scholarly journals Classification of the direct limits of involution simple associative algebras and the corresponding dimension groups

2013 ◽  
Vol 381 ◽  
pp. 73-95 ◽  
Author(s):  
A.A. Baranov
Keyword(s):  
Associative Algebras ◽  
Dimension Groups ◽  
Direct Limits

scholarly journals The Algebraic and Geometric Classification of Four Dimensional Superalgebras

2021 ◽  
Author(s):  
◽  
Aaron Armour
Keyword(s):  
Classification Problem ◽  
Associative Algebras ◽  
New Variety ◽  
Graded Algebras ◽  
Finite Representation ◽  
Algebraic Classification ◽  
Irreducible Components ◽  
Module Algebras ◽  
Sweedler’S Hopf Algebra

<p><b>The algebraic and geometric classification of k-algbras, of dimension fouror less, was started by Gabriel in “Finite representation type is open” [12].</b></p> <p>Several years later Mazzola continued in this direction with his paper “Thealgebraic and geometric classification of associative algebras of dimensionfive” [21]. The problem we attempt in this thesis, is to extend the resultsof Gabriel to the setting of super (or Z2-graded) algebras — our main effortsbeing devoted to the case of superalgebras of dimension four. Wegive an algebraic classification for superalgebras of dimension four withnon-trivial Z2-grading. By combining these results with Gabriel’s we obtaina complete algebraic classification of four dimensional superalgebras.</p> <p>This completes the classification of four dimensional Yetter-Drinfeld modulealgebras over Sweedler’s Hopf algebra H4 given by Chen and Zhangin “Four dimensional Yetter-Drinfeld module algebras over H4” [9]. Thegeometric classification problem leads us to define a new variety, Salgn —the variety of n-dimensional superalgebras—and study some of its properties.</p> <p>The geometry of Salgn is influenced by the geometry of the varietyAlgn yet it is also more complicated, an important difference being thatSalgn is disconnected. While we make significant progress on the geometricclassification of four dimensional superalgebras, it is not complete. Wediscover twenty irreducible components of Salg4 — however there couldbe up to two further irreducible components.</p>

Tensor Products of Dimension Groups and K0 of Unit-Regular Rings

1986 ◽  
Vol 38 (3) ◽  
pp. 633-658 ◽  
Author(s):  
K. R. Goodearl ◽  
D. E. Handelman
Keyword(s):  
Regular Ring ◽  
Hausdorff Space ◽  
Tensor Products ◽  
Matrix Algebras ◽  
Dimension Group ◽  
Grothendieck Groups ◽  
Dimension Groups ◽  
Direct Limits ◽  
Finite Products ◽  
Regular Rings

We study direct limits of finite products of matrix algebras (i.e., locally matricial algebras), their ordered Grothendieck groups (K0), and their tensor products. Given a dimension group G, a general problem is to determine whether G arises as K0 of a unit-regular ring or even as K0 of a locally matricial algebra. If G is countable, this is well known to be true. Here we provide positive answers in case (a) the cardinality of G is ℵ1, or (b) G is an arbitrary infinite tensor product of the groups considered in (a), or (c) G is the group of all continuous real-valued functions on an arbitrary compact Hausdorff space. In cases (a) and (b), we show that G in fact appears as K0 of a locally matricial algebra. Result (a) is the basis for an example due to de la Harpe and Skandalis of the failure of a determinantal property in a non-separable AF C*-algebra [18, Section 3].

scholarly journals Structure and classification of Hom-associative algebras

2020 ◽  
Vol 24 (1) ◽  
pp. 79-102
Author(s):  
Abdenacer Makhlouf ◽  
Ahmed Zahari
Keyword(s):  
Deformation Theory ◽  
Matrix Algebra ◽  
Algebraic Varieties ◽  
Associative Algebras ◽  
Irreducible Components

The purpose of this paper is to study the structure and the algebraic varieties of Hom-associative algebras. We characterize multiplicative simple Hom-associative algebras and give some examples deforming the 2 × 2-matrix algebra to simple Hom-associative algebras. We provide a classification of n-dimensional Hom-associative algebras for n ≤ 3. Then we study irreducible components using deformation theory.

scholarly journals Classification of direct limits of generalized Toeplitz algebras

1997 ◽  
Vol 181 (1) ◽  
pp. 89-140 ◽  
Author(s):  
Huaxin Lin ◽  
Hongbing Su
Keyword(s):  
Direct Limits

scholarly journals On the Classification of Bol-Moufang Type of Some Varieties of Quasi Neutrosophic Triplet Loop (Fenyves BCI-Algebras)

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 427 ◽  
Author(s):  
Tèmítọ́pẹ́ Jaíyéọlá ◽  
Emmanuel Ilojide ◽  
Memudu Olatinwo ◽  
Florentin Smarandache
Keyword(s):  
Research Finding ◽  
Research Work ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Structures ◽  
Associative Algebras ◽  
Necessary And Sufficient

In this paper, Bol-Moufang types of a particular quasi neutrosophic triplet loop (BCI-algebra), chritened Fenyves BCI-algebras are introduced and studied. 60 Fenyves BCI-algebras are introduced and classified. Amongst these 60 classes of algebras, 46 are found to be associative and 14 are found to be non-associative. The 46 associative algebras are shown to be Boolean groups. Moreover, necessary and sufficient conditions for 13 non-associative algebras to be associative are also obtained: p-semisimplicity is found to be necessary and sufficient for a F 3 , F 5 , F 42 and F 55 algebras to be associative while quasi-associativity is found to be necessary and sufficient for F 19 , F 52 , F 56 and F 59 algebras to be associative. Two pairs of the 14 non-associative algebras are found to be equivalent to associativity ( F 52 and F 55 , and F 55 and F 59 ). Every BCI-algebra is naturally an F 54 BCI-algebra. The work is concluded with recommendations based on comparison between the behaviour of identities of Bol-Moufang (Fenyves’ identities) in quasigroups and loops and their behaviour in BCI-algebra. It is concluded that results of this work are an initiation into the study of the classification of finite Fenyves’ quasi neutrosophic triplet loops (FQNTLs) just like various types of finite loops have been classified. This research work has opened a new area of research finding in BCI-algebras, vis-a-vis the emergence of 540 varieties of Bol-Moufang type quasi neutrosophic triplet loops. A ‘Cycle of Algebraic Structures’ which portrays this fact is provided.

A Classification of 2-Varieties

1976 ◽  
Vol 28 (2) ◽  
pp. 348-364 ◽  
Author(s):  
Tim Anderson ◽  
Erwin Kleinfeld
Keyword(s):  
Associative Algebras

The purpose of this paper is to give a classification of those varieties of power-associative algebras over a field F which satisfy the condition(1.1) For each A in and each ideal I of A, I2 is an ideal of A.

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