Tanaka structures modeled on extended Poincar� algebras

Altomani, Andrea; Santi, Andrea

doi:10.1512/iumj.2014.63.5186

Cited by 11 publications

(66 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the contrary, cases (3) and (4) are rigid: the Tanaka prolongation of n = g − coincides with g. This follows from Yamaguchi's theorem [48] and agrees with the classification [4]. Thus these cases are of finite type.…”

Section: Fii Fisupporting

confidence: 67%

“…An intimately related object to the pseudo H-type algebras that appears naturally in mathematical physics is the notion of extended (super-)Poincaré algebras, see [1,2,3,4]. Some of our results have analogues in this theory.…”

Section: Introductionmentioning

confidence: 70%

“…They are precisely the cases (2, 4), (3,4) and (4,4). The last one may not have any pseudo H-type as it follows from Table 1 in the next section.…”

Section: Digression: Rigidity Vs Pseudo H-typementioning

confidence: 93%

“…There are two irreducible Clifford modules V + and V − of dimension 8 that, in this case, are also admissible. They are isometric to R 4,4 and generate isomorphic pseudo H-type algebras, see [19,20]. The following operators are mutually commuting symmetric involutions: Table 2 shows the commutation relations of the operators P j and J z k , where 1 appears if they commute and −1 if they anti-commute.…”

Section: Pseudo H-type Algebras With J 2 -Conditionmentioning

confidence: 99%

“…Case (r, s) = (7, 0). Though this case is known in the literature [14], it can be treated similarly to the case (3,4) starting from the involutions…”

Section: Pseudo H-type Algebras With J 2 -Conditionmentioning

confidence: 99%

See 4 more Smart Citations

Rigidity of 2-Step Carnot Groups

et al. 2017

View full text Add to dashboard Cite

Abstract. In the present paper we study the rigidity of 2-step Carnot groups, or equivalently, of graded 2-step nilpotent Lie algebras. We prove the alternative that depending on bi-dimensions of the algebra, the Lie algebra structure makes it either always of infinite type or generically rigid, and we specify the bi-dimensions for each of the choices. Explicit criteria for rigidity of pseudo Hand J-type algebras are given. In particular, we establish the relation of the so-called J 2 -condition to rigidity, and we explore these conditions in relation to pseudo H-type algebras.

show abstract

Section: Fii Fisupporting

confidence: 67%

Section: Introductionmentioning

confidence: 70%

“…They are precisely the cases (2, 4), (3,4) and (4,4). The last one may not have any pseudo H-type as it follows from Table 1 in the next section.…”

Section: Digression: Rigidity Vs Pseudo H-typementioning

confidence: 93%

Section: Pseudo H-type Algebras With J 2 -Conditionmentioning

confidence: 99%

“…Case (r, s) = (7, 0). Though this case is known in the literature [14], it can be treated similarly to the case (3,4) starting from the involutions…”

Section: Pseudo H-type Algebras With J 2 -Conditionmentioning

confidence: 99%

See 3 more Smart Citations

Rigidity of 2-Step Carnot Groups

et al. 2017

View full text Add to dashboard Cite

show abstract

Spencer Cohomology and 11-Dimensional Supergravity

Figueroa-O’Farrill

Santi

2016

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

Abstract:We recover the classification of the maximally supersymmetric bosonic backgrounds of 11-dimensional supergravity by Lie algebraic means. We classify all filtered deformations of the Z-graded subalgebraswhich differ only in zero degree, that is h 0 ⊂ g 0 and h j = g j for j < 0. Aside from the Poincaré superalgebra itself and its Z-graded subalgebras, there are only three other Lie superalgebras, which are the symmetry superalgebras of the non-flat maximally supersymmetric backgrounds. In passing we identify the gravitino variation with (a component of) a Spencer cocycle.

show abstract

Complete classification of pseudo H-type Lie algebras: I

Furutani

Markina

2017

Geom Dedicata

View full text Add to dashboard Cite

Let N be a 2-step nilpotent Lie algebra endowed with non-degenerate scalar product . , . and let N = V ⊕ ⊥ Z, where Z is the centre of the Lie algebra and V its orthogonal complement with respect to the scalar product. We study the classification of the Lie algebras for which the space V arises as a representation space of a Clifford algebra Cl(R r,s ) and the representation map J : Cl(R r,s ) → End(V ) is related to the Lie algebra structure by Jzv, w = z, [v, w] for all z ∈ R r,s and v, w ∈ V . The classification is based on the range of parameters r and s and is completed for the Clifford modules V , having minimal possible dimension, that are not necessary irreducible. We find the necessary condition for the existence of a Lie algebra isomorphism according to the range of integer parameters 0 ≤ r, s < ∞. We present the constructive proof for the isomorphism map for isomorphic Lie algebras and defined the class of non-isomorphic Lie algebras.1 2 KENRO FURUTANI, IRINA MARKINA Lie algebras. For the Clifford algebras Cl(R r,0 ), generated by the Euclidean space R r , the H-type algebras N r,0 (V ) were introduced by Kaplan [24] and attracted a lot of attention [6,10,13,25,26,34,35]. The Lie algebras N r,0 (V ) is a typical example of a standard metric form. For the Clifford algebras Cl(R r,s ), generated by a sign indefinite non-degenerate scalar product space R r,s , the pseudo H-type Lie algebras N r,s (V ) were introduced in [11], and studied in [12,16,18,20,21].We study the isomorphism properties between the Lie algebras N r,s (V ). We show that the Lie algebras N r,s (V ) can not be isomorphic to N u,t (V ) unless r = t and s = u or r = u and s = t. The present paper is the first part of the complete classification, where we concentrate on the classification of the Lie algebras based on the Clifford modules of minimal possible dimensions (which are not necessarily irreducible), admitting a scalar product making the representation map J z skew symmetric. Then, by making use the Atiyah-Bott periodicity for underlying Clifford algebras we extend the study to an arbitrary dimension pseudo H-type algebras. We also show that the Lie algebras based on the non-equivalent irreducible Clifford modules are isomorphic. We stress, that the isomorphic relation between the Clifford algebras and the associated pseudo H-type Lie algebras is not functorial. In some cases the isomorphic Clifford algebras lead to isomorphic Lie algebras, in other cases not.Apart from being motivated by itself interesting mathematical question of the classification of Lie algebras, we want to mention here possible applications in other areas of mathematics. It was shown [10,16] that the pseudo H-type Lie algebras admit the integer structure constants that in its turn, according to the Malćev theorem [31], guarantees the existence of lattices on the corresponding Lie groups. The factorization of pseudo H-type Lie groups by lattices gives a vast of new examples of nilmanifolds, which type strongly depend on the classification of pseudo H-type alg...

show abstract

Tanaka structures modeled on extended Poincar� algebras

Cited by 11 publications

References 19 publications

Rigidity of 2-Step Carnot Groups

Rigidity of 2-Step Carnot Groups

Spencer Cohomology and 11-Dimensional Supergravity

Complete classification of pseudo H-type Lie algebras: I

Contact Info

Product

Resources

About