## ABC Algebra Workshop

- Start Date: 04/12/2008
- End Date: 04/14/2008

Georgia Benkart (Wisconsin)

Nils Bruin (SFU)

Najmuddin Fakhruddin (Tata Institute)

Eric Friedlander (Northwestern)

Kalle Karu (UBC)

Jochen Kuttler (Alberta)

Matilde Lalin (Alberta)

Dan Rogalski (San Diego)

Paul Smith (Washington)

Simon Fraser University

**Title**: Mahler measures in group rings (Lalin)

**Abstract**: We will consider a generalization of the Mahler measure elements in group rings, in terms of the Lueck-Fuglede-Kadison determinant. A natural question is the variation of the Mahler measure when the base group changes. We will consider finite and infinite groups and discuss the Mahler measure over infinite groups as limit of Mahler measures over finite groups.

**Title**: Classification of quasi-reductive group schemes (Najmuddin Fakhruddin)

**Abstract**: A quasi-reductive group scheme is a flat affine group scheme over a discrete valuation ring such that the generic fibre and the reduced geometric special fibre are reductive groups of the same dimension. The motivation for the definition, due to G. Prasad and J. Yu, comes from the work of Mirkovic and Vilonen on the geometric construction of the Langlands dual of a reductive group. I will give a short sketch of how quasi-reductive group schemes arise, state some of the results of Prasad and Yu, and then sketch the proof of a classification result for the generic fibre of such group schemes over strictly henselian discrete valuation rings.

**Title**: Mahler measures in group rings (Matilde Lalin)

**Abstract**: We will consider a generalization of the Mahler measure to elements in group rings, in terms of the Lueck-Fuglede-Kadison determinant. A natural question is the variation of the Mahler measure when the base group changes. We will consider finite and infinite groups and discuss the Mahler measure over infinite groups as limit of Mahler measures over finite groups. This is a joint work with O. Dasbach.

**Title**: Subalgebras of the Sklyanin algebra (Dan Rogalski)

**Abstract**: The Sklyanin algebra of dimension 3 is the generic example of a noncommutative projective plane. We study some interesting subalgebras of the 3-Veronese of the Sklyanin algebra which behave like blowups along a divisor of degree at most 7 on the elliptic curve embedded in the plane. We show that every subalgebra generated in degree one which is also a maximal order must be one of these algebras.

**Title**: An equivalence of categories involving the graded Weyl algebra and an algebraic quotient stack. (Paul Smith)

**Abstract**: Let A denote the first Weyl algebra over the complex field. It is isomorphic to the ring of differential operators with polynomial coefficients on the complex line. It is generated by elements x and y subject to the relation xy-yx=1. We consider it as a Z-graded ring with the degrees of x and y being +1 and -1 respectively. Let S denote the commutative algebra over the complex numbers generated by indeterminates x_n, n \in Z, modulo the relations x_n^2+n=x_m^2+m.

Let Z_{fin} be the abelian group whose elements are the finite subsets of Z (the integers) with group operation "exclusive or". We give S a grading by Z_{fin} by declaring degree(x_n)={n}. Let G be the affine group scheme Spec(CZ_{fin}) where CZ_{fin} is the complex group algebra for Z_{fin} endowed with its standard Hopf algebra structure. The grading on C corresponds to and action of G as automorphisms of S or, equivalently, as automorphisms of Spec(S).

Let X denote the stack-theoretic quotient [Spec(S)/G].

Theorem: The following three categories are equivalent:

(1) Gr(A,Z)

(2) Gr(S,Z_{fin})

(3) Qcoh X

(4) G-equivariant S-modules

where Gr(-,-) denotes the category of graded modules with degree preserving homomorphisms, and Qcoh X is the category of quasi-coherent sheaves on X.

The equivalence of the categories in (2), (3), and (4) is well-known. The equivalence with the category of graded modules over the Weyl algebra is new and is the focus of this talk.

**Title**: Hard Lefschetz theorem for barycentric subdivisions. (Kalle Karu)

**Abstract**: The well-known Hard Lefschetz theorem states that cup product with the first Chern class of an ample line bundle defines an isomorphism in certain degrees of cohomology. It has been conjectured by McMullen and Stanley that for toric varieties the Hard Lefschetz theorem holds even in the non-projective case, with ample line bundle replaced by a general line bundle. I will discuss this conjecture in the case where the toric variety corresponds to the barycentric subdivision of a fan.

**Title**: Representations and Cohomology of G(F_q) (Eric M. Friedlander)

**Abstract**: Let G be an algebraic group smooth over a prime field F_p. We present an extension to the prime power q = p^d of the comparison by J. Carlson, Z. Lin, and D. Nakano of the cohomology of the of the finite Chevalley group G(F_p) and the cohomology of the p-restricted Lie algebra g = Lie(G). We may also present comparisons of other invariants of G(F_q) and those of associated p-restricted Lie algebras.

**Title**: Equivariant tree models (Jochen Kuttler)

**Abstract**: A spaced tree is a tree together with a vector space at each vertex (plus some additional data). Spaced trees arise naturally in algebraic statistics and phylo-genetics. For example consider n species alive today. Assuming that (if properly aligned) the positions in the strings of nucleotides of these n species have evolved independently and according to some statistical process one obtains an empirical probability distribution on the set {A, C, G, T}^n. On the other hand, any hypothetical evolutionary tree with n leaves each one corresponding to one species gives rise to a family of distributions on {A, C, G, T}^n, parameterized by a distribution at the root of the tree, and transition matrices along the edges. The vector space at each vertex of the tree then is simply the free vector space generated by A, C, G, T , and the transition matrices become linear maps (what we call representations of the tree in line with the corresponding notion for quivers).

An important question is how to test these hypothetical distributions against the empirically observed one, i.e. to determine whether the empirical distribution can be obtained from a given tree by a suitable parameterization. In order to do this, as a first-order approach one could determine generators of the ideal of polynomial functions vanishing on the entire family of distributions, the so called phylogenetic identities, and evaluate the generators on the empirical distribution (this is of course not conclusive, since not all elements in the zero set of the ideal need to be parameterizable). In practice, a finite group often acts on the data (e.g. permutes {A, C, G, T }), and then the distributions are invariant, whence equivariant models. We introduce a unified and generalized approach to tree models in algebraic statistics with a somewhat surprising application of classical invariant theory, notably the First Fundamental Theorem for GL_n . In particular, we show how the computation of the ideal may be reduced to the computation of the ideal for smaller trees (i.e. stars). This proves a conjecture of Allman-Rhodes who showed the reduction works set-theoretically. I will give a short introduction to the sub ject and outline our approach.

This is joint work with Jan Draisma (Eindhoven Technical University).

**Title**: Explicit methods for determining the rational points on curves (Nils Bruin)

**Abstract**: Faltings proved in 1983 that curves of general type only have a finite number of rational points. However, his prove does not give us a clue on how to find them. An earlier, partial result by Chabauty in 1941 uses p-adic analysis is much better suited for explicit applications. In this talk I will discuss how Chabauty's method, combined with a combinatorial argument, is often able to produce sharp bounds on the number of rational points on a curve.

**Title**: Bases, Braids, and Beyond (Georgia Benkart)

**Abstract**: This talk will focus on different choices of bases for simple Lie algebras (particularly for sl_2), and their connections with the modular group, representations of the braid group, lattices, and hyperbolic Kac-Moody Lie algebras.

The event consists of a series of 9 talks from algebraists, some from

Alberta and British Columbia, and others from outside of Canada. We

will have a banquet on the evening of 12th of April.

**Workshops**

Saturday

9:00 - 9:15 Introductory remarks

9:15 - 10:00 Matilde Lalin

10:00 - 10:30 Coffee break

10:30 - 11:15 Paul Smith

11:30 - 12:15 Dan Rogalski

12:15 - 2:00 Lunch

2:00 - 2:45 Nils Bruin

2:45 - 3:15 Coffee break

3:15 - 4:00 Jochen Kuttler

5:00 Banquet

Sunday

9:00 - 9:45 Eric Friedlander

10:00 - 10:45 Najmuddin Fakhruddin

10:45 - 11:15 Coffee break

11:15 - 12:00 Georgia Benkart

12:15 - 1:00 Kalle Karu

Jason Bell, jpb @ cs.sfu.ca

**Financial Support **

Financial support will be available for a limited number of graduate students and postdocs. Applicants for support should indicate this on the registration form, and should provide:

- Brief letter of application
- CV
- Letter of support from the applicant's supervisor.

These should be emailed to jpb@math.sfu.ca giving the applicant's name in the subject line, as soon as possible. Plain text file is preferred.

**Registration**

To register, please click here.

**Directions**

The conference will be held at IRMACS. For directions, please see: http://irmacs.sfu.ca/visitor_info/directions.php