Lie superalgebras over gl2 Article uri icon

abstract

  • Finite-dimensional real and complex Lie superalgebras having either sl2 or gl2 = 〈I2〉 ⊕ sl2 as their underlying even Lie algebra g0, are classified up to isomorphism. It is shown that the number N(g) of non isomorphic Lie superalgebras defined in g = g0 ⊕ V associated to a given representation p : g0 → gl(V) only depends on the multiplicities of the irreducible modules of dimensions 1, 2, and 2l 1, when g0 = gl2; if g0 = sl2, it depends only on the multiplicities of the irreducible modules of dimensions 1, 2, and 3. In dealing with any finite-dimensional representation p : gl2 → gl(V), the representation space V can be decomposed in the form V1 ⊕ V2, with a completely reducible V1 associated to diagonal Jordan blocks of p(I2), and a noncompletely reducible V2 associated to nondiagonal Jordan blocks of p(I2). It is then proved that the given classification only depends on the representation in V1. The question of which isomorphism classes admit invariant orthogonal-like geometric structures is also settled; in other words, it is determined which superalgebras studied in this work are quadratic in the sense of [2]. It is shown in particular that the only quadratic Lie superalgebra of the form sl2 ⊕ V is osp1,2. © Taylor & Francis Group, LLC.
  • Finite-dimensional real and complex Lie superalgebras having either sl2 or gl2 = 〈I2〉 ⊕ sl2 as their underlying even Lie algebra g0, are classified up to isomorphism. It is shown that the number N(g) of non isomorphic Lie superalgebras defined in g = g0 ⊕ V associated to a given representation p : g0 → gl(V) only depends on the multiplicities of the irreducible modules of dimensions 1, 2, and 2l %2b 1, when g0 = gl2; if g0 = sl2, it depends only on the multiplicities of the irreducible modules of dimensions 1, 2, and 3. In dealing with any finite-dimensional representation p : gl2 → gl(V), the representation space V can be decomposed in the form V1 ⊕ V2, with a completely reducible V1 associated to diagonal Jordan blocks of p(I2), and a noncompletely reducible V2 associated to nondiagonal Jordan blocks of p(I2). It is then proved that the given classification only depends on the representation in V1. The question of which isomorphism classes admit invariant orthogonal-like geometric structures is also settled; in other words, it is determined which superalgebras studied in this work are quadratic in the sense of [2]. It is shown in particular that the only quadratic Lie superalgebra of the form sl2 ⊕ V is osp1,2. © Taylor %26amp; Francis Group, LLC.

publication date

  • 2011-01-01