Freudenthal Magic Square Research Articles

From octonions to composition superalgebras via tensor categories

The nontrivial unital composition superalgebras, of dimension 3 and 6, which exist only in characteristic 3, are obtained from the split Cayley algebra and its order 3 automorphisms, by means of the process of semisimplification of the symmetric tensor category of representations of the cyclic group of order 3. Connections with the extended Freudenthal magic square in characteristic 3, that contains some exceptional Lie superalgebras specific of this characteristic are discussed too. In the process, precise recipes to go from (nonassociative) algebras in this tensor category to the corresponding superalgebras are given.

Hyperkähler manifolds from the Tits–Freudenthal magic square

We suggest a way to associate to each Lie algebra of type $$G_2,D_4,F_4,E_6$$ , $$E_7,E_8$$ a family of polarized hyperkahler fourfolds, constructed as parametrizing certain families of cycles of hyperplane sections of certain homogeneous or quasi-homogeneous varieties. These cycles are modeled on the Legendrian varieties studied by Freudenthal in his geometric approach to the celebrated Tits–Freudenthal magic square of Lie algebras.

Sextonions, Zorn matrices, and $$\mathbf {e}_{\mathbf{7} \frac{\mathbf{1}}{\mathbf{2}}}$$ e 7 1 2

By exploiting suitably constrained Zorn matrices, we present a new construction of the algebra of sextonions (over the algebraically closed field $\mathbb{C}$). This allows for an explicit construction, in terms of Jordan pairs, of the non-semisimple Lie algebra $\mathbf{e_{7 \frac12}}$, intermediate between $\mathbf{e_{7}}$ and $\mathbf{e_{8}}$, as well as of all Lie algebras occurring in the sextonionic row and column of the extended Freudenthal Magic Square.

On the Road Map of Vogel’s Plane

We define “population” of Vogel’s plane as points for which universal character of adjoint representation is regular in the finite plane of its argument. It is shown that they are given exactly by all solutions of seven Diophantine equations of third order on three variables. We find all their solutions: classical series of simple Lie algebras (including an “odd symplectic” one), \({D_{2,1,\lambda}}\) superalgebra, the line of sl(2) algebras, and a number of isolated solutions, including exceptional simple Lie algebras. One of these Diophantine equations, namely \({knm=4k+4n+2m+12,}\) contains all simple Lie algebras, except so\({(2N+1).}\) Among isolated solutions are, besides exceptional simple Lie algebras, so called \({\mathfrak{e}_{7\frac{1}{2}}}\) algebra and also two other similar unidentified objects with positive dimensions. In addition, there are 47 isolated solutions in “unphysical semiplane” with negative dimensions. Isolated solutions mainly belong to the few lines in Vogel plane, including some rows of Freudenthal magic square. Universal dimension formulae have an integer values on all these solutions at least for first three symmetric powers of adjoint representation.

An Extended Freudenthal Magic Square in characteristic 3

Freudenthal's Magic Square, which in characteristic 0 contains the exceptional Lie algebras other than G 2 , is extended over fields of characteristic 3, through the use of symmetric composition superalgebras, to a larger square that contains both Lie algebras and superalgebras. With one exception, the simple Lie superalgebras that appear have no counterpart in characteristic 0.

The extended Freudenthal Magic Square and Jordan algebras

The Lie superalgebras in the extended Freudenthal Magic Square in characteristic 3 are shown to be related to some known simple Lie superalgebras, specific to this characteristic, constructed in terms of orthogonal and symplectic triple systems, which are defined in terms of central simple degree three Jordan algebras.

The Magic Square and Symmetric Compositions II

The construction of Freudenthal's Magic Square, which contains the exceptional simple Lie algebras of types F_4,E_6,E_7 and E_8 , in terms of symmetric composition algebras is further developed here. The para-Hurwitz algebras, which form a subclass of the symmetric composition algebras, will be defined, in the split case, in terms of the natural two dimensional module for the simple Lie algebra \mathfrak{sl}_2 . As a consequence, it will be shown how all the Lie algebras in Freudenthal's Magic Square can be constructed, in a unified way, using copies of \mathfrak{sl}_2 and of its natural module.

Universal characters from the MacDonald identities

The main result of [Kostant, Invent. Math. 158 (1) (2004) 181–226, arXiv:math.GR/0309232] shows that a sequence of complexes associated with the MacDonald identities gives a sequence of universal characters which extends and generalises some representations of the exceptional series given in [Deligne, C. R. Acad. Sci. Paris Sér. Math. 322(4) (1996) 321–326]. This proves the conjectures in [Macfarlane, Pfeiffer, J. Phys. A 36 (2003) 230 5–2317. arxiv.math-ph/0208014]. In this paper, we use a twisted version of this definition to define a sequence of complexes for the first row of the Freudenthal magic square. We also calculate the dimensions of these characters for the special linear groups and for the symplectic groups. This gives two new explicit expressions for the D’Arcais polynomials.

Integrable aspects of the scaling q-state Potts models I: bound states and bootstrap closure

We discuss the q-state Potts models for q⩽4, in the scaling regimes close to their critical or tricritical points. Starting from the kink S-matrix elements proposed by Chim and Zamolodchikov, the bootstrap is closed for the scaling regions of all critical points, and for the tricritical points when 4> q⩾2. We also note a curious appearance of the extended last line of Freudenthal's magic square in connection with the Potts models.

R-matrices and the magic square

The distinguished representations associated with the rows of the Freudenthal magic square have a uniform tensor product graph with edges labelled by linear functions of the dimension of the corresponding division algebra.

On the exceptional series, and its descendants

Many of the striking similarities which occur for the adjoint representation of groups in the exceptional series (cf. [1–3]) also occur for certain representations of specific reductive subgroups. The tensor algebras on these representations are easier to describe (cf. [4,5,7]), and may offer clues to the original situation. The subgroups which occur form a Magic Triangle, which extends Freudenthal's Magic Square of Lie algebras. We describe these groups from the perspective of dual pairs, and their representations from the action of the dual pair on an exceptional Lie algebra. To cite this article: P. Deligne, B.H. Gross, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 877–881.

The Projective Geometry of Freudenthal's Magic Square

We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial.