Monogenous algebras. Back to Kronecker

In this note we develop some properties of those algebras (called here locally simple) which can be generated by a single element after, if need be, a faithfully flat extension. For finite algebras, this is shown to be in fact a property of the geometric fibers. Morphisms between rings of algebraic integers are locally simple. Expanding an idea introduced by Kronecker we show that much of the properties (in particular local simplicity) of a finite and locally free A-algebra B can be read through the characteristic polynomial of the generic element of B.Comment: Note ecrite en septembre 2003. 13 page

Automorphisms of Higher Rank Lamplighter Groups

Let $\Gamma_d(q)$ denote the group whose Cayley graph with respect to a particular generating set is the Diestel-Leader graph $DL_d(q)$, as described by Bartholdi, Neuhauser and Woess. We compute both $Aut(\Gamma_d(q))$ and $Out(\Gamma_d(q))$ for $d \geq 2$, and apply our results to count twisted conjugacy classes in these groups when $d \geq 3$. Specifically, we show that when $d \geq 3$, the groups $\Gamma_d(q)$ have property $R_{\infty}$, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when $d=2$ the lamplighter groups $\Gamma_2(q)=L_q = {\mathbb Z}_q \wr {\mathbb Z}$ have property $R_{\infty}$ if and only if $(q,6) \neq 1$.Comment: 28 page

Knot invariants and higher representation theory

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel and Sussan for sl_n. Our technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is $\mathfrak{sl}_n$, we show that these categories agree with certain subcategories of parabolic category O for gl_k. We also investigate the finer structure of these categories: they are standardly stratified and satisfy a double centralizer property with respect to their self-dual modules. The standard modules of the stratification play an important role as test objects for functors, as Vermas do in more classical representation theory. The existence of these representations has consequences for the structure of previously studied categorifications. It allows us to prove the non-degeneracy of Khovanov and Lauda's 2-category (that its Hom spaces have the expected dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke algebras are symmetric Frobenius. In work of Reshetikhin and Turaev, the braiding and (co)evaluation maps between representations of quantum groups are used to define polynomial knot invariants. We show that the categorifications of tensor products are related by functors categorifying these maps, which allow the construction of bigraded knot homologies whose graded Euler characteristics are the original polynomial knot invariants.Comment: 99 pages. This is a significantly rewritten version of arXiv:1001.2020 and arXiv:1005.4559; both the exposition and proofs have been significantly improved. These earlier papers have been left up mainly in the interest of preserving references. v3: final version, to appear in Memoirs of the AMS. Proof of nondegeneracy moved to separate erratu

An Attempt to Enhance Buchberger's Algorithm by Using Remainder Sequences and GCDs (II) (Computer Algebra - Theory and its Applications)

Let F = {F, , ..., Fm+1} ⊂ ℚ[x, u] be a given system, where m+l 2: 3, (x) = (x, , ..., xm) and (u) = (u, , ...，叫）， with ∀xi >-- ∀uj. Let GB(F) = {G₁, G₂, ・・・}, with G₁ --< G₂ --< ・・・, be the reduced Grabner basis of F w.r.t. the lexicographic order. In a previous paper [10], one of the authors proposed a method of enhancing Buchberger's algorithm for computing GB(F). His idea_is to compute a set g':= {G1 , G2, ... } ⊂ ℚ[x, u], such that each Gi is either O or as mall multiple of Gi, and apply Buchberger's algorithm to F ∨ g'. He proposed a scheme of computing G₁, G₂, ... by the PRSs (polynomial remainder sequences) and the GCDs in "G₁ ⇒ G₂ ⇒ ・・・" order, without computing Spolynomials. The scheme is supported by two new useful theorems and one proposition to remove the extraneous factor. In fact, for a simple but never toy example, his scheme has computed G₁ successfully (G₁ became G₁ by the proposition mentioned above). However, an unexpected difficulty occurred in computing G₂; it contained a pretty large extraneous factor which was not removed by the proposition. In this paper, we find a surprising phenomenon with which we can remove the above mentioned extraneous factor in G₂ and obtain G₂. As for G₃ and G₄, we obtain very good "body doubles" of them, by eliminating variables in leading coefficients of intermediate remainders of the PRSs computed for G₁. For systems of many sub-variables, n ≥ 3, our method introduces an extra factor in ℚ[u3, ..，un]， into the "LCto W" polynomial; see the text for the LCtoW polynomial. Furthermore, we present several techniques to enhance the computation

Twisted Alexander Invariants of Twisted Links

Let L be an oriented (d+1)-component link in the 3-sphere, and let L(q) be the d-component link in a homology 3-sphere that results from performing 1/q-surgery on the last component. Results about the Alexander polynomial and twisted Alexander polynomials of L(q) corresponding to finite-image representations are obtained. The behavior of the invariants as q increases without bound is described.Comment: 21 pages, 6 figure