On maximal subalgebras

Racine, Michel L.

doi:10.1016/0021-8693(74)90198-7

Cited by 43 publications

(65 citation statements)

References 8 publications

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“…When dimensionally reducing from D = 11, in order to make manifest the e 6(6) structure of the scalar sector in D = 5, one must first dualise the 3-form potential terms appearing in (4.3), as described in detail in [57]. This gives a total of 42 scalars parametrising E 6(6) (R)/ USp (8).…”

Section: = 4 To D = 3 and E 7(7)mentioning

confidence: 99%

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A kind of magic

Borsten¹,

Marrani²

2017

Class. Quantum Grav.

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We introduce the extended Freudenthal-Rosenfeld-Tits magic square based on six algebras: the reals R, complexes C, ternions T, quaternions H, sextonions S and octonions O. The sextonionic row/column of the magic square appeared previously and was shown to yield the non-reductive Lie algebras, sp , for R, C, T, H, S and O respectively. The fractional ranks here are used to denote the semi-direct extension of the semi-simple Lie algebra in question by a unique (up to equivalence) nilpotent "Jordan" algebra. We present all possible real forms of the extended magic square. It is demonstrated that the algebras of the extended magic square appear quite naturally as the symmetries of supergravity Lagrangians. The sextonionic row (for appropriate choices of real forms) gives the noncompact global symmetries of the Lagrangian for the D = 3 maximal N = 16, magic N = 4 and magic non-supersymmetric theories, obtained by dimensionally reducing the D = 4 parent theories on a circle, with the graviphoton left undualised. In particular, the extremal intermediate non-reductive Lie algebrã e 7(7) 1 2 (which is not a subalgebra of e 8(8) ) is the non-compact global symmetry algebra of D = 3, N = 16 supergravity as obtained by dimensionally reducing D = 4, N = 8 supergravity with e 7(7) symmetry on a circle. On the other hand, the ternionic row (for appropriate choices of real forms) gives the non-compact global symmetries of the Lagrangian for the D = 4 maximal N = 8, magic N = 2 and magic nonsupersymmetric theories, as obtained by dimensionally reducing the parent D = 5 theories on a circle. In particular, the Kantor-Koecher-Tits intermediate non-reductive Lie algebra e 6(6) 1 4 is the non-compact global symmetry algebra of D = 4, N = 8 supergravity as obtained by dimensionally reducing D = 5, N = 8 supergravity with e 6(6) symmetry on a circle.

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Section: = 4 To D = 3 and E 7(7)mentioning

confidence: 99%

“…Here M is built from the 42 scalar fields parametrising the coset E 6(6) (R)/ USp (8). Under the global E 6(6) (R) group, M transforms linearly, M → U T MU, where U is in the 27 ′ representation of E 6(6) , and is invariant under the local USp(8).…”

Section: = 4 To D = 3 and E 7(7)mentioning

confidence: 99%

A kind of magic

Borsten¹,

Marrani²

2017

Class. Quantum Grav.

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“…In this paper maximal subalgebras of central separable algebras over a commutative ring R are considered. The ideal structure of such algebras is entirely determined by that of R so it is interesting to consider one-sided ideals and to a lesser extent subalgebras.We recall some results of [7]. Let F be a field and A a (finite dimensional) central simple algebra over F. So A se Mn(D), D a finite dimensional division algebra over its center F and A acts on an «-dimensional left D vector space V. By a subalgebra of A we understand an F subspace of A which is closed under multiplication, and by maximal subalgebra, a proper subalgebra which is maximal with respect to inclusion.…”

mentioning

confidence: 99%

“…We recall some results of [7]. Let F be a field and A a (finite dimensional) central simple algebra over F. So A se Mn(D), D a finite dimensional division algebra over its center F and A acts on an «-dimensional left D vector space V. By a subalgebra of A we understand an F subspace of A which is closed under multiplication, and by maximal subalgebra, a proper subalgebra which is maximal with respect to inclusion.…”

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confidence: 99%

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Maximal subalgebras of central separable algebras

Racine¹

1978

Proc. Amer. Math. Soc.

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Abstract. Let A be a central separable algebra over a commutative ring R. A proper Ä-subalgebra of A is said to be maximal if it is maximal with respect to inclusion.Theorem. Any proper subalgebra of A is contained in a maximal one. Any maximal subalgebra B of A contains a maximal ideal mA of A, m a maximal ideal of R, and B/mA is a maximal subalgebra of the central simple R/m algebra A/mA.More intrinsic characterizations are obtained when ii is a Dedekind domain.In [3] and [4] Dynkin studied maximal subalgebras of simple Lie algebras over an algebraically closed field of characteristic zero and the connection between these and the maximal subgroups of the classical groups. In [7] and [8] the maximal subalgebras of the following classes of central simple algebras were determined: associative, associative with involution, alternative and Jordan. In this paper maximal subalgebras of central separable algebras over a commutative ring R are considered. The ideal structure of such algebras is entirely determined by that of R so it is interesting to consider one-sided ideals and to a lesser extent subalgebras.We recall some results of [7]. Let F be a field and A a (finite dimensional) central simple algebra over F. So A se Mn(D), D a finite dimensional division algebra over its center F and A acts on an «-dimensional left D vector space V. By a subalgebra of A we understand an F subspace of A which is closed under multiplication, and by maximal subalgebra, a proper subalgebra which is maximal with respect to inclusion. If A = F then 0 is the unique maximal subalgebra of A. If A =£ F then all maximal subalgebras of A contain 1 and they are exactly the subalgebras of the form:(I)S(WO = {a E A\Wa c W), W any nonzero proper subspace of V.

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Structure of critical rings

Mal'tsev¹

1982

Sib Math J

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On maximal subalgebras

Cited by 43 publications

References 8 publications

A kind of magic

A kind of magic

Maximal subalgebras of central separable algebras

Structure of critical rings

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