Let L be a Lie algebra over the field F. A lattice automorphism of a?’ is an automorphism ??: a?’(L) a?’ a?’(L) of the lattice a?’(L) of all subalgebras of L. We seek to describe the lattice automorphisms in terms of maps ??: La?’L of the underlying algebra. A semi-automorphism ?? of L is an automorphism of the algebraic system consisting of the pair (F, L), that is, a pair of maps ??: Fa?’ F, ??: La?’ L preserving the operations. Thus ??: F a?’ F is an automorphism of F and (x + y)?? = x?? + y??, (xy)?? = x??y?? (??x)?? = ????x?? for all x, y ?μ L and ?? ?μ F. Clearly, any semi-automorphism of L induces a lattice automorphism. To study a given lattice automorphism ??, we select a semi-automorphism a such that ????-1 fixes certain subalgebras, and so we reduce the problem to the investigation of lattice automorphisms leaving these subalgebras fixed.
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机译：令L为场F上的一个Lie代数。a？'的晶格自同构是自同构??：所有晶格a？'（L）的a？'（L）a？'a？'（L）我们试图用图??来描述晶格自同构：基础代数的La？'L。半自同构L的代数是由（F，L）对组成的代数系统的自同构，即一对映射??：Fa？’F，??：La？’L保留了运算。因此??：F a？’F是F和（x + y）?? = x ?? + y ??，（xy）?? = x ?? y ?? （？？X）？？ = ???? x ??对于所有x，y？μL和？显然，L的任何半自同构都会引起晶格自同构。为了研究给定的晶格自同构ε，我们选择一个半自同构α，使得β-1固定某些子代数，因此我们将问题简化为研究使这些亚代数保持不变的晶格自同构。
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