1,903 research outputs found
An infinite-dimensional 2-generated primitive axial algebra of Monster type
No full textRehren proved in Axial algebras. Ph.D. thesis, University of Birmingham (2015), Trans Am Math Soc 369:6953–6986 (2017) that a primitive 2-generated axial algebra of Monster type (,) over a field of characteristic other than 2, has dimension at most 8 if ∉{2,4} if α∉{2β,4β}. In this note, we show that Rehren’s bound does not hold in the case α=4β by providing an example (essentially the unique one) of an infinite-dimensional 2-generated primitive axial algebra of Monster type (2,1/2) over an arbitrary field
of characteristic other than 2 and 3. We further determine its group of automorphisms and describe some of its relevant features
2-generated axial algebras of Monster type (2β,β)
No full textAxial algebras of Monster type (α,β) are a class of non-associative algebras that includes, besides associative algebras, other important examples such as the Jordan algebras and the Griess algebra. 2-generated primitive axial algebras of Monster type (α,β) naturally split into three cases: the case when α∉{2β,4β}, the case α=4β and α=2β. In this paper we give a complete classification all 2-generated primitive axial algebras of Monster type (2β,β).</p
2-generated axial algebras of Monster type
No full textAbstract. We provide the basic setup for the project, initiated by Felix
Rehren, aiming at classifying all 2-generated axial algebras of Monster type
(α, β) over a field F. Using this, we first show that every such algebra has
dimension at most 8, except for the case (α, β) = (2,1/2), where the Highwater
algebra provides examples of dimension n, for all n ∈ N∪{∞}. We then classify
all 2-generated axial algebras of Monster type (α, β) over Q(α, β), for α and β
algebraically independent over Q. Finally, we generalise the Norton-Sakuma
Theorem to every primitive 2-generated axial algebra of Monster type (1/4, 1/32)
over a field of characteristic zero, dropping the hypothesis on the existence of
a Frobenius form
Miyamoto involutions in axial algebras of Jordan type half
No full textNonassociative commutative algebras A, generated by idempotents e whose adjoint operators ad e : A → A, given by x ↦ xe, are diagonalizable and have few eigenvalues, are of recent interest. When certain fusion (multiplication) rules between the associated eigenspaces are imposed, the structure of these algebras remains rich yet rather rigid. For example, vertex operator algebras give rise to such algebras. The connection between the Monster algebra and Monster group extends to many axial algebras which then have interesting groups of automorphisms.Axial algebras of Jordan type η are commutative algebras generated by idempotents whose adjoint operators have a minimal polynomial dividing (x-1)x(x-η), where η ∉ {0, 1} is fixed, with well-defined and restrictive fusion rules. The case of η ≠1/2 was thoroughly analyzed by Hall, Rehren and Shpectorov in a recent paper, in which axial algebras were introduced. Here we focus on the case where η = 1/2, which is less understood and is of a different nature
New flag-transitive geometries for the groups {Mathematical expression} and {Mathematical expression}
No full textIn Pralle and Shpectorov (Adv Geom 7(1):1-17, 2007) the class of ovoidal hyperplanes in dual polar spaces of rank 4 is described. In this paper we observe that by removing such a hyperplane and a related second hyperplane one obtains a nice geometry for the group stabilising the ovoidal hyperplane. We show that this group acts flag-transitively and that the geometry is simply connected.</p
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No full text[no abstract available
SERGEY YURIEVICH PREOBRAZHENSKY, SCHOLAR AND AUTHOR OF VERSES
No full textThe article is written in memoria of Sergey Yurievich Preobrahzhensky, linguist and poet. It tackles upon the scientific interests of the scholar, deals with his principal concepts in prosody and other spheres of poetics. Besides, It also contains a brief characteristics and examples of his own poetic work
Axial algebras for sporadic simple groups HS and Suz.
No full textDoctoral Degree. University of KwaZulu-Natal, Durban.Motivated by the construction of the Monster sporadic simple group as a
group of automorphisms of an algebra and the recent development of ax- ial algebras as a
generalization of Majorana representations, we construct axial algebras for the sporadic simple
groups HS and Suz in different ways analogous to the Norton algebra construction. We study how
these algebras decompose as direct sums of the adjoint action of an axis. Fusion rules, that is the
rules with which eigenvectors from various eigenspaces of the adjoint action multiply, are found.
This places these groups in a general framework of groups acting on algebras hence giving a common
theme for their origin
2-generated axial algebras of Monster type (2β,β)
No full textAxial algebras of Monster type are a class of non-associative algebras that includes, besides associative algebras, other important examples such as the Jordan algebras and the Griess algebra. 2-generated primitive axial algebras of Monster type (a, β) naturally split into three cases: the case when a\not \in {2β,4β}, the case a=4β and a=4β. In this paper we give a complete classification all 2-generated primitive axial algebras of Monster type (2β,β)
Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 1: General simple connectedness
No full textAbstractWe extend the Phan theory described in [C. Bennett, R. Gramlich, C. Hoffman, S. Shpectorov, Curtis–Phan–Tits theory, in: A.A. Ivanov, M.W. Liebeck, J. Saxl (Eds.), Groups, Combinatorics, and Geometry, World Scientific, River Edge, 2003, pp. 13–29] to the last remaining infinite series of classical Chevalley groups over finite fields. Namely, we prove that the twin buildings for the group Spin(2n+1,q2), q odd, admit a unique unitary flip and that the corresponding flipflop geometry is simply connected for almost all finite fields Fq2. Applying standard methods from amalgam theory, this results in a characterization of central quotients of the group Spin(2n+1,q) by a Phan system of rank one and rank two subgroups. In the present first part of a series of two articles we present simple connectedness results for sufficiently large fields or sufficiently large rank. To be precise, the result stated in the present paper is proved for all cases but n=3 and q∈{3,5,7,9}, the remaining cases are dealt with in the sequel [R. Gramlich, M. Horn, W. Nickel, Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 2: Machine computations, submitted for publication] computationally
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