The Algebra Seminar

SPRING 2003

The seminar meets Tuesdays in room LN 2205 at 2:50 p.m. There will be refreshments in the Anderson Reading Room at 4:00.

Organizer: Alex Feingold

• September 9:
  Title: Organizational meeting



• January 28 : Luise Kappe
  Title: ``From groups to loops: variations on a theme by Bernhard Neumann''
  Abstract: A group is said to have a covering by subgroup if it is the set-theoretic union of proper subgroups, and if the set of subgroups is finite we say the covering is finite. Results on finite coverings appear for the first time in a 1941 book by Scorza, among them the theorem that a group is the set theoretic union of three proper subgroups if and only if the group has a homomorphic image isomorphic to the Klein 4-group, a result rediscovered by other authors later. The contributions of Bernhard Neumann in this area were fundamental for the development of the subject, foremost the following result which has been called Neumann's Lemma: If a group is the union of finitely many cosets of subgroups, then the cosets of subgroups of infinite index can be omitted from the union and the remaining cosets still cover the group. Neumann's Lemma leads to a characterization of groups having a finite covering by subgroups or normal subgroups, and being the union of finitely many abelian subgroups. It appears to be only natural to look at covering questions in other algebraic structures. This has been done for rings, semigroups, and most recently for loops. The focus of this talk is on coverings of loops by subloops. The starting point is an example of an infinite loop which is the union of three commutative subloops, but has no finite homomorphic image and has a trivial center. This shows that the results for loops cannot be as general as those for groups and justifies the additional assumptions we have to pose on the loops or the covering to arrive at analogues of the results for groups.



• February 4 : John Donnelly
  Title: ``Conditions for the amenability or non-amenability of R. Thompson's group F''
  Abstract: Let G be a finitely generated group, and let H be a finite generating set for G. We say that G is amenable if for each k in (0,1), there exists a nonempty, finite subset E of G such that for each h in H, $| E \cap Eh | / | E | > k$, that is, the intersection of E and Eh has order greater than k times the order of E. R. Thompson's group F is the group with generators x_j for j = 0,1,2,... and relations (x_n) (x_m) = (x_m) (x_{n+1}) for all n > m. The speaker will give some conditions for F to be amenable.



• February 11 : Stephanie Reifferscheid
  Title: ``On the join of subgroup-closed Fitting classes of finite solvable groups''
  Abstract: In the theory of finite solvable groups Fitting classes, that is classes of groups closed under taking subnormal subgroups and products of normal subgroups, play an important role. Since the intersection of Fitting classes is again a Fitting class, we can define the Fitting class generated by a given set S of groups to be the intersection of all Fitting classes containing S. Unfortunately, this class is very hard to deal with - for instance even the problem of finding an effective description of the Fitting class generated by the symmetric group on three elements is still unsolved. Thus it seems unlikely to get strong results about the Fitting class generated by arbitrary many Fitting classes. In this talk we will see that the situation is totally different if we restrict ourselves to the case that all classes under consideration are s-closed Fitting classes, that is, Fitting classes closed under taking subgroups. We will show that the join of arbitrary many s-closed Fitting classes of finite solvable groups behaves nicely with respect to intersection and certain extensions, this leading to strong results concerning the lattice of all s-closed Fitting classes of finite solvable group.



• February 18 : Alex Feingold
  Title: ``Fusion Algebras and Tensor Algebras: Geometrical Aspects of Computational Algorithms''
  Abstract: For any finite dimensional simple Lie algebra, g, and any positive integer, k, there is a commutative, associative algebra, F(g,k), with a distinguished basis indexed by a finite set of dominant integral weights of g, determined by k. The structure constants with respect to this distinguished basis are non-negative integers with special properties. The tensor algebra of g is a commutative, associative algebra, T(g), with a distinguished basis indexed by all the dominant integral weights of g. The structure constants of T(g) are dimensions of certain subspaces, so they are certainly non-negative integers. Algorithms for computing these structure constants for F(g,k) and for T(g) will be discussed, giving some relations between them.



• February 25 : Charles Ragozzine
  Title: ``From Lie Algebras to Quantum Enveloping Algebras''
  Abstract: In the last twenty years, quantum groups have emerged as an area of interest for mathematical research. The term quantum group is reserved for special Hopf algebras that are nontrivial deformations of the algebra of regular functions of an algebraic group or nontrivial deformations of the enveloping algebras of semisimple Lie algebras. We will examine a specific example of the latter type, following the general theory by building up from the algebraic group to the quantum group. The end result will be a construction of the quantum enveloping algebra of the Lie algebra sl(2).



• March 4 : Joseph Evan
  Title: ``Permutable Subgroups of Direct Products: An Overview''
  Abstract: A subgroup M of a group G is a permutable subgroup of G, or is permutable in G, if for all subgroups X of G, MX=XM. This is precisely the condition under which MX is itself a subgroup of G. In this talk, we will provide necessary and sufficient conditions for an arbitrary subgroup of the direct product GH to be permutable, and then discuss an example which shows us that in some way, this characterization is the best one possible. We will then give a summary of which questions regarding permutability in a direct product GH have and have not been answered.



• March 11 : Spring Recess, No Meeting
  Title: ``''
  Abstract:



• March 18 : Alex Feingold
  Title: ``Fusion Algebras and Tensor Algebras: Geometrical Aspects of Computational Algorithms (Continued from Feb. 18)''
  Abstract: See Abstract of Feb. 18.



• March 25 : Rigoberto Florez
  Title: ``The Relationship Between Full Algebraic Matroids and Lattice Properties, Part II: Pseudomodularity and Full Algebraic Matroids''
  Abstract:



• April 1 : Gail Yamskulna
  Title: ``C_2 - cofiniteness of the vertex operator algebra V_L^+ when L is a rank one lattice''
  Abstract: Let L be a rank one positive definite even lattice. We prove that the vertex operator algebra (VOA) V_L^+ satisfies the C_2 condition. Here, V_L^+ is the fixed point sub-VOA of the VOA V_L associated with the automorphism lifted from the -1 isometry of L.



• April 8 : Arthur Lubovsky
  Title: ``Groups and rings in which the commutator operation is associative''
  Abstract: The commutator operation is a binary operation on groups and rings, but not necessarily associative. We look at various characterizations of groups and rings with associative commutator operation.



• April 15 : Joshua Palmatier
  Title: ``An Introduction to M-Zeroids and Implicators on Finite Chains''
  Abstract: This is the first of two talks being presented for the admission to candidacy exam. The second talk, titled: ``Connections Between M-Zeroids and Implicators on Finite Chains", will be presented on Wednesday, April 16, at 1:10 PM in LN-2205.



• April 22 : Dani Wise, McGill University
  Title: ``Nonpositive immersions, local indicability and coherence''
  Abstract: A group G is locally indicable if every nontrivial finitely generated subgroup of G has an infinite cyclic quotient. A group G is coherent if every finitely generated subgroup of G is finitely presentable. I will discuss a certain Euler characteristic condition on a 2-complex X which implies that the fundamental group of X is both locally indicable and coherent.



• April 29 : Ivonne Ortiz
  Title: ``The lower algebraic K-theory of $\Gamma_3$''
  Abstract: In this dissertation we compute the lower algebraic K-theory of $\Gamma_3$ a discrete subgroup of the group of isometries of hyperbolic 3-space. This group forms part of a family of hyperbolic, non-cocompact, n-simplex reflection groups from which to study the problem of computing the K-theory of infinite groups with torsion. The main result is that for $\Gamma_3$, the Whitehead group of $\Gamma_3$ is zero, $\tilde{K}_0(\mathbb Z\Gamma_3)=\mathbb Z/4 \oplus \tilde{K}_0(\mathbb Z[\mathbb Z/2 \times S_4])$, $K_{-1}(\mathbb Z\Gamma_3)=\mathbb Z \oplus \mathbb Z$ and $K_n(\mathbb Z\Gamma_3)$ is zero for all $n < -1$.



• May 6 : Adam McCaffery
  Title: ``On cyclic commutator subgroups''
  Abstract:

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