Strictly regular elements in Freudenthal triple systems

Ferrar, J. C.

doi:10.1090/s0002-9947-1972-0374223-1

Cited by 56 publications

(95 citation statements)

References 14 publications

(6 reference statements)

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“…This vector has q(v) = 0, and so by [Fer72,3.7] there are two uniquely determined (up to scalar multiples) "strictly regular" elements e 1 and e 2 such that v lies in their span. These are e 1 = ( 1 0 0 0 ) and e 2 = ( 0 0 0 1 ).…”

Section: Small Representationsmentioning

confidence: 99%

The Rost invariant has trivial kernel for quasi-split groups of low rank

Garibaldi¹

2001

Commentarii Mathematici Helvetici

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Abstract. For G a simple simply connected algebraic group defined over a field F , Rost has shown that there exists a canonical map R G : (2)). This includes the Arason invariant for quadratic forms and Rost's mod 3 invariant for exceptional Jordan algebras as special cases. We show that R G has trivial kernel if G is quasi-split of type E 6 or E 7 . A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank.

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Section: Small Representationsmentioning

confidence: 99%

The Rost invariant has trivial kernel for quasi-split groups of low rank

Garibaldi¹

2001

Commentarii Mathematici Helvetici

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“…The FTS comes equipped with a non-degenerate bilinear antisymmetric quadratic form, a quartic form and a trilinear triple product [1,2,28,47,48]:…”

Section: B the Freudenthal Triple System And 4d Black Holesmentioning

confidence: 99%

“…The rank of an arbitrary element x ∈ M(J) is uniquely defined using the relations in Table II [28,48]. The rank of any element is invariant under Aut(M(J)) [28].…”

Section: Triple Productmentioning

confidence: 99%

Black holes admitting a Freudenthal dual

et al. 2009

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The quantised charges x of four dimensional stringy black holes may be assigned to elements of an integral Freudenthal triple system whose automorphism group is the corresponding U-duality and whose U-invariant quartic norm ∆(x) determines the lowest order entropy. Here we introduce a Freudenthal duality x →x, for whichx = −x. Although distinct from U-duality it nevertheless leaves ∆(x) invariant. However, the requirement thatx be integer restricts us to the subset of black holes for which ∆(x) is necessarily a perfect square. The issue of higher-order corrections remains open as some, but not all, of the discrete U-duality invariants are Freudenthal invariant. Similarly, the quantised charges A of five dimensional black holes and strings may be assigned to elements of an integral Jordan algebra, whose cubic norm N (A) determines the lowest order entropy. We introduce an analogous Jordan dual A ⋆ , with N (A) necessarily a perfect cube, for which A ⋆⋆ = A and which leaves N (A) invariant. The two dualities are related by a 4D/5D lift.

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“…In [2] and [3], it is shown that every algebra of type FC, can be constructed from a ternary algebra 1/ which has no zero divisors, which is a module over a central Jordan division algebra J, and which possesses a skew map V x \J -^ J. If the highest root space of the Lie algebra has dimension 1, the corresponding ternary algebras have been studied in [4], [5] and [6]. If this dimension is greater than 1 and the module is not irreducible, the ternary algebras (and hence the Lie algebras) have been completely described in [3].…”

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confidence: 99%

“…Using this equation it is straightforward to check that (6) <x, y, z*e> = + 2(xy7)z -2(yz')x -2(xzJ)y and thus by (5) (2) gives…”

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confidence: 99%

Lie algebras of type 𝐵𝐶₁

Allison¹

1976

Trans. Amer. Math. Soc.

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ABSTRACT.' Let L be a central simple Lie algebra of type BC, with highest root space of dimension greater than one over a field of characteristic zero. It is shown that either I is isomorphic to the simple Lie algebra associated with a skew hermitian form of index one or L can be constructed from the tensor product of two composition algebras. This result is obtained by completing the description (begun in [3]) of the corresponding class of ternary algebras.

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Strictly regular elements in Freudenthal triple systems

Cited by 56 publications

References 14 publications

The Rost invariant has trivial kernel for quasi-split groups of low rank

The Rost invariant has trivial kernel for quasi-split groups of low rank

Black holes admitting a Freudenthal dual

Lie algebras of type 𝐵𝐶₁

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