Vinberg's T-Algebras. From Exceptional Periodicity to Black Hole Entropy

We introduce the so-called Magic Star (MS) projection within the root lattice of finite-dimensional exceptional Lie algebras, and relate it to rank-3 simple and semi-simple Jordan algebras. By relying on the Bott periodicity of reality and conjugacy properties of spinor representations, we present the so-called Exceptional Periodicity (EP) algebras, which are finite-dimensional algebras, violating the Jacobi identity, and providing analternative with respect to Kac-Moody infinite-dimensional Lie algebras. Remarkably, also EP algebras can be characterized in terms of a MS projection, exploiting special Vinberg T-algebras, a class of generalized Hermitian matrix algebras introduced by Vinberg in the '60s within his theory of homogeneous convex cones. As physical applications, we highlight the role of the invariant norm of special Vinberg T-algebras in Maxwell-Einstein-scalar theories in 5 space-time dimensions, in which the Bekenstein-Hawking entropy of extremal black strings can be expressed in terms of the cubic polynomial norm of the T-algebras.


Projecting root lattices onto the magic star
Within the r-dimensional root lattice of g 2 , f 4 , e 6 , e 7 and e 8 (with r = 2, 4, 6, 7, 8, resp.),one can find a plane (defined by the two Cartans of an a 2 subalgebra) on which the projection of the roots results into the so-called "Magic Star" (MS) (reported in Fig. 1).To the best of our knowledge, the MS was firstly observed in late '90s by Mukai1 [2], and later re-discovered and treated in some detail by Truini [3] (see also [4]), within a different approach relying Jordan Pairs [5]; see also [1].

≡
are its partial and total diagonal degenerations, respectively.
Within this report, we will consider things over .In this case, there are at least two noncompact real forms of the "enlarged" exceptional sequence qconf J q 3 q=8,4,2,1,0,−1/3,−2/3,−1 which can be easily interpreted in terms of symmetries of rank-3 real Jordan algebras: they are given in Tables 2 and Table 3. and they both pertain to the following non-compact, real form of (2)): qconf e 8 qconf J Table 2: The split real form of the exceptional sequence.In this case, for q = 8, 4, 2, 1, , where s is the split form of = , , , respectively.The following maximal, Jordan algebraic embeddings enjoy the following matrix realization as (r i ∈ , A i ∈ or s , i = 1, 2, 3)

035.3
where the bar denotes the conjugation in or in s .Usually, the matrix elements r 1 and r 2 are associated to lightcone degrees of freedom, i.e.
Furthermore, the following algebraic isomorphisms hold (cf.e.g.[15]): where Γ 1,q+1 and Γ q/2+1,q/2+1 are (generally simple) Jordan algebras of rank 2 with a quadratic form of (Lorentian resp.Kleinian) signature (1, q + 1) resp.(q/2 + 1, q/2 + 1), i.e. the Clifford algebras of O (1, q + 1) resp.O(q/2 + 1, q/2 + 1); for this reason, it is customary to refer to (4) as to the the spin-factor embeddings.By setting = , i.e. q = 8, in (4), and considering the various symmetries of Jordan algebras, one obtains the graded structure of suitable real forms of finite-dimensional exceptional Lie algebras with respect to the corresponding pseudo-orthogonal Lie algebras, thus obtaining the spinor content of the exceptional algebras themselves: 1.For what concerns the derivations der (namely, the Lie algebra of the automorphism group) of the rank-3 Jordan algebras, one obtains the 2-graded structure of the real, compact form of f 4 , namely: where 16 is the Majorana spinor irrepr. of so 9 , and the upperscripts "m" and "s" respectively indicate maximality and symmetric nature.The fact that the 2-graded vector space so 9 ⊕ 16 can be endowed with the structure of a (simple, exceptional) Lie algebra, and thus satisfies the Jacobi identity (in particular, for three elements in 16), relies on a remarkable Fierz identity for so 9 gamma matrices.
2. At the level of the reduced structure Lie algebra str 0 , one obtains the 3-graded structure of the real, minimally non-compact form of e 6 , namely: where 16 and 16 ′ are the Majorana-Weyl (MW) spinors of so 9,1 , which constitute an example of Jordan pair which is not a pair of Jordan algebras (see e.g.[5], as well as [3,4] for a recent treatment); also, the indeterminacy denoted by "or" depends on the spinor polarization of the embedding [16].The fact that the 3-graded vector space(s) in the r.h.s. of (10) can be endowed with the structure of a (simple, exceptional) Lie algebra, and thus satisfies the Jacobi identity (in particular, for three elements in 16 ⊕ 16 ′ ), relies on a remarkable Fierz identity for so 9,1 gamma matrices.Note that str J 3 ≃ str 0 J 3 ⊕ is isomorphic to the Lie algebra of the automorphism group Aut J 3 , J ′ 3 of the Jordan pair J 3 , J ′ 3 : 035. 4 3.At the level of the conformal Lie algebra conf, one obtains where 32 is the MW spinor of so 10,2 , and the possible priming (denoting spinor conjugation) depends on the choice of the spinor polarization [16].By further branching the sl 2, , one obtain a 5-grading of contact type (recently reconsidered within the classification worked out in [17]) of the real, minimally non-compact form of e 7 , namely: The fact that the 5-graded vector space(s) in the r.h.s. of ( 13) can be endowed with the structure of a (simple, exceptional) Lie algebra, and thus satisfies the Jacobi identity (in particular, for three elements in 32 (′) ⊕ 32 (′) ), relies on a remarkable Fierz identity for so 10,2 gamma matrices.Note that conf J 3 is isomorphic to the Lie algebra of the automorphism group Aut F J 3 of the reduced Freudenthal triple system constructed over J 3 : 4. Finally, at the level of the quasi-conformal Lie algebra4 qconf [13,14], one obtains the 2-graded structure of the real, minimally non-compact form of e 8 , namely: where 128 is the MW spinor of so 12,4 , and, again, the possible priming (standing for spinorial conjugation) relates to the choice of the spinor polarization [16].Further decomposition of so 12,4 yields to a 5-grading of "extended Poincaré" type [17]: where 64 is the MW spinor of so 11,3 and the "or" indeterminacy depends on the spinor polarization [16].The fact that the 2-graded vector space so 12,4 ⊕128 (′) can be endowed with the structure of a (simple,exceptional) Lie algebra, and thus satisfies the Jacobi identity (in particular, for three elements in 128 (′) ), relies on a remarkable Fierz identity for so 12,4 gamma matrices.Equivalently, the fact that the 5-graded vector space(s) in the r.h.s. of ( 16) can be endowed with the structure of a (simple, exceptional) Lie algebra, and thus satisfies the Jacobi identity (in particular, for three elements in 64 ⊕ 64 ′ ), relies on a remarkable Fierz identity for so 11,3 gamma matrices.

From Bott periodicity to exceptional periodicity
Thus, we have related the existence of (finite-dimensional, simple) exceptional Lie algebras to some remarkable Fierz identities holding in q + 8 dimensions (in particular, with signature 9 + 0, 9 + 1, 10 + 2,and 12 + 4, for q = 1, 2, 4 and 8, respectively).Now, by observing that the reality properties of spinors and the existence and symmetry of invariant spinor bilinears are periodic mod 8 (Bott periodicity), one can define some algebras which (for the moment, formally) generalize the spinor content of the real forms of exceptional Lie algebras discussed above: these are the so-called "Exceptional Periodicity" (EP) algebras [1,18], and, as vector spaces, they are defined as follows (n ∈ ∪ {0} throughout 5 ): 1. Level der: f (n) where ψ so 9+8n ≡ 2 4+4n is the Majorana spinor of so 9+8n .
A rigorous algebraic definition of the above EP algebras has been given in [18] (see also [1]) by introducing the notion of generalized roots, and by defining the structure constants in terms of (a suitably generalized) Kac's asymmetry function [19,20].In this report, we confine ourselves to remark that EP algebras are not simply non-reductive nor semisimple, spinor-affine extensions of (pseudo-)orthogonal Lie algebras, but their spinor generators are non-translational (i.e., non-Abelian), as are the spinor generators of 6 f 4(−52) ≡ f  24) .This yields to the violation of the Jacobi identity when considering three spinor generators as input in the Jacobiator [18].As of today, a rigorous, axiomatic treatment of EP algebras is missing: can EP algebras be defined in terms of some characterizing identities, going beyond Jacobi?This remains an open problem. 5Note that there has been a shift of unity with respect to the notation of [1] and [18]: the index n used here is actually n − 1 of such Refs. 6The treatment on given here is based on the EP generalization of the various symmetry Lie algebras of the Albert algebra J 3 , and it yielded to some specific real forms of f 7 and e (n) 8 .Starting from , a rigorous definition of all real forms of EP algebras, by means of the introduction of suitable involutive morphisms within the corresponding EP generalized root lattices [18], will be the object of forthcoming works.The crucial result, which motivates and renders all the above construction and the corresponding construction in the EP lattices non-trivial, is the following [18]: for n > 0, all EP algebras admit a a 2 subalgebra, such that the projection of their generalized root lattices onto the 2dimensional plane defined by the Cartans of such a 2 has a Magic Star structure, with those generalized roots corresponding to the degeneracies on the tips of such EP-generalized Magic Star which can be endowed with an algebraic structure, denoted by T q,n 3 , generalizing the rank-3 simple Jordan algebras J q 3 ≡ J 3 ≡ H 3 ( ) mentioned above.The resulting, EP-generalized Magic Star is depicted in Fig. 2. Remarkably, such a generalization is 7 the unique possible one, and it is provided by the Hermitian part of (a class of) rank-3 T-algebras of special type.Such algebras were introduced some time ago by Vinberg [22], and they recently appeared in [23][24][25], in which they have been named Vinberg special T-algebras.

Vinberg special T-algebras and Bekenstein-Hawking entropy
The real forms of EP algebras resulting from the treatment given above, i.e. f 8(−24) (corresponding to der, str 0 , conf and qconf levels, or, equivalently -by the symmetry of the Freudenthal-Tits Magic Square [11,12] -to q = 1, 2, 4 and 8, respectively), the 3 × 3 generalized matrix algebras T q,n 3 corresponding to the set of generalized roots degenerating to a point on each of the tips of the EP-generalized Magic Star (depicted in Fig. 2) can be realized as follows: 7 Within a set of reasonable and intuitive assumptions [22].
Actually, S q is related to the reality properties of the spinors of so q+8n , and in Physics it is named R-symmetry.Furthermore, Fund S q denotes the smallest non-trivial representation of the simple Lie algebra S q (if any): with real dimension fund q := dim Fund S q = 1, 2, 2, 1 , for q = 1, 2, 4, 8 .
(30) Thus, the total real dimension of T q,n 3 is dim (T q,n 3 ) = q + 3 + 8n + fund q • 2 [(q+1)/2]+4n+δ q,1 .( As mentioned above, T q,n 3 (21) is the Hermitian part of a certain class of generalized matrix algebras going under the name of rank-3 T-algebras, introduced sometime ago by Vinberg as a unique,consistent generalization of rank-3, simple Jordan algebras, within its theory of homogeneous convex cones [22]: more precisely, T q,n 3 has been dubbed exceptional T-algebra in Sec.4.3 of [1].Upon a slight generalization (i.e., by including P + Ṗ copies of spinor irreprs., and correspondingly extending S q to the "full-fledged" R-symmetry S q P, Ṗ ), T q,n 3 gets generalized to T q,n,P, Ṗ 3 (with P, Ṗ ∈ ∪ {0}), which occur in the study of so-called homogeneous real special manifolds. 9These are non-compact Riemannian spaces occurring as (vector multiplets') scalar manifolds of N = 2-extended Maxwell-Einstein supergravity theories in D = 4+1 space-time dimensions, firstly discussed to some extent by Cecotti [28].More recently, T q,n,P, Ṗ 3 have appeared under the name of Vinberg special T-algebras in works on Vinberg's theory of homogeneous cones (and generalizations thereof) and on its relation to the entropy of extremal black holes in N = 2-extended Maxwell-Einstein supergravity theories in D = 3 + 1 space-time dimensions [23][24][25].

Table 3 :
Another (non-split) non-compact real form of the exceptional sequence.