Related papers: The N-eigenvalue Problem and Two Applications

For a given abelian group G, we classify the isomorphism classes of G-gradings on the simple restricted Lie algebras of types W(m;1) and S(m;1) (m>=2), in terms of numerical and group-theoretical invariants. Our main tool is automorphism…

We present some problems related to the conjugacy classes of $\mathsf{Aut}(\mathbb{N}^*)$.

We study $\Bbb Z_2^{\otimes N}$ graded contractions of the real compact simple Lie algebra $so(N+1)$, and we identify within them the Cayley-Klein algebras as a naturally distinguished subset.

It is shown that Maeda's conjecture on eigenforms of level 1 implies that for every positive even d and every p in a density-one set of primes, the simple group PSL_2(F_{p^d}) occurs as the Galois group of a number field ramifying only at…

We solve the long standing problem of classification of standard compact Clifford-Klein forms of homogeneous spaces of simple non-compact real Lie groups under the extra assumption that $G$, $H$, $L$ are simple and absolutely simple. Then…

We determine, in an inductive framework, the vertices of the polytope $P(s,K)$ controlling the conjugacy classes of elements which product to one in the maximal compact subgroup $K$ of a simple complex algebraic group $G$. This extends…

Let $M$ be a factor with separable predual and $G$ a compact group of automorphisms of $M$ whose action is minimal, i.e. $M^{G^\prime}\cap M = C$, where $M^G$ denotes the $G$-fixed point subalgebra. Then every intemediate von Neumann…

The conjugacy problem in braid groups has been extensively studied, particularly from an algorithmic perspective. Established methods based on Garside structures, such as initial summit sets and super summit sets, provide effective…

We consider a certain modification of the group $G^3_n$ which describes dynamics of point configurations, in particular braids, and define a representation of the modified $G^3_n$. The braid representation induced is powerful enough to…

We provide new group presentations for surface braid groups which are positive. We study some properties of such presentations and we solve the conjugacy problem in a particular case.

For the finite groups GU(3), SU(3), GL(3), SL(3) over a finite field we solve the class product problem, i.e., we give a complete list of $m$-tuples of conjugacy classes whose product does not contain the identity matrix.

Let p be a prime number and f an overconvergent p-adic automorphic form on a definite unitary group which is split at p. Assume that f is of "classical weight" and that its Galois representation is crystalline at places dividing p, then f…

After explaining the concepts of Langlands dual and miniscule representations, we define an analog of the Gauss sum for any compact, simple Lie group with a simply laced Lie algebra. We then show a reciprocity property when a Lie group is…

Using our recent results on eigenvalues of invariants associated to the Lie superalgebra gl(m|n), we use characteristic identities to derive explicit matrix element formulae for all gl(m|n) generators, particularly non-elementary…

We solve two problems in representation theory for the periplectic Lie superalgebra pe(n), namely the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category O into…

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d,n be positive integers, and let N be a d-fold covering space of M. We show that the covering map induces an embedding of the n-th…

In the band theory for non-Hermitian systems, the energy eigenvalues, which are complex, can exhibit non-trivial topology which is not present in Hermitian systems. In one dimension, it was recently noted theoretically and demonstrated…

Hilbert--Lie groups are Lie groups whose Lie algebra is a real Hilbert space whose scalar product is invariant under the adjoint action. These infinite-dimensional Lie groups are the closest relatives to compact Lie groups. Here we study…

Let $A$ be a unital associative algebra over a field $k$. All unital associative algebras containing $A$ as a subalgebra of a given codimension $\mathfrak{c}$ are described and classified. For a fixed vector space $V$ of dimension…

For a differential operator $L$ of order $n$ over $C(z)$ with a finite (differential) Galois group $G\subset {\rm GL}(C^n)$, there is an algorithm, by M. van Hoeij and J.-A.~Weil, which computes the associated evaluation of the invariants…