MATH-F-303: Algebra and Geometry II
Algebra is part of mathematics dealing with formal
structures and relations between them. In essence,
algebra is a language of mathematics, studied not
only for its own beauty, but used almost everywhere.
I would explain basic concepts of group theory
and linear algebra, focusing on those which are
used in geometry and physics.
0. Basic concepts of algebra: groups, rings, fields,
vector spaces.
1. Definition of an algebra. Examples: matrices, quaternions,
field extensions. Algebra of polynomials. Hypercomplex numbers.
2. Group of rotations of R^2 and complex numbers.
Groups of rotations in R^3, R^4 and their quaternionic
interpretation.
3. Structure of the groups SO(3), SO(4).
4. Bilinear forms: symmetric, antisymmetric bilinear
forms. Orthogonal complement and orthonormal basis.
Signature as invariant of a bilinear symmetric form.
Quadratic forms and their polarizations.
5. Tensor product and its basic properties.
Duality between tensor product and the space of bilinear forms.
Space of homomorphisms and its interpretation as a tensor product.
6. Tensor algebra. Algebra homomorphisms.
Ideals in an algebra. Left and right ideals.
Tensor algebra as free algebra.
7. Algebras defined by generators and relations.
Clifford algebra. Complex numbers, quaternions,
Mat(2) as Clifford algebras.
8. Graded algebras and graded ideals.
Grassmann algebra and its basic properties.
Antisymmetrization. Sign rule. Dimension of
the Grassmann algebra.
Useful literature:
B. L. van der Waerden, "Modern algebra"
Serge Lang, "Algebra"
E. B. Vinberg, "A Course in Algebra"
I. M. Gelfand and A. Shen, "Algebra"
Igor R. Shafarevich, "Basic Notions of Algebra"