Literature Survey on Non-Associative Rings and Developments
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Quantum Groupoids Acting on Semiprime Algebras
Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2011, Article ID 546058, 9 pages doi:10.1155/2011/546058 Research Article Quantum Groupoids Acting on Semiprime Algebras Inesˆ Borges1 and Christian Lomp2 1 Instituto Superior De Contabilidade e Administrac¸ao˜ de Coimbra, Quinta Agr´ıcola-Bencanta, 3040-316 Coimbra, Portugal 2 Departamento de Matematica,´ Faculdade de Ciencias,ˆ Universidade do Porto, Rua Campo Alegre 687, 4169-007 Porto, Portugal Correspondence should be addressed to Christian Lomp, [email protected] Received 30 March 2011; Accepted 5 June 2011 Academic Editor: Olaf Lechtenfeld Copyright q 2011 I. Borges and C. Lomp. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Following Linchenko and Montgomery’s arguments we show that the smash product of an involutive weak Hopf algebra and a semiprime module algebra, satisfying a polynomial identity, is semiprime. 1. Introduction Group actions, Lie algebras acting as derivations and finite group gradings are typical examples of Hopf algebra actions which have been studied for many years. Several generali- zations of Hopf algebras have emerged in recent years, like weak Hopf algebras or quantum groupoids introduced by Bohm¨ et al. 1. The action of such objects on algebras, as given by ∗ quantum groupoids acting on C -algebras, 2 or weak Hopf algebras arising from Jones towers 3 are particularly interesting. New examples of weak Hopf algebras arose from double groupoids 4, which were also used to find new weak Hopf actions see 2. -
DERIVATIONS and PROJECTIONS on JORDAN TRIPLES an Introduction to Nonassociative Algebra, Continuous Cohomology, and Quantum Functional Analysis
DERIVATIONS AND PROJECTIONS ON JORDAN TRIPLES An introduction to nonassociative algebra, continuous cohomology, and quantum functional analysis Bernard Russo July 29, 2014 This paper is an elaborated version of the material presented by the author in a three hour minicourse at V International Course of Mathematical Analysis in Andalusia, at Almeria, Spain September 12-16, 2011. The author wishes to thank the scientific committee for the opportunity to present the course and to the organizing committee for their hospitality. The author also personally thanks Antonio Peralta for his collegiality and encouragement. The minicourse on which this paper is based had its genesis in a series of talks the author had given to undergraduates at Fullerton College in California. I thank my former student Dana Clahane for his initiative in running the remarkable undergraduate research program at Fullerton College of which the seminar series is a part. With their knowledge only of the product rule for differentiation as a starting point, these enthusiastic students were introduced to some aspects of the esoteric subject of non associative algebra, including triple systems as well as algebras. Slides of these talks and of the minicourse lectures, as well as other related material, can be found at the author's website (www.math.uci.edu/∼brusso). Conversely, these undergraduate talks were motivated by the author's past and recent joint works on derivations of Jordan triples ([116],[117],[200]), which are among the many results discussed here. Part I (Derivations) is devoted to an exposition of the properties of derivations on various algebras and triple systems in finite and infinite dimensions, the primary questions addressed being whether the derivation is automatically continuous and to what extent it is an inner derivation. -
Moufang Loops and Alternative Algebras
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 2, Pages 313{316 S 0002-9939(03)07260-5 Article electronically published on August 28, 2003 MOUFANG LOOPS AND ALTERNATIVE ALGEBRAS IVAN P. SHESTAKOV (Communicated by Lance W. Small) Abstract. Let O be the algebra O of classical real octonions or the (split) algebra of octonions over the finite field GF (p2);p>2. Then the quotient loop O∗=Z ∗ of the Moufang loop O∗ of all invertible elements of the algebra O modulo its center Z∗ is not embedded into a loop of invertible elements of any alternative algebra. It is well known that for an alternative algebra A with the unit 1 the set U(A) of all invertible elements of A forms a Moufang loop with respect to multiplication [3]. But, as far as the author knows, the following question remains open (see, for example, [1]). Question 1. Is it true that any Moufang loop can be imbedded into a loop of type U(A) for a suitable unital alternative algebra A? A positive answer to this question was announced in [5]. Here we show that, in fact, the answer to this question is negative: the Moufang loop U(O)=R∗ for the algebra O of classical octonions over the real field R andthesimilarloopforthe algebra of octonions over the finite field GF (p2);p>2; are not imbeddable into the loops of type U(A). Below O denotes an algebra of octonions (or Cayley{Dickson algebra) over a field F of characteristic =2,6 O∗ = U(O) is the Moufang loop of all the invertible elements of O, L = O∗=F ∗. -
The Center of Topologically Primitive Exponentially Galbed Algebras
THE CENTER OF TOPOLOGICALLY PRIMITIVE EXPONENTIALLY GALBED ALGEBRAS MART ABEL AND MATI ABEL Received 29 September 2004; Revised 11 December 2005; Accepted 18 December 2005 Let A be a unital sequentially complete topologically primitive exponentially galbed Haus- dorff algebra over C, in which all elements are bounded. It is shown that the center of A is topologically isomorphic to C. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1. Introduction (1) Let A be an associative topological algebra over the field of complex numbers C with separately continuous multiplication. Then A is an exponentially galbed algebra (see, e.g., [1–4, 19, 20]) if every neighbourhood O of zero in A defines another neighbourhood U of zero such that n a k a ... a ∈ U ⊂ O k : 0, , n (1.1) k=0 2 for each n ∈ N. Herewith, A is locally pseudoconvex,ifithasabase{Uλ : λ ∈ Λ} of neigh- bourhoods of zero consisting of balanced and pseudoconvex sets (i.e., of sets U for which μU ⊂ U,whenever|μ| 1, and U + U ⊂ ρU for a ρ 2). In particular, when every Uλ in {Uλ : λ ∈ Λ} is idempotent (i.e., UλUλ ⊂ Uλ), then A is called a locally m-pseudoconvex algebra, and when every Uλ in {Uλ : λ ∈ Λ} is A-pseudoconvex (i.e., for any a ∈ A there is a μ>0suchthataUλ,Uλa ⊂ μUλ), then A is called a locally A-pseudoconvex algebra.Itiswell known (see [21,page4]or[6, page 189]) that the locally pseudoconvex topology on A is given by a family {pλ : λ ∈ Λ} of kλ-homogeneous seminorms, where kλ ∈ (0,1] for each λ ∈ Λ.Thetopologyofalocallym-pseudoconvex (A-pseudoconvex) algebra -
An Introduction to Operad Theory
AN INTRODUCTION TO OPERAD THEORY SAIMA SAMCHUCK-SCHNARCH Abstract. We give an introduction to category theory and operad theory aimed at the undergraduate level. We first explore operads in the category of sets, and then generalize to other familiar categories. Finally, we develop tools to construct operads via generators and relations, and provide several examples of operads in various categories. Throughout, we highlight the ways in which operads can be seen to encode the properties of algebraic structures across different categories. Contents 1. Introduction1 2. Preliminary Definitions2 2.1. Algebraic Structures2 2.2. Category Theory4 3. Operads in the Category of Sets 12 3.1. Basic Definitions 13 3.2. Tree Diagram Visualizations 14 3.3. Morphisms and Algebras over Operads of Sets 17 4. General Operads 22 4.1. Basic Definitions 22 4.2. Morphisms and Algebras over General Operads 27 5. Operads via Generators and Relations 33 5.1. Quotient Operads and Free Operads 33 5.2. More Examples of Operads 38 5.3. Coloured Operads 43 References 44 1. Introduction Sets equipped with operations are ubiquitous in mathematics, and many familiar operati- ons share key properties. For instance, the addition of real numbers, composition of functions, and concatenation of strings are all associative operations with an identity element. In other words, all three are examples of monoids. Rather than working with particular examples of sets and operations directly, it is often more convenient to abstract out their common pro- perties and work with algebraic structures instead. For instance, one can prove that in any monoid, arbitrarily long products x1x2 ··· xn have an unambiguous value, and thus brackets 2010 Mathematics Subject Classification. -
ON FULLY IDEMPOTENT RINGS 1. Introduction Throughout This Note
Bull. Korean Math. Soc. 47 (2010), No. 4, pp. 715–726 DOI 10.4134/BKMS.2010.47.4.715 ON FULLY IDEMPOTENT RINGS Young Cheol Jeon, Nam Kyun Kim, and Yang Lee Abstract. We continue the study of fully idempotent rings initiated by Courter. It is shown that a (semi)prime ring, but not fully idempotent, can be always constructed from any (semi)prime ring. It is shown that the full idempotence is both Morita invariant and a hereditary radical property, obtaining hs(Matn(R)) = Matn(hs(R)) for any ring R where hs(−) means the sum of all fully idempotent ideals. A non-semiprimitive fully idempotent ring with identity is constructed from the Smoktunow- icz’s simple nil ring. It is proved that the full idempotence is preserved by the classical quotient rings. More properties of fully idempotent rings are examined and necessary examples are found or constructed in the process. 1. Introduction Throughout this note each ring is associative with identity unless stated otherwise. Given a ring R, denote the n by n full (resp. upper triangular) matrix ring over R by Matn(R) (resp. Un(R)). Use Eij for the matrix with (i, j)-entry 1 and elsewhere 0. Z denotes the ring of integers. A ring (possibly without identity) is called reduced if it has no nonzero nilpotent elements. A ring (possibly without identity) is called semiprime if the prime radical is zero. Reduced rings are clearly semiprime and note that a commutative ring is semiprime if and only if it is reduced. The study of fully idempotent rings was initiated by Courter [2]. -
Math 210B. Finite-Dimensional Commutative Algebras Over a Field Let a Be a Nonzero finite-Dimensional Commutative Algebra Over a field K
Math 210B. Finite-dimensional commutative algebras over a field Let A be a nonzero finite-dimensional commutative algebra over a field k. Here is a general structure theorem for such A: Theorem 0.1. The set Max(A) of maximal ideals of A is finite, all primes of A are maximal and minimal, and the natural map Y A ! Am m is an isomorphism, with each Am having nilpotent maximal ideal. Qn In particular, if A is reduced then A ' i=1 ki for fields ki, with the maximal ideals given by the kernels of the projections A ! ki. The assertion in the reduced case follows from the rest since if A is reduced then so is each Am (and hence its nilpotent unique maximal ideal vanishes, implying Am must be a field). Note also that the nilpotence of the maximal ideal mAm implies that for some large n we have n n n Am = Am=m Am = (A=m )m = A=m (final equality since m is maximal in A), so the isomorphism in the Theorem can also be expressed Q n as saying A ' m A=m for large n. Most of the proof of this result is worked out in HW1 Exercise 7, and here we just address one point: the nilpotence of the maximal ideal of the local ring Am at each maximal ideal m of A. That is, we claim that the maximal ideal M := mAm is nilpotent. To establish such nilpotence, note that M is finitely generated as an Am-module since Am is noetherian (as A is obviously noetherian!). -
Irreducible Representations of Finite Monoids
U.U.D.M. Project Report 2019:11 Irreducible representations of finite monoids Christoffer Hindlycke Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Mars 2019 Department of Mathematics Uppsala University Irreducible representations of finite monoids Christoffer Hindlycke Contents Introduction 2 Theory 3 Finite monoids and their structure . .3 Introductory notions . .3 Cyclic semigroups . .6 Green’s relations . .7 von Neumann regularity . 10 The theory of an idempotent . 11 The five functors Inde, Coinde, Rese,Te and Ne ..................... 11 Idempotents and simple modules . 14 Irreducible representations of a finite monoid . 17 Monoid algebras . 17 Clifford-Munn-Ponizovski˘ıtheory . 20 Application 24 The symmetric inverse monoid . 24 Calculating the irreducible representations of I3 ........................ 25 Appendix: Prerequisite theory 37 Basic definitions . 37 Finite dimensional algebras . 41 Semisimple modules and algebras . 41 Indecomposable modules . 42 An introduction to idempotents . 42 1 Irreducible representations of finite monoids Christoffer Hindlycke Introduction This paper is a literature study of the 2016 book Representation Theory of Finite Monoids by Benjamin Steinberg [3]. As this book contains too much interesting material for a simple master thesis, we have narrowed our attention to chapters 1, 4 and 5. This thesis is divided into three main parts: Theory, Application and Appendix. Within the Theory chapter, we (as the name might suggest) develop the necessary theory to assist with finding irreducible representations of finite monoids. Finite monoids and their structure gives elementary definitions as regards to finite monoids, and expands on the basic theory of their structure. This part corresponds to chapter 1 in [3]. The theory of an idempotent develops just enough theory regarding idempotents to enable us to state a key result, from which the principal result later follows almost immediately. -
Malcev Algebras
MALCEVALGEBRAS BY ARTHUR A. SAGLEO) 1. Introduction. This paper is an investigation of a class of nonassociative algebras which generalizes the class of Lie algebras. These algebras satisfy certain identities that were suggested to Malcev [7] when he used the com- mutator of two elements as a new multiplicative operation for an alternative algebra. As a means of establishing some notation for the present paper, a brief sketch of this development will be given here. If ^4 is any algebra (associative or not) over a field P, the original product of two elements will be denoted by juxtaposition, xy, and the following nota- tion will be adopted for elements x, y, z of A : (1.1) Commutator, xoy = (x, y) = xy — yx; (1.2) Associator, (x, y, z) = (xy)z — x(yz) ; (1.3) Jacobian, J(x, y, z) = (xy)z + (yz)x + (zx)y. An alternative algebra A is a nonassociative algebra such that for any elements xi, x2, x3 oí A, the associator (¡Ci,x2, x3) "alternates" ; that is, (xi, x2, xs) = t(xiv xh, xh) for any permutation i%,i2, i%of 1, 2, 3 where « is 1 in case the permutation is even, — 1 in case the permutation is odd. If we introduce a new product into an alternative algebra A by means of a commutator x o y, we obtain for any x, y, z of A J(x, y, z)o = (x o y) o z + (y o z) o x + (z o y) o x = 6(x, y, z). The new algebra thus obtained will be denoted by A(~K Using the preceding identity with the known identities of an alternative algebra [2] and the fact that the Jacobian is a skew-symmetric function in /l(-), we see that /1(_) satis- fies the identities (1.4) ïoï = 0, (1.5) (x o y) o (x o z) = ((x o y) o z) o x + ((y o z) o x) o x + ((z o x) o x) o y Presented to the Society, August 31, 1960; received by the editors March 14, 1961. -
MOUFANG LOOPS THAT SHARE ASSOCIATOR and THREE QUARTERS of THEIR MULTIPLICATION TABLES 1. Introduction Moufang Loops, I.E., Loops
MOUFANG LOOPS THAT SHARE ASSOCIATOR AND THREE QUARTERS OF THEIR MULTIPLICATION TABLES ALES· DRAPAL¶ AND PETR VOJTECHOVSK· Y¶ Abstract. Two constructions due to Dr¶apalproduce a group by modifying exactly one quarter of the Cayley table of another group. We present these constructions in a compact way, and generalize them to Moufang loops, using loop extensions. Both constructions preserve associators, the associator subloop, and the nucleus. We con- jecture that two Moufang 2-loops of ¯nite order n with equivalent associator can be connected by a series of constructions similar to ours, and o®er empirical evidence that this is so for n = 16, 24, 32; the only interesting cases with n · 32. We further investigate the way the constructions a®ect code loops and loops of type M(G; 2). The paper closes with several conjectures and research questions concerning the distance of Moufang loops, classi¯cation of small Moufang loops, and generalizations of the two constructions. MSC2000: Primary: 20N05. Secondary: 20D60, 05B15. 1. Introduction Moufang loops, i.e., loops satisfying the Moufang identity ((xy)x)z = x(y(xz)), are surely the most extensively studied loops. Despite this fact, the classi¯cation of Moufang loops is ¯nished only for orders less than 64, and several ingenious constructions are needed to obtain all these loops. The purpose of this paper is to initiate a new approach to ¯nite Moufang 2-loops. Namely, we intend to decide whether all Moufang 2-loops of given order with equivalent associator can be obtained from just one of them, using only group-theoretical constructions. -
Split Octonions
Split Octonions Benjamin Prather Florida State University Department of Mathematics November 15, 2017 Group Algebras Split Octonions Let R be a ring (with unity). Prather Let G be a group. Loop Algebras Octonions Denition Moufang Loops An element of group algebra R[G] is the formal sum: Split-Octonions Analysis X Malcev Algebras rngn Summary gn2G Addition is component wise (as a free module). Multiplication follows from the products in R and G, distributivity and commutativity between R and G. Note: If G is innite only nitely many ri are non-zero. Group Algebras Split Octonions Prather Loop Algebras A group algebra is itself a ring. Octonions In particular, a group under multiplication. Moufang Loops Split-Octonions Analysis A set M with a binary operation is called a magma. Malcev Algebras Summary This process can be generalized to magmas. The resulting algebra often inherit the properties of M. Divisibility is a notable exception. Magmas Split Octonions Prather Loop Algebras Octonions Moufang Loops Split-Octonions Analysis Malcev Algebras Summary Loops Split Octonions Prather Loop Algebras In particular, loops are not associative. Octonions It is useful to dene some weaker properties. Moufang Loops power associative: the sub-algebra generated by Split-Octonions Analysis any one element is associative. Malcev Algebras diassociative: the sub-algebra generated by any Summary two elements is associative. associative: the sub-algebra generated by any three elements is associative. Loops Split Octonions Prather Loop Algebras Octonions Power associativity gives us (xx)x = x(xx). Moufang Loops This allows x n to be well dened. Split-Octonions Analysis −1 Malcev Algebras Diassociative loops have two sided inverses, x . -
On Ring Extensions for Completely Primary Noncommutativerings
ON RING EXTENSIONS FOR COMPLETELY PRIMARY NONCOMMUTATIVERINGS BY E. H. FELLER AND E. W. SWOKOWSKI 0. Introduction. It is the authors' purpose in this paper to initiate the study of ring extensions for completely N primary noncommutative rings which satisfy the ascending chain condition for right ideals (A.C.C.). We begin here by showing that every completely N primary ring R with A.C.C. is properly contained in just such a ring. This is accomplished by first showing that R[x~\, x an indeterminate where ax = xa for all aeR,isN primary and then constructing the right quotient ring QCR[x]). The details of these results appear in §§1,7 and 8. The correspond- ing results for the commutative case are given by E. Snapper in [7] and [8]. If Re: A, where A is completely JVprimary with A.C.C. then,from the discussion in the preceding paragraph, it would seem natural to examine the structure of R(o) when oeA and ao = era for all aER in the cases where a is algebraic or transcendental over R. These structures are determined in §§ 6 and 8 of the present paper. The definitions and notations given in [2] will be used throughout this paper. As in [2], for a ring R, JV or N(R) denotes the union of nilpotent ideals^) of R, P or P(R) denotes the set of nilpotent elements of R and J or J(R) the Jacobson radical of R. The letter H is used for the natural homomorphism from R to R/N = R.