** Generalised linear algebra **

In the first note we introduced one of the fundamental structures of generalised geometry, namely the generalised tangent bundle . In the extension of the standard tangent bundle to , we are simultaneously extending linear algebra to some notion of *generalised linear algebra*. This is a point nicely emphasised in Roberto Rubio’s lecture notes.

Think of it this way. In standard differential geometry we have the tangent bundle , whose sections are endowed with a Lie bracket satisfying the Leibniz rule

In generalised geometry, on the other hand, we are motivated to extend linear algebra to generalised linear algebra such that we replace the standard vector space of differential geometry with the generalised vector space

with a d-dimensional vector space and its dual. We can denote the elements of this space with and .

For intuition, realise that this extension can be motivated at its most basic when studying symplectic forms that represent a skew-symmetric analogue of an inner product: i.e., skew-symmetric isomorphisms or . This also takes us into a study of linear presymplectic and Poisson structures, but such preliminary discussion is dropped in these notes.

**Definition 1 (Canonical pairing and inner product signature)**

*Given the generalised vector space , then for two generalised tangent vectors there exists a canonical metric naturally equipped on this space such that*

*
which is a maximally indefinite inner product of signature . Note that denotes contraction by . The factor of is purely convention with no geometric meaning. *

Next, define a subspace . Then the orthogonal compliment for this pairing can be defined in the standard sense

It is said that is isotropic if . The idea of maximally isotropic subspaces raises interesting considerations, particularly globally defined structures with an integrability condition, which are often studied under the heading of linear Dirac structures. As a matter of introduction, if is isotropic but not constrained in an isotropic subspace than it is maximally isotropic.

The bilinear form (3) can be verified by taking a basis for with dual basis for . Then we can define as the total basis for , wherein the inner product admits a diagonal form .

Explicitly, consider the basis with the canonical pairing given by

in which the signature is with an orthogonal basis of vectors of positive length and vectors of negative length. It can then be shown that the dimension of the maximally isotropic subspace must be at most . In fact, this statement is true for any maximally isotropic subspace.

**Proposition 2**

*For a vector space , we have that is the dimension of any maximally isotropic subspace of .*

*Proof:* Let be a maximally isotropic subspace. Consider to be semidefinite such that or for all . Otherwise, consider a compliment so that , then if contains two vectors with and , a suitable linear combination would be null. As a result would be isotropic containing . Additionally, as we have

On the other hand, given that the pairing is non-degenerate, we write such that we now have

which is to say so .

What we have found is that, as is a vector space of dimension with a symmetric pairing of signature , there is a choice two maximally isotropic subspaces.

**Remark 1 (Motivation)**

*There is another way to motivate the inner product (3) by simply noting that and are vector spaces, and, as they are related by a duality transformation (this will be made clear later) we obtain a natural bilinear symmetric form.*

** Brief remark: Clifford E modules **

The bilinear form (3), or what is also referred to as the canonical metric or canonical pairing, should not be confused with a metric in the standard sense. To obtain an ordinary metric, we must introduce extra structure.

To make sense of this claim, consider the generalised bundle from note 1 [LINK]. Each fibre of has an action on the corresponding fibre of the exterior bundle given by

It follows that

which gives description of the exterior algebra in terms of the structure of a bundle of Clifford modules with respect to the natural bilinear on . Typically, in the literature the corresponding Clifford algebras generated by is denoted Cliff(E).

To conclude: as we’ll see, the form on enables the reduction of the structure group of to . In fact, in the maths literature we see that further reduction to the structure is possible. This will be the topic of the next note, as we focus on symmetries and the structure group generators.