Past Algebra Seminar

A mathematical structure is `axiomatizable' if it is completely determined by some family of sentences in a suitable first-order language. This idea has been explored for various kinds of structure, but I will concentrate on groups. There are some general results (not many) about which groups are or are not axiomatizable; recently there has been some interest in the sharper concept of 'finitely axiomatizable' or FA - that is, when only a finite set of sentences (equivalently, a single sentence) is allowed.

While an infinite group cannot be FA, every finite group is so, obviously. A profinite group is kind of in between: it is infinite (indeed, uncountable), but compact as a topological group; and these groups share many properties of finite groups, though sometimes for rather subtle reasons. I will discuss some recent work with Andre Nies and Katrin Tent where we prove that certain kinds of profinite group are FA among profinite groups. The methods involve a little model theory, and quite a lot of group theory.

The notion of colour Lie algebra, introduced by Ree (1960), generalises notions of Lie algebra and Lie superalgebra. From an orthogonal representation V of a quadratic colour Lie algebra g, we give various ways of constructing a colour Lie algebra g’ whose bracket extends the bracket of g and the action of g on V. A first possibility is to consider g’=g⊕V and requires the cancellation of an invariant studied by Kostant (1999). Another construction is possible when the representation is ``special’’ and in this case the extension is of the form g’=g⊕sl(2,k)⊕V⊗k^2. Covariants are associated to special representations and satisfy to particular identities generalising properties studied by Mathews (1911) on binary cubics. The 7-dimensional fundamental representation of a Lie algebra of type G_2 and the 8-dimensional spinor representation of a Lie algebra of type so(7) are examples of special representations.

Combinatorial anabelian geometry is a modern branch of anabelian geometry which deals with those aspects of anabelian geometry which manifest themselves over algebraically closed fields of characteristic zero. The origin of combinatorial anabelian geometry is in S. Mochizuki’s pioneering papers from 2007, in which he reinterpreted and generalised some key components of his earlier famous proof of the Grothendieck conjecture. S. Mochizuki  discovered that one can separate arguments which work over algebraically closed fields from arithmetic arguments, and study the former by using combinatorial methods. This led to a very nontrivial development of the theory of combinatorial anabelian geometry by S. Mochizuki and Y. Hoshi and other mathematicians. In this talk, after introducing the theory of combinatorial anabelian geometry I will discuss  applications of combinatorial anabelian geometry to the study of the absolute Galois group of number fields and of p-adic local fields and to the study of the Grothendieck-Teichmueller group. In particular, I will talk about the recent construction of a splitting of the natural inclusion of the absolute Galois group of p-adic numbers to the (largest) p-adic Grothendieck–Teichmueller group and a splitting of the natural embedding of the absolute Galois group of rationals into the commensurator of the absolute Galois group of the maximal abelian extension of rationals in the Grothendieck–Teichmueller group.
 

In 1985, McDowell introduced a family of parabolically induced Whittaker modules over a complex semisimple Lie algebra, which includes both Verma modules and the nondegenerate Whittaker modules studied by Kostant. Many classical results for Verma modules and the Bernstein--Gelfand--Gelfand category O have been generalized to the category of Whittaker modules introduced by Milicic--Soergel, including the classification of irreducible objects and the Kazhdan--Lusztig conjectures. Contravariant forms on Verma modules are unique up to scaling and play a key role in the definition of the Jantzen filtration. In this talk I will discuss a classification of contravariant forms on parabolically induced Whittaker modules. In a recent result, joint with Anna Romanov, we show that the dimension of the space of contravariant forms on a parabolically induced Whittaker module is given by the cardinality of a Weyl group. This result illustrates a divergence from classical results for Verma modules, and gives insight to two significant open problems in the theory of Whittaker modules: the Jantzen conjecture and the absence of an algebraic definition of duality.

A group acting on a regular rooted tree has the congruence subgroup property if every subgroup of finite index contains a level stabilizer. The congruence subgroup problem then asks to quantitatively describe the kernel of the surjection from the profinite completion to the topological closure as a subgroup of the automorphism group of the tree. We will study the congruence subgroup property for a family of branch groups whose construction generalizes that of the Hanoi Towers group, which models the game “The Towers of Hanoi".

 

Deformation quantization modules or DQ-modules where introduced by M. Kontsevich to study the deformation quantization of complex Poisson varieties. It has been advocated that categories of DQ-modules should provide invariants of complex symplectic varieties and in particular a sort of complex analog of the Fukaya category. Hence, it is natural to aim at describing the functors between such categories and relate them with categories appearing naturally in algebraic geometry. Relying, on methods of homotopical algebra, we obtain an analog of Orlov representation theorem for functors between categories of DQ-modules and relate these categories to deformations of the category of quasi-coherent sheaves.
 

Let $p$ be an odd prime and $n$ a natural number. We determine the irreducible constituents of the permutation module induced by the action of the symmetric group $S_n$ on the cosets of a Sylow $p$-subgroup $P_n$. In the course of this work, we also prove a symmetric group analogue of a well-known result of Navarro for $p$-solvable groups on a conjugacy action of $N_G(P)$. Before describing some consequences of these results, we will give an overview of the background and recent related results in the area.

Unitary highest weight modules were classified in the 1980s by Enright-Howe-Wallach and independently by Jakobsen. The classification is based on a version of the Dirac inequality, but the proofs also require a number of other techniques and are quite involved. We present a much simpler proof based on a different version of the Dirac inequality. This is joint work with Vladimir Soucek and Vit Tucek.
 

For $G$ connected, reductive algebraic group defined over $\mathbb{C}$ the Springer Correspondence gives a bijection between the irreducible representations of the Weyl group $W$ of $G$ and certain pairs comprising a $G$-orbit on the nilpotent cone of the Lie algebra of $G$ and an irreducible local system attached to that $G$-orbit. These irreducible representations can be concretely realised as a W-action on the top degree homology of the fibres of the Springer resolution. These Springer fibres are geometrically very rich and provide interesting Weyl group combinatorics: for instance, the irreducible components of these Springer fibres form a basis for the corresponding irreducible representation of $W$. In this talk, I'll give a general survey of the Springer Correspondence and then discuss recent joint projects with Daniele Rosso, Vinoth Nandakumar and Arik Wilbert on Kato's Exotic Springer correspondence.

Formally, the Fourier coefficients of Eisenstein series on Kac-Moody groups contain as yet mysterious automorphic L-functions relevant to open conjectures such as that of Ramanujan and Langlands functoriality. In this talk, we will consider the constant Fourier coefficient, if it even makes sense rigorously, and its relationship to the geometry and combinatorics of a Kac-Moody group. Joint work with Kyu-Hwan Lee.

Let k be a characteristic $p>0$ perfect field, V be a complete DVR whose residue field is $k$ and fraction field $K$ is of characteristic $0$. We denote by $\mathcal{E}  _K$ the Amice ring with coefficients in $K$, and by $\mathcal{E} ^\dagger _K$ the bounded Robba ring with coefficients in $K$. Berthelot's classical theory of Rigid Cohomology over varieties $X/k((t))$ gives $\mathcal{E}  _K$-valued objects.  Recently, Lazda and Pal developed a refinement of rigid cohomology,
i.e. a theory of $\mathcal{E} ^\dagger _K$-valued Rigid Cohomology over varieties $X/k((t))$. Using this refinement, they proved a semistable version of the variational Tate conjecture. 

The purpose of this talk is to introduce to a theory of arithmetic D-modules with $\mathcal{E} ^\dagger _K$-valued cohomology which satisfies a formalism of Grothendieck’s six operations. 
 

In the 1990s Moy and Prasad revolutionized the representation theory of p-adic groups by showing how to use Bruhat-Tits theory to assign invariants to representations of p-adic groups. The tools they introduced resulted in rapid advancements in both representation theory and harmonic analysis -- areas of central importance in the Langlands program. A crucial ingredient for many results is an explicit construction of (types for) representations of p-adic groups. In this talk I will indicate why, survey what constructions are known (no knowledge about p-adic groups assumed) and present recent developments based on a refinement of Moy and Prasad's invariants.​

Roughly speaking, a commutative-by-finite Hopf algebra is a Hopf
algebra which is an extension of a commutative Hopf algebra by a
finite dimensional Hopf algebra.
There are many big and significant classes of such algebras
(beyond of course the commutative ones and the finite dimensional ones!).
I'll make the definition precise, discuss examples
and review results, some old and some new.
No previous knowledge of Hopf algebras is necessary.
 

For a polynomial $h(x)$ in $F[x]$, where $F$ is any field, let $A$ be the
$F$-algebra given by generators $x$ and $y$ and relation $[y, x]=h$.
This family of algebras include the Weyl algebra, enveloping algebras of
$2$-dimensional Lie algebras, the Jordan plane and several other
interesting subalgebras of the Weyl algebra.

In a joint work in progress with Samuel Lopes, we computed the Hochschild
cohomology $HH^*(A)$ of $A$ and determined explicitly the Gerstenhaber
structure of $HH^*(A)$, as a Lie module over the Lie algebra $HH^1(A)$.
In case $F$ has characteristic $0$, this study has revealed that $HH^*(A)$
has finite length as a Lie module over $HH^1(A)$ with pairwise
non-isomorphic composition factors and the latter can be naturally
extended into irreducible representations of the Virasoro algebra.
Moreover, the whole action can be understood in terms of the partition
formed by the multiplicities of the irreducible factors of the polynomial
$h$.
 

Given a stability condition defined over a category, every object in this category
is filtered by some distinguished objects called semistables. This
filtration, that is unique up-to-isomorphism, is know as the
 Harder-Narasimhan filtration.
One less studied property of stability conditions, when defined over an
 abelian category, is the fact that each of them induce a chain of torsion
classes that is naturally indexed.
 In this talk we will study arbitrary indexed chain of torsion classes. Our
first result states that every indexed chain of torsion classes induce a
 Harder-Narasimhan filtration. Following ideas from Bridgeland we
 show that the set of all indexed chains of torsion classes satisfying a mild 
 technical condition forms a topological space. If time we
 will characterise the neighbourhood or some distinguished points. 

In combinatorics, the 'nicest' way to prove that two sets have the same size is to find a bijection between them, giving more structure to the seeming numerical coincidences. In representation theory, many of the outstanding conjectures seem to imply that the characteristic p of the ground field can be allowed to vary, and we can relate different groups and different primes, to say that they have 'the same' representation theory. In this talk I will try to make precise what we could mean by this

The classical Dirac operator is part of an osp(1|2) realisation inside the Weyl-Clifford algebra which is Pin-invariant. This leads to a multiplicity-free decomposition of the space of spinor-valued polynomials in irreducible modules for this Howe dual pair. In this talk we review an abstract generalisation A of the Weyl algebra that retains a realisation of osp(1|2) and we determine its centraliser algebra explicitly. For the special case where A is a rational Cherednik algebra, the centralizer algebra provides a refinement of the previous decomposition whose analogue was no longer irreducible in general. As an example, for the  group S3 in specific, we will examine the finite-dimensional irreducible modules of the centraliser algebra.

The celebrated localisation theorem of Beilinson-Bernstein asserts that there is an equivalence between representations of a Lie algebra and modules over the sheaf of differential operators on the corresponding flag variety. In this talk we discuss certain analogues of this result in various contexts. Namely, there is a localisation theorem for quantum groups due to Backelin and Kremnizer and, more recently, Ardakov and Wadsley also proved a localisation theorem working with certain completed enveloping algebras of p-adic Lie algebras. We then explain how to combine the ideas involved in these results to construct
a p-adic analytic quantum flag variety and a category of D-modules on it, and we show that the global section functor on these D-modules yields an equivalence of categories.

Polyfree groups are defined as groups having a series of normal
subgroups such that each sucessive quotient is free. This property
imples locally indicability and therefore also right orderability. Right
angled Artin groups are known to be polyfree (a result shown
independently by Duchamp-Krob, Howie and Hermiller-Sunic). Here we show
that Artin FC-groups for which all the defining relation are of even
type  are also polyfree. This is a joint work with Ruven Blasco and Luis
Paris.

I will explain how Dirac operators provide precious information about geometric and algebraic aspects of representations of real Lie groups. In particular, we obtain an explicit realisation of representations, leading terms in the asymptotics of characters and a precise connection with nilpotent orbits.

We will discuss some recent results with Martin Bridson about 
Sidki's construction X(G). In particular, if G is a finitely presented
group then X(G) is a finitely presented group. We will discuss as well the
result that if G has polynomial isoperimetric function and the maximal
metabelian quotient of G is virtually nilpotent then X(G) has polynomial
isoperimetric function. Part of the arguments we will use have homological
nature.

Abstract regular polytopes are finite quotients of Coxeter complexes
with string diagram, satisfying a natural intersection property, see
e.g. [MMS2002]. They arise in a number of geometric and group-theoretic
contexts. The first class of such objects, beyond the
well-understood examples coming from finite and affine Coxeter groups,
are locally toroidal cases, e.g.  extensions of quotients of the affine
F_4 complex [3,3,4,3].  In 1996 P.McMullen & E.Schulte constructed a
number of examples of locally toroidal abstract regular polytopes of
type [3,3,4,3,3], and conjectured completeness of their list. We
construct counterexamples to the conjecture using a Y-shaped
presentation for a subgroup of the Monster, and discuss various
related questions.
 

For a reductive group $ G $, Steinberg established a map from the Weyl group to nilpotent $ G $-orbits using momentmaps on double flag varieties.  In particular, in the case of the general linear group, he re-interpreted the Robinson-Schensted correspondence between the permutations and pairs of standard tableaux of the same shape in terms of product of complete flags.

We generalize his theory to the case of symmetric pairs $ (G, K) $, and obtained two different maps.  In the case where $ (G, K) = (\GL_{2n}, \GL_n \times \GL_n) $, one of the maps is a generalized Steinberg map, which induces a generalization of the RS correspondence for degenerate permutations.  The other is an exotic moment map, which maps degenerate permutations to signed Young diagrams, i.e., $ K $-orbits in the Cartan space $ (\lie{g}/\lie{k})^* $.

We explain geometric background of the theory and combinatorial procedures which produces the above mentioned maps.

This is an on-going joint work with Lucas Fresse.