Hans Havlicek: Multilinear Algebra

This is a series of lectures and exercises for graduate students of mathematics. We present an introduction to the basic concepts of multilinear algebra.

Contents

The subsequent list refers to my lectures in summer 2000 (chapters 1-3) and winter 2001 (chapters 4 and 5).

The first part is due to be read in summer 2008.

1. Tensor Products

• Multilinear mappings of vector spaces
• Existence and universal property of the tensor product
• Commutativity and associativity of the tensor product
• The tensor product in terms of coordinates
• Tensor products and spaces of linear mappings
• Kronecker product of linear mappings (matrices)
• Contraction
• Lowering and raising of indices
• Duality

2. Tensor Algebras

• Covariant, contravariant and mixed tensors
• Classical definition and notation of a tensor in terms of coordinates
• Structure tensor of an algebra
• Mixed tensor algebra
• Universal property of the tensor algebra

3. Exterior Algebras

• Exterior powers and p-vectors
• Grassmann coordinates of subspaces
• Alternation operator
• Exterior powers of linear mappings
• Exterior algebra
• Duality and p-forms
• Exterior algebra
• Decomposable p-vectors

4. Clifford Algebras

• Quadratic forms
• Clifford mappings
• Clifford mappings and exterior algebra
• Clifford Algebras
• Dimension of a Clifford algebra
• Examples of Clifford algebras
• Structures on a Clifford algebra

5. Spinors

• Clifford groups and spin groups
• Isometries
• The centre of a Clifford algebra
• Semisimple modules and representation of algebras
• Spaces of spinors