Quaternion Type Research Articles

𝐵𝑉-structure on Hochschild cohomology for exceptional local algebras of quaternion type. The case of an even parameter

A full description is provided for the B V BV -structure on the Hochschild cohomology of exceptional local algebras of quaternion type determined by parameters ( k , 0 , d ) (k,0,d) in the case of an even parameter k ≥ 3 k \geq 3 , according to Erdmann’s classification. The method of comparison morphisms and the weak self-homotopy method are developed and used. This article may be viewed as a generalization of similar results about the B V BV -structures on the Hochschild cohomology of algebras of quaternion type.

Accurate and fast quaternion fractional-order Franklin moments for color image analysis

Quaternion type moments (QTM) based on hypergeometric functions and harmonic functions have attracted interest in the imaging field. The Franklin system is a complete orthonormal system consisting of piecewise linear continuous functions with Haar wavelet collocation points. However, due to the lack of efficient computational methods and explicit expressions, research on piecewise polynomials with Haar wavelet collocation points remains underexplored. Furthermore, the impact of magnitude and phase of quaternion on the color image features extracted by QTM has not been comprehensively explored. To address these problems, a fast and accurate computation method for Franklin functions is proposed. Quaternion fractional-order Franklin moments (QFFM) are proposed and using the Minkowski distance as a prior parameter for selecting unit pure quaternions. Experiments are conducted to demonstrate the superior performance of QFFM in image reconstruction and color image classification tasks compared with QTM-based and deep learning-based methods. Moreover, with the help of experiments, it is confirmed the effectiveness of using the Minkowski distance to select magnitude and phase of quaternion.

A New Decision Making Model Based on Complex Intuitionistic Fuzzy Soft Lattice for Traffic Monitoring in the Pandemic Scenarios

Various representations of real‐life scenarios by algebraic structures are of vital importance in the modern era of Industrial Revolution 4.0 as it enables such data to be interpreted and processed by artificial intelligence, thus bridging the interaction gap between humans and machines. In response to that, this article proposes a new decision making model for traffic control among shopping complexes utilizing the novel concept of the complex intuitionistic fuzzy soft lattice of the circular type (CIFSL‐C), which takes the time within a day as truly directional entities. As a result, CIFSL‐C possesses the combined structural complexity and the level of sophistication of all the previous algebraic models which they are derived from, enabling them to be utilized in representing many real‐life scenarios, including those related to the pandemic, which will be rigorously discussed in this article. In addition, the complex intuitionistic fuzzy soft lattice of the quaternion type (CIFSL‐Q) will also be established.

Hochschild Cohomology of Algebras of Quaternion Type. IV: Cohomology Algebra for Exceptional Local Algebras

The description of the Hochschild cohomology algebra for a family of local algebras of quaternion type is given in terms of generators and relations. This family appears in the famous K. Erdmann’s classification only in the case where the characteristic of the base field is equal to 2.

Algebras of generalized quaternion type

We introduce and study the algebras of generalized quaternion type, being natural generalizations of algebras which occurred in the study of blocks of group algebras with generalized quaternion defect groups. We prove that all these algebras, with 2-regular Gabriel quivers, are periodic algebras of period 4 and very specific deformations of the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles. The main result of the paper forms an important step towards the Morita equivalence classification of all periodic symmetric tame algebras of non-polynomial growth. Applying the main result, we establish existence of wild periodic algebras of period 4, with arbitrary large number (at least 4) of pairwise non-isomorphic simple modules. These wild periodic algebras arise as stable endomorphism rings of cluster tilting Cohen-Macaulay modules over one-dimensional hypersurface singularities.

From Brauer graph algebras to biserial weighted surface algebras

We prove that the class of Brauer graph algebras coincides with the class of indecomposable idempotent algebras of biserial weighted surface algebras. These algebras are associated with triangulated surfaces with arbitrarily oriented triangles, investigated recently in Erdmann and Skowroński (J Algebra 505:490–558, 2018, Algebras of generalized dihedral type, Preprint. arXiv:1706.00688, 2017). Moreover, we prove that Brauer graph algebras are idempotent algebras of periodic weighted surface algebras, investigated in Erdmann and Skowroński (Algebras of generalized quaternion type, Preprint. arXiv:1710.09640, 2017).

Külshammer ideals of algebras of quaternion type

For a symmetric algebra [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] Külshammer constructed a descending sequence of ideals of the center of [Formula: see text]. If [Formula: see text] is perfect, this sequence was shown to be an invariant under derived equivalence and for algebraically closed [Formula: see text] the dimensions of their image in the stable center were shown to be invariant under stable equivalence of Morita type. Erdmann classified algebras of tame representation type which may be blocks of group algebras, and Holm classified Erdmann’s list up to derived equivalence. In both classifications, certain parameters occur in the classification, and it was unclear if different parameters lead to different algebras. Erdmann’s algebras fall into three classes, namely of dihedral, semidihedral and of quaternion type. In previous joint work with Holm, we used Külshammer ideals to distinguish classes with respect to these parameters in case of algebras of dihedral and semidihedral type. In the present paper, we determine the Külshammer ideals for algebras of quaternion type and distinguish again algebras with respect to certain parameters.

A reduction theorem for tau -rigid modules

We prove a theorem which gives a bijection between the support tau -tilting modules over a given finite-dimensional algebra A and the support tau -tilting modules over A / I, where I is the ideal generated by the intersection of the center of A and the radical of A. This bijection is both explicit and well-behaved. We give various corollaries of this, with a particular focus on blocks of group rings of finite groups. In particular we show that there are tau -tilting-finite wild blocks with more than one simple module. We then go on to classify all support tau -tilting modules for all algebras of dihedral, semidihedral and quaternion type, as defined by Erdmann, which include all tame blocks of group rings. Note that since these algebras are symmetric, this is the same as classifying all basic two-term tilting complexes, and it turns out that a tame block has at most 32 different basic two-term tilting complexes. We do this by using the aforementioned reduction theorem, which reduces the problem to ten different algebras only depending on the ground field k, all of which happen to be string algebras. To deal with these ten algebras we give a combinatorial classification of all tau -rigid modules over (not necessarily symmetric) string algebras.

Weighted surface algebras

A finite-dimensional algebra A over an algebraically closed field K is called periodic if it is periodic under the action of the syzygy operator in the category of A-A-bimodules. The periodic algebras are self-injective and occurred naturally in the study of tame blocks of group algebras, actions of finite groups on spheres, hypersurface singularities of finite Cohen–Macaulay type, and Jacobian algebras of quivers with potentials. Recently, the tame periodic algebras of polynomial growth have been classified and it is natural to attempt to classify all tame periodic algebras. We introduce the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular tetrahedral algebras, are symmetric tame periodic algebras of period 4. Moreover, we describe the socle deformations of the weighted surface algebras and prove that all these algebras are also symmetric tame periodic algebras of period 4. The main results of this paper form an important step towards a classification of all periodic symmetric tame algebras of non-polynomial growth, and lead to a complete description of all algebras of generalized quaternion type with 2-regular Gabriel quivers [36].

Higher Tetrahedral Algebras

We introduce and study the higher tetrahedral algebras, an exotic family of finite-dimensional tame symmetric algebras over an algebraically closed field. The Gabriel quiver of such an algebra is the triangulation quiver associated to the coherent orientation of the tetrahedron. Surprisingly, these algebras occurred in the classification of all algebras of generalized quaternion type, but are not weighted surface algebras. We prove that a higher tetrahedral algebra is periodic if and only if it is non-singular.

Clifford Algebras and Their Applications to Lie Groups and Spinors

In these lectures, we discuss some well-known facts about Clifford algebras: matrix representations, Cartan's periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in $n$ dimensions, etc. We also present our point of view on some problems. Namely, we discuss the generalization of the Pauli theorem, the basic ideas of the method of averaging in Clifford algebras, the notion of quaternion type of Clifford algebra elements, the classification of Lie subalgebras of specific type in Clifford algebra, etc.

The BV-Algebra Structure on the Hochschild Cohomology of Local Algebras of Quaternion Type in Characteristic 2

This paper is a sequel of the joint paper by the author with S. O. Ivanov, Yu. Volkov, and G. Zhou. In the present paper, the BV -structure, and therefore, the Gerstenhaber algebra structure on the Hochschild cohomology of local algebras of generalized quaternion type is completely described over a field of characteristic 2. The family of algebras under investigation contains group algebras of generalized quaternion groups for which the case of characteristic 2 is the only one where the calculation of Hochschild cohomology and structures on it is a highly nontrivial problem. Also the group algebras of generalized quaternion groups represent classes of Morita-equivalence of tame group blocks from K. Erdmann’s classification. In particular, the BV -structure on the Hochschild cohomology of group algebras of some noncommutative groups is described.

Construction of a two unique product semigroup defined by permutation relations of quaternion type

For a regular representation H⊆Symn of the generalized quaternion group of order n=4k, with k≥2, the monoid Sn(H) presented with generators a1,a2,…,an and with relations a1a2⋯an=aσ(1)aσ(2)⋯aσ(n), for all σ∈H, is investigated. It is shown that Sn(H) has the two unique product property. As a consequence, for any field K, the monoid algebra K[Sn(H)] is a domain with trivial units which is semiprimitive.

Self-Injective algebras of stable Calabi-Yau dimension three

In the present paper, a class of algebras that allows the so-called DTI-family of relations is introduced. With few exceptions, the stable Calabi-Yau dimension of these algebras is equal to 3. We prove that all algebras of quaternion type are contained in this class, and we give some other examples of such algebras. Furthermore, minimal projective bimodule resolutions for algebras from this class are described. Bibliography: 12 titles.

Correlations of multiplicities in length spectra for congruence subgroups

Bogomolny-Leyvraz-Schmit (1996) and Peter (2002) proposed an asymptotic formula for the correlation of the multiplicities in length spectrum on the modular surface, and Lukianov (2007) extended its asymptotic formula to the Riemann surfaces derived from the congruence subgroup $\Gam_0(n)$ and the quaternion type co-compact arithmetic groups. The coefficients of the leading terms in these asymptotic formulas are described in terms of Euler products over prime numbers, and they appear in eigenvalue statistic formulas found by Rudnick (2005) and Lukianov (2007) for the Laplace-Beltrami operators on the corresponding Riemann surfaces. In the present paper, we further extend their asymptotic formulas to the higher level correlations of the multiplicities for any congruence subgroup of the modular group.

Hochschild cohomology of algebras of quaternion type: the family $Q{\left( {2{\mathcal B}} \right)_1}$ in characteristic 3

In the present paper, the Hochschild cohomology is investigated for algebras of quaternion type from the family $Q{\left( {2{\mathcal B}} \right)_1}$ over an algebraically closed field of characteristic 3. Using the previously constructed 4-periodic bimodule resolution, we describe the Hochschild cohomology algebra of the above algebras in terms of generators and relations. Bibliography: 11 titles.

Classifying tame blocks and related algebras up to stable equivalences of Morita type

We contribute to the classification of finite dimensional algebras under stable equivalence of Morita type. More precisely we give a classification of Erdmann’s algebras of dihedral, semi-dihedral and quaternion type and obtain as byproduct the validity of the Auslander–Reiten conjecture for stable equivalences of Morita type between two algebras, one of which is of dihedral, semi-dihedral or quaternion type.

Quaternion Typification of Clifford Algebra Elements

We present a new classification of Clifford algebra elements. Our classification is based on the notion of quaternion type. Using this classification we develop a method for analyzing of commutators and anticommutators of Clifford algebra elements. This method allows us to find out and prove a number of new properties of Clifford algebra elements.

Hochschild cohomology of algebras of quaternion type. III. Algebras with a small parameter

For algebras of quaternion type in the family $Q(2{\cal B})_1$ , which have a “small” natural parameter, the Hochschild cohomology algebra is described. A beforehand constructed 4-periodic bimodule resolution for the algebras under consideration is used. As a result, the description of the Hochschild cohomology algebra for all algebras of quaternion type with two simple modules in characteristic 2 is complete. Bibliography: 7 titles.

Hochschild cohomology of algebras of quaternion type. II. The family % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb % qeguuDJXwAKbacfiGae8xuaeLaaiikaiaaikdatuuDJXwAK1uy0Hwm % aeXbfv3ySLgzG0uy0Hgip5wzaGGbaiab+XsicjaacMcadaWgaaWcba % GaaGymaaqabaaaaa!4997! $$ Q(2\mathcal{B})_1 $$ in characteristic 2

In the present paper, the Hochschild cohomology is investigated for algebras of quaternion type from the family \( Q(2\mathcal{B})_1 \) over an algebraically closed field of characteristic 2. A 4-periodic bimodule resolution is constructed for the most part of algebras in this family. Using this resolution, a description in terms of generators and relations is given for the Hochschild cohomology algebra of the algebras under consideration. Bibliography: 20 titles.