if restricted to local families, the constructed families are non-equivalent to the trivial family. In particular, we show that the current algebra is geometrically…

Holonomy Lie algebra of a geometric lattice

Motivated by Kohno's result on the holonomy Lie algebra of a hyperplane arrangement, we define the holonomy Lie algebra of a finite geometric lattice in a combinatorial way. For a solvable pair of lattices, we show that the holonomy Lie algebra is an almost-direct product of the holonomy Lie algebra of the sublattice and a free Lie subalgebra. This yields the structure of the holonomy Lie algebra of a finite hypersolvable (including supersolvable) lattice. As applications, we obtain the structure of the holonomy Lie algebra of (the Salvetti complex of) a supersolvable oriented matroid, and tha…

通用几何簇代数

Universal geometric cluster algebras

本文研究了基于交换矩阵B的几何簇代数及其系数特化，探讨了通用几何系数的构造，通过突变风扇F_B与g向量的关系，讨论了有限类型和仿射类型的构造方法。