The correspondence between del Pezzo surfaces and field theory models, discussed in [1] and in [2] over the complex numbers or for split real forms, is extended to other real forms, in particular to those compatible with supersymmetry. Specifically, all theories of the Magic triangle [3] that reduce to the pure supergravities in four dimensions correspond to singular real del Pezzo surfaces and the same is true for the Magic square of = 2 SUGRAS [4]. A real del Pezzo surface is the invariant set under an antilinear involution of a complex one. This conjugation induces an involution of the Picard group that preserves the anticanonical class and the intersection form. The known non-split U-duality algebras are embedded into Borcherds superalgebras defined by their Cartan matrix (minus the intersection form) and fixed by the anti-involution. These data may be described by Tits-Satake bicoloured superdiagrams. As in the split case, oxidation results from blowing down disjoint real 1's of self-intersection -1. The singular del Pezzo surfaces of interest are obtained by degenerating regular surfaces upon contraction of real curves of self-intersection -2. We use the finite classification of real simple singularities to exhibit the relevant normal surfaces. We also give a general construction of more magic triangles like a type-I split magic triangle and prove their (approximate) symmetry with respect to their diagonal, this symmetry argument was announced in our previous paper for the split case.

Real borcherds superalgebras and M-theory

In this article we characterize the fields over which connected split semisimple algebraic groups and split Kac-Moody groups admit an Iwasawa decomposition.

/pdf/iwasawa-decompositions-of-split-kac-moody-groups-543x5qljzw.pdf

Iwasawa decompositions of split Kac-Moody groups

We say that an indecomposable Cartan matrix A with entries in the ground field is almost affine if the Lie (super)algebra determined by it is not finite dimensional or affine (Kac–Moody) but the Lie sub(super)algebra determined by any submatrix of A, obtained by striking out any row and any column intersecting on the main diagonal, is the sum of finite dimensional or affine Lie (super)algebras. A Lie (super)algebra with Cartan matrix is said to be almost affine if it is not finite dimensional or affine (Kac–Moody), and all of its Cartan matrices are almost affine. We list all almost affine Lie superalgebras over complex numbers with indecomposable Cartan matrix correcting two earlier claims of classification.

https://www.tandfonline.com/doi/pdf/10.1142/S1402925110000829

The classification of almost affine (hyperbolic) lie superalgebras

The correspondence between del Pezzo surfaces and field theory models over the complex numbers or for split real forms is extended to other real forms, in particular to those compatible with supersymmetry. Specifically, all theories of the Magic triangle that reduce to the pure supergravities in four dimensions correspond to singular real del Pezzo surfaces and the same is true for the Magic square of N=2 SUGRAS. A real del Pezzo surface is the invariant set under an antilinear involution of a complex one. This conjugation induces an involution of the Picard group that preserves the anticanonical class and the intersection form. The known non-split U-duality algebras are embedded into superBorcherds algebras defined by their Cartan matrix (minus the intersection form) and fixed by the anti-involution. These data may be described by Tits-Satake bicoloured diagrams. As in the split case, oxidation results from blowing down disjoint real P^1's of self-intersection -1. The singular del Pezzo surfaces of interest are obtained by degenerating regular surfaces upon contraction of real curves of self-intersection -2. We use the finite classification of real simple singularities to exhibit the relevant normal surfaces. We also give a general construction of more magic triangles like a type I split magic triangle and prove their (approximate) symmetry with respect to their diagonal, this symmetry argument was announced in our previous paper for the split case.

Real Borcherds Superalgebras and M-theory

We say that an indecomposable Cartan matrix A with entries in the ground field of characteristic 0 is almost affine if the Lie sub(super)algebra determined by it is not finite dimensional or affine but the Lie (super)algebra determined by any submatrix of A, obtained by striking out any row and any column intersecting on the main diagonal, is the sum of finite dimensional or affine Lie (super)algebras. A Lie (super)algebra with Cartan matrix is said to be almost affine if it is not finite dimensional or affine, and all of its Cartan matrices are almost affine. 
We list all almost affine Lie superalgebras over complex numbers correcting two earlier claims of classification and make available the list of almost affine Lie algebras obtained by Li Wang Lai.

The classification of almost affine (hyperbolic) Lie superalgebras

DC–DC converter finds widespread use in variety of applications. DC–DC converter inherently possess nonlinear dynamics, therefore different linearization schemes are used to get a linearized model ...

Modeling of DC–DC Converter Using Exact Feedback Linearization Method: A Discussion

Satake superdiagrams corresponding to the Lie superalgebras , , , C(n), together with the exceptional ones are constructed from their Dynkin diagrams with a view to determine the real forms of these superalgebras. The involutive automorphisms are computed from a modified formula to account for all pertinent cases in a consistent way. This mechanism is also used to furnish a general treatment of the Iwasawa and Langlands decompositions of these superalgebras. In particular, we compute these decompositions for for illustrational purposes and obtain the corresponding induced representations using the method of Schmidt construction.

Satake superdiagrams, real forms and Iwasawa decomposition of classical Lie superalgebras

Isoclinism of Lie superalgebras has been defined and studied currently. In this article it is shown that for finite dimensional Lie superalgebras of same dimension, the notation of isoclinism and isomorphism are equivalent. Furthermore we show that covers of finite dimensional Lie superalgebras are isomorphic using isoclinism concept.

Some properties of isoclinism in Lie superalgebras

We carry out the Iwasawa decomposition of the affine Kac–Moody algebras within the general framework of involutive automorphisms which are determined with the help of the Satake diagrams corresponding to these algebras. The parabolic subalgebras essential for the construction of induced representations are computed from the Langlands decomposition. In particular, we consider the untwisted versions, viz., A1(1) and A2(1), as representative examples for the purpose of illustration.

Iwasawa decomposition of affine Kac–Moody algebras using Satake diagrams

Isoclinism of Lie superalgebras has been defined and studied currently. In this article, it is shown that for finite dimensional Lie superalgebras of same dimension, the notation of isoclinism and ...

K. C. Pati

Papers

Modeling of DC–DC Converter Using Exact Feedback Linearization Method: A Discussion

Satake superdiagrams, real forms and Iwasawa decomposition of classical Lie superalgebras

Some properties of isoclinism in Lie superalgebras

Iwasawa decomposition of affine Kac–Moody algebras using Satake diagrams

Some properties of isoclinism in Lie superalgebras