Majorana Algebras Research Articles

From forbidden configurations to a classification of some axial algebras of Monster type

Ivanov introduced the shape of a Majorana algebra as a record of the 2-generated subalgebras arising in that algebra. As a broad generalisation of this concept and to free it from the ambient algebra, we introduce the concept of an axet and shapes on an axet. A shape can be viewed as an algebra version of a group amalgam. Just like an amalgam, a shape leads to a unique algebra completion which may be non-trivial or it may collapse. Then for a natural family of shapes of generalised Monster type we classify all completion algebras and discover that a great majority of them collapse, confirming the observations made in an earlier paper [12].

Majorana algebra for the Hoffman–Singleton graph

Majorana theory is an axiomatic tool for studying the Monster group M and its subgroups through the 196,884-dimensional Conway–Griess–Norton algebra. The theory was introduced by A. A. Ivanov in 2009 and since then it experienced a remarkable development including the classification of Majorana representations for small (and not so small) groups. The group U_3(5) is (isomorphic to) the socle of the centralizer in M of a subgroup of order 25. The involutions of this U_3(5)-subgroup are 2A-involutions in the Monster. Therefore, U_3(5) possesses a Majorana representation (based on the embedding in the Monster). We prove that this is the unique Majorana representation of U_3(5), calculate its dimension, which is 798, and obtain a description in terms of the Hoffman–Singleton graph of which the automorphism group has U_3(5) as an index 2 subgroup.

Double axes and subalgebras of Monster type in Matsuo algebras

Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov (in 2015) as a broad generalization of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster sporadic simple group. The class of axial algebras of Monster type includes Majorana algebras for the Monster and many other sporadic simple groups, Jordan algebras for classical and some exceptional simple groups, and Matsuo algebras corresponding to 3-transposition groups. Thus, axial algebras of Monster type unify several strands in the theory of finite simple groups. It is shown here that double axes, i.e., sums of two orthogonal axes in a Matsuo algebra, satisfy the fusion law of Monster type Primitive subalgebras generated by two single or double axes are completely classified and 3-generated primitive subalgebras are classified in one of the three cases. These classifications further lead to the general flip construction outputting a rich variety of axial algebras of Monster type. An application of the flip construction to the case of Matsuo algebras related to the symmetric groups results in three new explicit infinite series of such algebras.

Decomposition algebras and axial algebras

We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras:(1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category.We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions.We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples.We also take the opportunity to fix some terminology in this rapidly expanding subject.

Minimal 3-generated Majorana algebras

Majorana theory was introduced by A.A. Ivanov as the axiomatization of certain properties of the 2A-axes of the Griess algebra. Since its inception, Majorana theory has proved to be a remarkable tool with which to study objects related to the Griess algebra and the Monster simple group. We introduce the definition of a minimal 3-generated Majorana algebra and begin the first steps towards classifying such algebras.In particular, we give a complete classification of finite minimal 3-generated 6-transposition groups. We then use an algorithm developed in GAP by M. Pfeiffer and M. Whybrow, together with some additional computational tools, to give an almost complete description of all minimal 3-generated Majorana algebras arising from this list of groups.

Majorana algebras generated by a 2⁢A{2A} algebra and one further axis

Abstract We consider Majorana algebras generated by three Majorana axes a 0 {a_{0}} , a 1 {a_{1}} and a 2 {a_{2}} such that a 0 {a_{0}} and a 1 {a_{1}} generate a dihedral algebra of type 2 ⁢ A {2A} . We show that such an algebra must occur as a Majorana representation of one of 27 groups. These 27 groups coincide with the subgroups of the Monster that are generated by three 2 ⁢ A {2A} -involutions a, b and c such that ab is also a 2 ⁢ A {2A} -involution, which were classified by S. P. Norton in 1985. Our work relies on that of S. Decelle and consists of showing that certain groups do not admit Majorana representations.

Generalised dihedral subalgebras from the Monster

The conjugacy classes of the Monster which occur in the McKay observation correspond to the isomorphism types of certain 2 2 -generated subalgebras of the Griess algebra. Sakuma, Ivanov and others showed that these subalgebras match the classification of vertex algebras generated by two Ising conformal vectors, or of Majorana algebras generated by two axes. In both cases, the eigenvalues α , β \alpha ,\beta parametrising the theory are fixed to 1 4 \frac {1}{4} , 1 32 \frac {1}{32} . We generalise these parameters and the algebras which depend on them, in particular finding the largest possible (nonassociative) axial algebras which satisfy the same key features, by working extensively with the underlying rings. The resulting algebras admit an associating symmetric bilinear form and satisfy the same 6 6 -transposition property as the Monster; 1 4 \frac {1}{4} , 1 32 \frac {1}{32} turns out to be distinguished.

Primitive axial algebras of Jordan type

An axial algebra over the field F is a commutative algebra generated by idempotents whose adjoint action has multiplicity-free minimal polynomial. For semisimple associative algebras this leads to sums of copies of F. Here we consider the first nonassociative case, where adjoint minimal polynomials divide (x−1)x(x−η) for fixed 0≠η≠1. Jordan algebras arise when η=12, but our motivating examples are certain Griess algebras of vertex operator algebras and the related Majorana algebras. We study a class of algebras, including these, for which axial automorphisms like those defined by Miyamoto exist, and there classify the 2-generated examples. Always for η≠12 and in identifiable cases for η=12 this implies that the Miyamoto involutions are 3-transpositions, leading to a classification.

Associative subalgebras of low-dimensional Majorana algebras

A Majorana algebra is a commutative nonassociative real algebra generated by a finite set of idempotents, called Majorana axes, that satisfy some of the properties of the 2A-axes of the Monster Griess algebra. The term was introduced by A.A. Ivanov in 2009 inspired by the work of S. Sakuma and M. Miyamoto. In the present paper, we revisit Mayer and Neutsch's theorem on associative subalgebras of the Griess algebra in the context of Majorana theory. We apply this result to determine all the maximal associative subalgebras of some low-dimensional Majorana algebras; namely, the Majorana algebras generated by two Majorana axes and the Majorana representations of the symmetric group of degree 4 involving 3C-algebras.

Level 1 WZW superselection sectors

The superselection structure of the Wess-Zumino-Witten theory based on the affine Lie algebra\(\widehat{so}(N)\) at level one is investigated for arbitraryN. By making use of the free fermion representation of the affine algebra, the endomorphisms which represent the superselection sectors on the observable algebra can be constructed as endomorphisms of the underlying Majorana algebra. These endomorphisms do not close on the chiral algebra of the theory, but we are able to obtain a larger algebra on which the endomorphisms close. The composition of equivalence classes of the endomorphisms reproduces the WZW fusion rules.

Physical algebras in four dimensions. II. The Majorana algebra

We show that, contrary to the traditional assumption, the Majorana matrices define an algebra that is distinct from the algebra of the Dirac matrices. Furthermore, both algebras are in turn distinct from the Clifford algebra in Minkowski spacetime. This result is discussed within the context of field descriptions in four-dimensional spacetime. It is pointed out why these three algebras are almost impossible to distinguish in the representation space of 4 × 4 matrices.

Realization, extension, and classification of certain physically important groups and algebras

An associative algebra of differential forms with division has been constructed. The algebra of forms in each different space provides a practical realization of the universal Clifford algebra of that space. A classification of all such algebras is given in terms of two distinct types of algebras Nk and Sk. The former include the dihedral, quaternion, and Majorana algebras; the latter include the complex, spinor, and Dirac algebras. The associative product expresses Hodge duality as multiplication by a basis element. This makes possible the realization of higher order algebras in a calculationally useful algebraic setting. The fact that the associative algebras, as well as the enveloped Lie algebras, are precisely those arising in physics suggests that this formalism may be a convenient setting for the formulation of basic physical laws.