Related papers: Lie Elements and Knuth Relations

Following our approach to metric Lie algebras developed in math.DG/0312243 we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semi-simple. We introduce cohomology sets (called quadratic cohomology) associated…

We re-examine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic…

The symmetric coinvariant algebra $C[x_1, dots, x_n]_{S_n}$ is the quotient algebra of the polynomial ring by the ideal generated by symmetric polynomials vanishing at the origin. It is known that the algebra is isomorphic to the regular…

Let $(G,\Omega)$ be a symplectic Lie group, i.e, a Lie group endowed with a left invariant symplectic form. If $\G$ is the Lie algebra of $G$ then we call $(\G,\omega=\Om(e))$ a symplectic Lie algebra. The product $\bullet$ on $\G$ defined…

A well known recurrence relation for the 6j-symbol of the quantum group su_q(2) is realized as a tridiagonal, symmetric eigenvalue problem. This formulation can be used to implement an efficient numerical evaluation algorithm, taking…

These are lecture notes of a course on symmetry group analysis of differential equations, based mainly on P. J. Olver's book 'Applications of Lie Groups to Differential Equations'. The course starts out with an introduction to the theory of…

We semiclassicalise the standard notion of differential calculus in noncommutative geometry on algebras and quantum groups. We show in the symplectic case that the infinitesimal data for a differential calculus is a symplectic connection,…

The subalgebra of diagonal elements of a quantum matrix group has been conjectured by Daniel Krob and Jean-Yves Thibon to be isomorphic to a cubic algebra, coined the quantum pseudo-plactic algebra. We present a functorial approach to the…

Let G be a complex semisimple Lie group and \tau a complex antilinear involution that commutes with the Cartan involution. If H denotes the connected subgroup of \tau-fixed points in G, and K is maximally compact, each H-orbit in G/K can be…

We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical Symplectic reflection algebras of Etingof and Ginzburg. We…

Let $n$ be a positive integer, $\sigma$ be an element of the symmetric group $\mathcal{S}_n$ and let $\sigma$ be a cycle of length $n$. The elements $\alpha ,\beta \in \mathcal{S}_n$ are $\sigma$-equivalent, if there are natural numbers $k$…

This text presents some basic notions in symplectic geometry, Poisson geometry, Hamiltonian systems, Lie algebras and Lie groups actions on symplectic or Poisson manifolds, momentum maps and their use for the reduction of Hamiltonian…

We initiate a study of the representation of the symmetric group on the multilinear component of an $n$-ary generalization of the free Lie algebra, which we call a free LAnKe. Our central result is that the representation of the symmetric…

It is shown that the quantised enveloping algebra of sl(n) contains a quantum Lie algebra, defined by means of axioms similar to Woronowicz's., This gives rise to Lie algebra-like generators and relations for the locally finite part of the…

We study relations between the two-parameter $\U_q(sl(n))$-invariant deformation quantization on $sl^*(n)$ and the reflection equation algebra. The latter is described by a quantum permutation on $\End(\C^n)$ given explicitly. The…

In this paper, we show the relation between $sp(4,\mathbb{R})$, the Lie algebra of the symplectic group, and the elements of $Sp(4,\mathbb{R})$. We use this result to obtain some special cases of symplectic matrices relevant to the study of…

In a recent paper, Pardo and the first named author introduced a class of C*-algebras which which are constructed from an action of a group on a graph. This class was shown to include many C*-algebras of interest, including all Kirchberg…

We introduce coordinates on the spaces of framed and decorated representations of the fundamental group of a surface with nonempty boundary into the symplectic group $Sp(2n,\mathbf R)$. These coordinates provide a noncommutative…

We introduce a bi-Hamiltonian hierarchy on the cotangent bundle of the real Lie group ${\mathrm{GL}}(n,{\mathbb{C}})$, and study its Poisson reduction with respect to the action of the product group ${{\mathrm U}(n)} \times {{\mathrm…

A quandle is an algebraic structure whose axioms correspond to the Reidemeister moves of knot theory. S. Kamada introduced the notion of a quandle with a good involution, which is later called a symmetric quandle. We are interested in the…