Multiple Lie Theory

Multiple Lie theory is the application of the methods of Lie group and Lie algebra theory to double, triple and $n$-fold structures arising in differential geometry.

Lie groups owe their power and importance to two principal properties: they can be linearized by differentiation to Lie algebras, which admit an algebraic classification, and they embody the most immediately recognizable and important notion of symmetry.

Lie groupoids extend this paradigm to the symmetry properties of bundle structures and to other more subtle forms of symmetry. The elements of Lie groupoids are abstract elements of length' and those of Lie algebroids are abstract tangent vectors --- they thus extend the basic tools of standard differential geometry and many of the basic processes of standard differential geometry are encompassed by the Lie theory of Lie groupoids and Lie algebroids.

Double Lie groupoids correspondingly consist of abstract elements of area'. Applying known constructions to obtain a `double Lie algebroid' is straightforward, but the abstract concept of double Lie algebroid is nontrivial and depends on the duality theory of double vector bundles. Double Lie algebroids may be regarded as abstract forms of the double (= iterated) tangent bundle, but also generalize the classical Drinfel'd double of a Lie bialgebra.

Double Lie structures arise naturally in Poisson geometry. For example, the cotangent of a Poisson Lie group possesses a Lie groupoid and a Lie algebroid structure which are compatible in a categorical sense; such structures are known as LAgroupoids and stand midway between double Lie groupoids and double Lie algebroids. Differentiating the cotangent LAgroupoid of a Poisson Lie group gives the classical Drinfel'd double of the Lie bialgebra.

Understanding triple and $n$-fold Lie algebroids requires an understanding of the duality theory of triple and $n$-fold vector bundles. This is a curious theory, of independent interest: the two dualizations possible in a double vector bundle generate the symmetric group $S_3$ while the three dualizations of a triple vector bundle generate a nonsplit extension of $S_4$ by a Klein group. Some results for the $n$-fold case are known, but there are basic unanswered questions in this theory.

For an idea of this work, the best place to start is now my 2011 Crelle paper.

A good alternative is still an announcement article from 1998.

A very elegant and conceptually simple formulation of the concept of double Lie algebroid has been given by Th. Voronov.

An overview of my early work can be found in the Introduction to my 2005 book..