On the Geometric Classification of Lie Super Algebras

Plática dada por María Alejandra Alvarez (Universidad Antofagast) en el marco del evento Graded Algebras, Geometry and related Topics para celebrar el 60 aniversario de Adolfo Sánchez Valenzuela el jueves 17 de noviembre del 2016 en la Universidad Autónoma de Yucatán (UADY)

Abstract:
Given a (m,n)-dimensional mathbb{K}-vector superspace V=V_0oplus V_1, one can consider the space of all bilinear products [cdot,cdot]:Vtimes Vrightarrow V such that (V,[cdot,cdot]) is a Lie superalgebra, i.e., those satisfying
· [V_i, V_j] subset V_{i+j} , for i, j in mathbbZ_2.
· The super skew-symmetry: [x,y]= -(-1)^{|x||y|}[y, x].
· The super Jacobi identity:
(-1)^{|x||z|}[ [ x,y ], z ] +(-1)^{|x||y|}[ [ y,z ], x ] + (-1)^{|y||z|}[ [ z,x ], y ] =0
for x,y,z in (V_0 cup V_1) setminus { 0 }.

In a given basis of V, a Lie superalgebra product is defined by the set of structure constants in the space mathbb{K}^{m^3+2nm^2} verifying the polynomial equations given by the super skew-symmetry and the super Jacobi identity. So, the set mathcal{LS}^{(m,n)} of Lie superalgebra products in the space mathbb{K}^{m^3+2nm^2} is an affine algebraic variety. Moreover, the G=GL_m(mathbb{K})oplusGL_n(mathbb{K}) acts on mathcal{LS}^{(m,n)} by “change of basis”: gcdot[x,y]=g[g^{-1}x,g^{-1}y], for gin G and x,yin V.

Given two Lie superalgebras mathfrak g and mathfrak h, we say that mathfrak g degenerates to mathfrak h if mathfrak h lies in the Zariski closure of the G-orbit mathcal{O}(mathfrakg). By studying the orbits closures in the variety of Lie superalgebras, one obtains the geometric classification.

In this work we consider the variety of complex Lie superalgebras of dimension (2,2) and obtain the algebraic and geometric classifications.

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