On algebras similar to the Griess algebra Sergey Shpectorov School of Mathematics, University of Birmingham Conference in honour of V.D. Mazurov Novosibirsk, 19th July 2013
The Monster The existence of the Monster sporadic simple group M was conjectured by Fischer and Griess in 1973 and it was proved in 1982 when Griess constructed M by hand in its action on a commutative non-associative real algebra of dimension 196 , 883.
The Griess algebra The modern version of the Griess algebra adds one dimension in order for it to have a multiplicative identity element. Thus the Griess algebra V is a 196,884-dimensional commutative, non-associative algebra with 1. Naturally, its full group of automorphisms is M . Additionally, V admits an inner product satisfying ( uv, w ) = ( u, vw ) for all u, v, w ∈ V .
Axes The Monster contains two classes of involutions, 2 A and 2 B . For z ∈ 2 A , C = C M ( z ) = 2 · B , where B is the Baby Monster sporadic simple group. Norton noticed that C V ( C ) is 2-dimensional, hence it is an associative subalgebra R ⊕ R . In particular, it contains exactly three idempotents: 1, u , and v = 1 − u . One of them, say u , has the additional property that ad u has 1-dimensional 1-eigenspace, namely, � u � . Other eigenvalues of ad u are 0, 1 1 4 , and 32 . This u is called the axis of z .
Coincidence? In Number Theory there is a famous modular invariant j ( τ ) = 1 q +744+196 , 884 q +21 , 493 , 760 q 2 +864 , 299 , 970 q 3 + . . . , where q = e 2 πiτ . McKay noticed the coincidence and later Thompson (and Norton) verified that the “coincidence” persists for other coefficients and it even extends to other Hauptmodul functions.
Monstrous Moonshine Conway and Norton conjectured that M acts on an infinite dimensional graded module and that j ( τ ) and other Hauptmoduls form the graded character related to this action. In particular, the coefficients of j ( τ ) provide the dimensions of the components of this graded module.
Monster VOA V ♮ The Moonshine module was constructed explicitly by Frenkel, Lepowsky, and Meurman in a book published in 1988. However, a final solution (and explanation) to the conjecture was given by Borcherds in 1992 and he was awarded a Fields medal for it. Namely, Borcherds built a vertex operator algebra V ♮ on the graded module predicted by Thompson and constructed by Frenkel, Lepowsky, and Meurman.
V natural A vertex operator algebra is a graded algebra with infinitely many products satisfying some rather complicated axioms. These structures were first discovered by physicists working in String Theory, a part of Quantum Field Theory. One of the products on the VOA V ♮ turns its weight 2 component V ♮ 2 into the Griess algebra V . Note that dim V ♮ 0 = 1 1 = 0, so V ♮ belongs to the class of the so-called OZ and V ♮ VOAs. Naturally, the full group of automorphisms of V ♮ is the Monster M .
Ising vectors The next breakthrough came in 1998 when Miyamoto observed that a special kind of idempotent (conformal vector) in the weight 2 component (the “Griess algebra”) of an arbitrary VOA leads to an involutive automorphism of the VOA. Miyamoto called these idempotents Ising vectors after physicist Ernst Ising who developed a very similar structure in his model of certain quantum phenomena. Inside the VOA, an Ising vector generates a subalgebra isomorphic to a Virasoro algebra and Miyamoto’s result follows from the representation theory of Virasoro algebras. In the case of V ♮ , the Ising vectors are the familiar 2 A -axes of Norton and, naturally, the involution corresponding to the Ising vector u is the 2 A -involution z ∈ M .
Sakuma’s theory Miyamoto also attempted in the early 2000s to classify the OZ VOAs generated by two Ising vectors. He completed one of the cases. A complete solution was given by a student of his, Sakuma, in 2007. It turns out there exist exactly eight different VOAs generated by two Ising vectors. All eight are subalgebras of V ♮ .
One operation algebras? At the time, Ivanov was writing his book on the Monster. He extracted from Sakuma’s proof the necessary properties of the Griess algebra of an OZ VOA and turned them into the axioms of a new class of single product finite dimensional algebras, which he called Majorana algebras , after another physicist, Ettore Majorana.
Majorana algebras A Majorana algebra is a commutative, non-associative finite dimensional real algebra V admitting a bilinear form associating with the algebra product: ( uv, w ) = ( u, vw ) for all u, v, w ∈ V . The algebra V must be generated by a collection of Majorana vectors v , which are idempotents, have length 1 (that is, ( v, v ) = 1) and also the following further properties: The adjoint action of v on V (that is, ad v : u �→ vu ) only has eigenvalues in the set { 1 , 0 , 1 4 , 1 32 } and V admits a basis of eigenvectors: V = V 1 ⊕ V 0 ⊕ V 1 4 ⊕ V 1 32 , where V λ is the eigenspace of ad v for the eigenvalue λ . We require V 1 = � v � .
Fusion rules Finally, the following fusion rules must hold 1 1 1 0 4 32 1 1 1 1 0 4 32 1 1 0 0 0 4 32 1 1 1 1 1 + 0 4 4 4 32 1 1 1 1 1 + 0 + 1 32 32 32 32 4 For example, the product of any element of V 0 with any element of V 1 32 must lie in V 1 32 , while the product of any element of V 1 4 with any other element of V 1 4 must lie in V 1 ⊕ V 0 .
Miyamoto involutions Every Majorana algebra has symmetries!! For every Majorana vector v , the linear transformation that acts as identity on V 1 ⊕ V 0 ⊕ V 1 4 (even part!) and as minus identity on V 1 32 (odd part!) is an automorphism of the algebra. This is the Miyamoto involution! When V is the Griess algebra, we get the 2 A -involution z in this way, taking v to be the axis of z . Note that if V 1 32 happens to be trivial then we can play a similar game with V 1 4 now being the odd part. Involutions obtained in this way are called σ -involutions, as opposed to the Miyamoto involutions which are called the τ -involutions, because of the notation z = τ v . If both V 1 4 and V 1 32 are trivial for each Majorana vector then the algebra is associative, and it is not interesting. Thus, every Majorana algebra has a group of symmetries attached to it.
Classifying Majorana algebras In a joint paper of Ivanov, Pasechnik, Seress and Shpectorov, Sakuma’s proof was transformed into a classification of 2-generated Majorana algebras. It can also be viewed as the classification of Majorana algebras associated with the dihedral groups. We also attempted some classifications of 3-generated algebras, starting from the group. Theorem (Under stricter axioms of Majorana algebras) there are exactly four Majorana algebras corresponding to the group S 4 , of dimensions 6, 9, 13, and 13. Also the cases of A 5 (Ivanov and Seress, with some help from Shpectorov) and L 3 (2) (Ivanov and Shpectorov) were completed. Note that a lot of groups arise in the 3-generated case. For example, the Griess algebra for M is 3-generated.
Computational tools There are several programs for computing Majorana algebras. An efficient program computing 2-closed algebras was developed by Seress. He was able to compute some algebras for groups as big as L 2 (11), A 7 , and even M 11 .
Universal enumerators There are also universal programs, similar in spirit to Todd-Coxeter algorithm. I developed one of those, or rather a toolbox of routines allowing to compute Majorana algebras of arbitrary kind. For example, I was able to complete all 12 cases for S 4 , arising under the weakest set of axioms. Rehren created another program, based on similar ideas. He is currently going through the list of small groups in GAP and computing all Majorana algebras for them.
3-Transposition Groups It follows from Sakuma’s theorem that every group associated with a Majorana algebra is a 6-transposition group. That is, the product of any two Miyamoto involutions has order at most 6. There are two cases where this property can be strengthened. First, in the σ -involution case (that is, where V 1 32 is trivial) the group associated with the algebra is a 3-transposition group. 3-Transposition groups were studied by Fischer and later completely classified by Cuypers and Hall. Based on their result, the σ -involution case was completely classified by Matsuo in the context of VOAs. However, his result uses properties of conformal vectors in VOAs that do not follow from our axioms.
3-Transposition Groups, II I recently noticed that if V 1 4 is trivial for all Majorana vectors (but V 1 32 isn’t trivial!), the group is again a 3-transposition group. In a joint project with Hall (paper in preparation) we classified all Majorana algebras arising in this way. In fact all of them are 1-closed. For a positive definite form case, we found two infinite classes of Majorana algebras, one related with symmetric groups S n , and the other with certain soluble groups of order 2 · 3 n , and a large number of “sporadic” Majorana algebras.
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