Algebra Research Group
|Professor; Graduate Program Chair
|Mathematical Sciences - General
|Eng & Math Sciences E461
Algebra at UWM
The department offers a bridge course in algebra (631-632) covering undergraduate material at a more advanced level and pace and a “true” graduate algebra course (731-732) every year. Almost every semester, we also offer an advanced course in algebra; some recent examples include Ring Theory, Homological Algebra, and special topics courses including Lie Algebras (and Lie Groups), Algebraic Groups, Algebraic Geometry, Noetherian rings, Lie Superalgebras, Combinatorics. Each semester there is a seminar in algebra. Sometimes seminars cover diverse topics; other times they focus on a particular topic. Recent topics have included Toric varieties, exceptional groups and Lie algebras, algebraic geometry, Hopf algebras and quantum groups, homological algebra, non-commutative algebraic geometryá la Artin, van der Bergh, et al., Gröbner bases, symmetric functions and representations of the symmetric group, Kac-Moody Lie algebras, quantum computing.
Below are some of the main research areas among the group, along with the researchers involved in these areas.
- Quantum Groups and Hopf Algebras: Bell, Musson, Zou
- Lie Theory & Algebraic Groups: Musson, Willenbring, Zou
- Representation Theory: Musson, Willenbring, Zou
- Noetherian Rings (including Rings of Differential Operators, Enveloping Algebras, Noncommutative Algebraic Geometry): Bell, Musson
Quantum Groups and Hopf Algebras
What is a quantum group? What field of mathematics does the study of quantum groups belong to? Quantum groups are studied by many mathematicians including topologists (because of their links to knot theory), mathematical physicists (because of their link to quantum theory and field theory, including the quantum inverse scattering method and the Yang-Baxter equation), geometers (because of their connection to non-commutative geometry), workers in Lie theory and algebraic groups (because of their connection to the study of algebraic groups in positive characterstic), and ring theorists (because they are rings — Hopf algebras in fact).
A quantum group is not a group! It is the (generally non-commutative) ring of “functions” on something like a group. It is a Hopf algebra probably with some extra structure (some sort of triangular or co-triangular structure?); in analytic studies, it may be a C*-algebra.
Lie Theory & Algebraic Groups
Lie groups are topological groups which are simultaneously (and compatibly) differentiable manifolds. The tangent space at the identity of a Lie group is a Lie algebra. Lie algebras may also be defined axiomatically as non-associative algebras satisfying certain identities (the identities being those satisfied when the new product [x,y]=xy-yx is introduced on an associative algebra). The finite-dimensional representations of the Lie group and its associated Lie algebra are essentially the same: this was the initial motivation for the study of Lie algebras (where one can employ more algebraic as opposed to analytic tools).
An algebraic group is a group that is simultaneously (and compatibly) an algebraic set (i.e., a set with group operations defined by polynomial equations). To each algebraic group there is also a Lie algebra associated.
Research in this department focuses on representations of Lie algebras and the structure of their enveloping algebras, in both the classical (characteristic 0) and the modular (positive characteristic) case. The study of quantum groups is also closely linked to the study of Lie algebras and algebraic groups.
Representing algebraic objects in a concrete way (say as matrices or permutations) is one of the ways in which abstract algebra attacks concrete problems. Here we are concerned with representations as matrices (equivalently, linear operators on finite dimensional vector spaces). Most members in the algebra group have studied such represenations in some cases (rings, Lie algebras, etc.). A particular form of representation theory is the study of finite-dimensional algebras (sometimes generalized to artinian rings). See here for a brief account of this theory.
The main thrust of the theory of commutative rings is intimately related to the theory of rings of polynomial functions (and rings derived from them such as quotients and localizations). Such rings are noetherian, that is, every ascending chain of ideals eventually becomes stationary. [For non-commutative rings, we must assume this not just for two-sided ideals but for one-sided ideals as well.]
Non-commutative rings are a much more varied species, but they tend to be related to rings of linear operators, which unlike functions, do not commute with each other. The study of non-commutative rings is a field begun in the 20th century, and much of the early work concentrated on division rings and algebras that were finite dimensional over a field. Such algebras are always artinian, that is, every descending chain of left (or right) ideals eventually becomes stationary. An artinian ring is always noetherian, but the converse is not true: artinian rings form a much more restricted class. The concepts of noetherian and artinian rings were abstracted from specific commutative examples in the 1920’s. Natural examples of non-commutative rings need not be noetherian; nevertheless, the noetherian hypothesis is very useful and fortunately does hold in many cases.
It is well-known that any commutative integral domain has a field of fractions; this need not be true for a non-commutative ring lacking zero divisors. O. Ore in 1931 gave necessary and sufficient conditions for a division ring of fractions to exist. While many interesting ring theoretic results were proven in between, it is probably fair to say that the modern study of non-commutative noetherian rings began with A. W. Goldie’s work in 1958-1960 giving necessary and sufficient conditions for a ring to have a semisimple ring of fractions. This result is much deeper than Ore’s 1931 result, and the techniques and subsidiary results gave researchers technical tools that are still in use today. [Goldie showed a noetherian ring has a semisimple ring of fractions if and only if it is semiprime and it has a division ring of fractions if and only if it has no zero divisors.]
Here are some examples of non-commutative noetherian rings that have been studied by members of the algebra group:
- Rings of differential operators on algebraic sets
- Enveloping algebras of finite dimensional Lie algebras and Lie superalgebras
- Group rings of polycyclic-by-finite groups
- Non-commutative polynomial rings
Here are some topics connected with noetherian rings that have been studied by members of the algebra group:
- Classification of simple modules
- Classification of primitive ideals and the structure of primitive factor rings
- Structure of rings connected to algebraic varieties (e.g., rings of differential operators and quantum groups)
- Homological and “dimension” properties
- Localization and the second layer condition
- Structure of injective modules
- Prime ideals in noetherian rings