NONCOMMUTATIVE ALGEBRA
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- Versione italiana
- Academic year
- 2020/2021
- Teacher
- CLAUDIA MENINI
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/02
Training objectives
- The aim of the course is to provide some basic concepts of the Theory of Hopf Algebras by developing some specific subjects.
First of all, starting from the notion of algebra, the concept of coalgebra is established and developed. Several examples are provided. The Sweedler's notation is given and practised. The categories of comodules and rational modules are presented and investigated.
The notion of Hopf algebra is introduced and analyzed, meaningful examples are given.
The following topics are developed: integrals, semisimplicity, cosemisemplicity and the coradical.
At the end of the course the student will be aware of the concept of Hopf algebra and will be trained in some specific techniques. This will allow him/her to address further basic topics concerning Hopf algebras, independently. Prerequisites
- Basic knowledge in Module Theory and in Categories.
Course programme
- The course consists in 42 hours of frontal lessons.
Definition of algebras and coalgebras. Examples of coalgebras. Generalized coassociativity and other formulae concerning Delta_n. Sweedler's notation. Tensor product of two coalgebras. The opposite coalgebra. Subcoalgebras, right coideals, left coideals and coideals. The fundamental Theorem of the quotient coalgebra. The algebra structure on Hom(C,A) where C is a coalgebra and A is an algebra. Examples, the dual C* of a coalgebra C. Grouplike elements of the dual of an algebra A are algebra morphisms from A into K. Linear independence of grouplike elements. (12 hours)
Comodules and rational C*-modules. Properties of the category of rational comodules. The right adjoint of the forgetful functor from the category of rational modules to the category of vector spaces. C is an injective cogenerator of the category of comodules. (6 hours)
Equivalent definitions of bialgebra. Definition of Hopf Algebra. Properties of the antipode. Exercises: practising the Sweedler notation.
Some properties of Hopf Algebras. The dual of a finite dimensional Hopf Algebra. Quotient bialgebras and Hopf algebras and their universal property.
Examples of Hopf algebras: The group algebra kG and its dual in the finite case.
The Tensor algebra, the Symmetric algebra and the Enveloping algebra. (8 hours)
Hopf modules. The equivalence between the category of Hopf modules over a Hopf algebra and the category of vector spaces. (3 hours)
Integrals in a Hopf algebra and in its dual and their properties. The space of integrals in finite dimensional Hopf algebras.
Semisimple and cosemisimple Hopf algebras.
Maschke's Theorem and dual Maschke's Theorem. Every semisimple Hopf algebra is of finite dimension. The dual of a semisimple Hopf algebra (resp. of a finite dimensional cosemisimple coalgebra) is cosemisimple (semisimple). The group algebra is always
cosemisimple. The Classical Maschke's Theorem. (6 hours)
The coradical. Some useful results. Simple coalgebras and simple comodules.
The correspondence between subcoalgebras of a coalgebras and the ideals of its dual.
Pointed coalgebras, connected coalgebras, irreducible coalgebras and their relations. Some results on semisimple modules and semisimple algebras. Schur's Lemma. Every ring is dense in the biendomorphism ring of a simple module over itself.
Every simple left artinian ring is isomorphic to a ring of finite matrices over a division ring. The Jacobson radical of a finite dimensional algebra is the intersection of all maximal two-sided ideals.
The coradical of a coalgebra C is the socle ot the left C*-module C.
The relations between the coradical of a finite dimensiona coalgebra C and the Jacobson radical of C*. The coradical of the tensor product of two coalgebras. (7 hours)
. Didactic methods
- All the topics of the course are deeply developed during lectures. In particular, the full proof of all stated results is given. The theoretical discussion is supported by examples and exercises. Students are asked to take part actively in lessons also by solving given exercises.
Learning assessment procedures
- The aim of the final exam consists in verifying the level of knowledge of the formative objectives previously stated.
The final exam consists in an oral discussion on all subjects of the course.
The first question concerns a topic chosen by the student that must be exposed in a detailed way. The other questions on the remaining part of the program have the objective to verify the comprehension of the basic concepts and the ability to link and compare different aspects discussed in the course. For this reason details are not required. The answer to first question is equivalent of 40% of the final mark of the exam.
By agreement with the students, the oral examination might consist also of a seminar given by the student on a subject connected to the course but not developed in the course itself.
The student that exposed a topic in class with teacher's positive evaluation is exempted from the first question and this evaluation is equivalent of 60% of the final mark of the exam. Reference texts
- Teacher’s Notes that can be found online. Stenström, Rings of quotients, Die Grundlehren der Mathematischen Wissenschaften, Band 217.
M. Sweedler, Hopf algebras. Mathematics Lecture Note Series W. A. Benjamin, Inc., New York 1969.
S. Montgomery, Hopf algebras and their actions on rings. CBMS Regional Conference Series in Mathematics, 82. A.M.S.Providence, RI,1993.
S. Dascalescu, C. Nastasescu, S. Raianu, Hopf algebras. An introduction. Monographs and Textbooks in Pure and Applied Mathematics, 235. Marcel Dekker, Inc., New York, 2001.
