Lie algebras from a categorical perspective

The variety of Lie algebras is the central object of study in the realm of non-associative algebras. In this talk we explain, through a series of recent results, why this is no accident: working in the framework of semi-abelian categories, which every variety of non-associative algebras over a field K inherits, a number of a priori unrelated categorical and structural properties each single out Lie algebras as the unique non-trivial such variety. After an overview of these categorical-algebraic methods, we will discuss (some of) the following characterisations. Over a field of characteristic zero, the variety Lie_K is:

• the only non-abelian variety that is locally algebraically cartesian closed, a non- abelian form of cartesian closedness, or algebraic exponentiation;
• the only non-abelian variety whose representations are representable, with representing object the Lie algebra gl(V ) of linear endomorphisms;
• the only non-abelian variety admitting a universal Kaluzhnin–Krasner embedding theorem, expressing every extension through a wreath product;
• over a field of characteristic zero, the only non-trivial variety in which every subalgebra of a free algebra is free and the square of every ideal is again an ideal.

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There will be coffee and cake after the seminar in the common room