Lectures related to Epsilonization: Algebra Homomorphisms and Derivation Numbers

Representation Theory: Algebras and Homomorphisms

Covers the goals and motivations of representation theory, focusing on associative algebras and homomorphisms.

Explores Schur's lemma and its applications in representations of an associative algebra over an algebraically closed field.

Ring Homomorphisms and Ideals

Explores ring homomorphisms, bilateral ideals, ring features, and ideal operations.

Group Cohomology

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

Pushouts in Group Theory: Universal Properties Explained

Covers the construction and universal properties of pushouts in group theory.

Polynomials and Endomorphisms

Covers the fundamentals of polynomials, endomorphisms, division, roots, matrices, and algebraic homomorphisms.

Morphisms between Affine Varieties

Covers defining morphisms between affine algebraic varieties and constructing morphisms using algebraic homomorphisms.

Algebras and Field Extensions

Introduces algebras over a field, k-linear endomorphisms, and commutative algebras.

Algebra: Chain Complexes

Explores chain complexes, abelian groups, homomorphisms, homology groups, and free abelian groups.

Group Homomorphisms

Explores group homomorphisms, isomorphisms, and generators in abstract algebra.

Group Theory - Part 2

Explores Cayley tables, group operations, homomorphisms, Lie groups, and differentiable groups.

Groups: Definitions, Properties, and Homomorphisms

Introduces the basic concepts of groups, including definitions, properties, and homomorphisms, with a focus on subgroup properties and normal subgroups.

Algebra of Matrices: Properties and Fields

Covers the properties of the algebra of matrices and related concepts.

Push-out: Categorical Perspective on Quotients

Explores the concept of push-out in group homomorphisms.

Chinese Remainder Theorem: Rings and Fields

Covers the Chinese remainder theorem for commutative rings and integers, polynomial rings, and Euclidean domains.

Amalgams: Group Pushouts

Covers the concept of amalgams, defining pushouts and quotient groups in group theory.

First Isomorphism Theorem

Covers the first isomorphism theorem in group theory and its applications.

Group Actions on Sets

Explores group actions on sets through homomorphisms and Cartesian products, illustrating their properties and equivalent definitions.

Homomorphisms: Birational Maps and Affine Varieties

Covers homomorphisms between affine varieties, birational maps, regular groups, and connectedness.

Lie Algebras: Introduction and Structure

Introduces Lie algebras, vector spaces with a special bracket operation.