Structure of basic Lie superalgebras and of their affine extensions

Abstract

We generalize to the case of superalgebras several properties of simple Lie algebras involving the use of Dynkin diagrams. If to a simple Lie algebra can be associated one Dynkin diagram, it is a finite set of nonequivalent ones which can be constructed for a basic superalgebra (or B.S.A.). The knowledge of these diagrams, which can be obtained for each B.S.A. in a systematic way, allows us to deduce the regular subsuperalgebras of a B.S.A. The symmetries of the Dynkin diagrams are related to outer automorphisms of B.S.A. and lead to some singular subsuperalgebras. Finally we consider the extended Dynkin diagrams in order to classify the affine B.S.A. and use their symmetries to construct the twisted basic superalgebras.

At the horizon of a supersymmetricAdS5black hole: Isometries and half-BPS giants

“…, k (4) , there are also bosonic isometries of this solution which are not associated with the supergroup. To this end it can also be verified that the left-invariant vector fields (which generate right translations) 27) satisfying [ξ R i , ξ R j ] = −ǫ ijk ξ R k are Killing, reflecting the su(2) R isometries of the squashed S 3 of the near-horizon region. In addition to these we expect there to be more bosonic isometries coming from the S 5 part of the geometry that preserve the 1-form i 3 k=1z i dz i where z i = lµ i e iξ i as before with µ 1 = sin α, µ 2 = cos α sin β and µ 3 = cos α cos β.…”

The quantum superstring as a WZNW model with N=2 superconformal symmetry

“…(The factor (−) P in this expression is needed because the contraction of the indices P does not follow our northeast-southwest convention). The only simple superalgebras with nondegenerate Killing metric are SU (m|n) for m = n, Osp(m|n) except Osp(2m|2m + 2), and F (4) and G(3) [27]. In our case we are of course dealing with a nilpotent algebra (all triple-(anti)commutators vanish), but we have explicitly exhibited an invariant metric.…”

Reflection K-matrices for a nineteen vertex model with Uq[osp(2|2)Vieira1, Santos2 2017Physics Letters A2030View full textAdd to dashboardCite

“…We highlight that vertex-models associated with Lie superalgebras, in particular those associated with twisted Lie superalgebras, are generally the most complex ones [1][2][3][4][5][53][54][55][56][57][58][59][60][61][62][63][64][65]. In fact, the classification of the reflection K-matrices for models associated with Lie superalgebras is not yet complete.…”

“…Since we are dealing here with a supersymmetric system, it will be useful to review the basics of Lie superalgebras [54][55][56][57][58][59][60][61][62][63][64][65]. In terms of the Weyl matrices e ij ∈ End (V ) − matrices whose elements are all zero, except that one on the ith line and jth column, which equals 1 − and always considering a sum on the repeated indices, we can define, in a Z 2 -graded Lie algebra, the graded tensor product of two homogeneous even elements A ∈ End (V ) and B ∈ End (V )…”