GROUPS OF TYPEE₇OVER ARBITRARY FIELDS

Abstract

Freudenthal triple systems come in two flavors, degenerate and nondegenerate. The best criterion for distinguishing between the two which is available in the literature is by descent. We provide an identity which is satisfied only by nondegenerate triple systems. We then use this to define algebraic structures whose automorphism groups produce all adjoint algebraic groups of type E 7 over an arbitrary field of characteristic = 2, 3.The main advantage of these new structures is that they incorporate a previously unconsidered invariant (a symplectic involution) of these groups in a fundamental way. As an application, we give a construction of adjoint groups with Tits algebras of index 2 which provides a complete description of this involution and apply this to groups of type E 7 over a real-closed field.1991 Mathematics Subject Classification. 17A40 (11E72 14L27 17B25 20G15).

Degeneration of groups of type E 7 and minimal coupling in supergravity

“…As recalled in Example 1.2 of [21] and proved in [16,23], all degenerate Freudenthal triple systems are isomorphic to the degenerate triple system in which the resulting quartic invariant polynomial I 4 is the square of a quadratic invariant polynomial I 2 which, as pointed out above, also corresponds to the case relevant for D = 4 supergravity with symmetric scalar manifold (see the treatment of section 2, as well). The degeneration of a U -duality group G 4 of type E 7 is also confirmed by the fact that the fundamental identity characterizing simple, non-degenerate groups of type E 7 (proved in section 2 of [21] for E 7 , and generalized in formula (2.19) further below at least for all groups listed in table 1) does not hold in these cases; see section 2. The cases of U -duality groups as semi-simple, non-degenerate groups of type E 7 relevant to D = 4 supergravity theories with symmetric (vector multiplets') scalar manifolds are also analyzed in subsection 2.4.…”

“…In the particular case of E 7 (see tables 1 and 2), it holds τ = 1/12 ⇒ β = 24, and the identity proved in Theorem 2.3 of [21] is retrieved. It is worth remarking that, by defining the parameter q as specified in JHEP06(2012)074 where A denotes the division algebra on which the corresponding rank-3 simple Jordan algebra J A 3 is constructed (q = 8, 4, 2, 1 for A = O, H, C, R, respectively).…”

“…As pointed out in section 2 of [21], the story changes for degenerate groups of type E 7 . Confining ourselves to the ones relevant in D = 4 supergravity with symmetric scalar manifold, they are nothing but G 4 = U(r, s) with r = 1 (N = 2 minimally coupled to s vector multiplets [22]) or r = 3 (N = 3 coupled to s vector multiplets [43]), and the relevant (complex) symplectic representation is R(G 4 ) = r + s. In these cases, it can be computed that…”

“…Focussing on the relevance to supergravity theories in D = 4, in the remaining part of this section we will characterize various classes of groups of type E 7 in terms of (tensor and) scalar identities, along the lines of [21] and exploiting results of previous investigations, such as [26] and [30].…”

“…Kähler geometry (C ijk = 0), in which the U -duality group G 4 = U(1, n) is a degenerate 4 group of type E 7 [21], with the rank-4 completely symmetric invariant q-structure reducible, as pointed out above. As recalled in Example 1.2 of [21] and proved in [16,23], all degenerate Freudenthal triple systems are isomorphic to the degenerate triple system in which the resulting quartic invariant polynomial I 4 is the square of a quadratic invariant polynomial I 2 which, as pointed out above, also corresponds to the case relevant for D = 4 supergravity with symmetric scalar manifold (see the treatment of section 2, as well).…”