We generalise the notion of coherent states to arbitrary Lie algebras by making an analogy with the GNS construction in C∗-algebras. The method is illustrated with examples of semisimple and non-semisimple finite dimensional Lie algebras as well as loop and Kac-Moody algebras. A deformed addition on the parameter space is also introduced simplifying some expressions and some applications to con...

We constructed a multi-parametric deformation of the Brauer algebra representation related with symplectic Lie algebras. The notion Manin matrix type C was generalised to case by using this and corresponding quadratic derived pairing operators for these algebras minors considered matrices. rank dimensions components were calculated.

We study the C∗-algebra An,θ generated by the Anzai flow on the n-dimensional torus T. It is proved that this algebra is a simple quotient of the group C∗-algebra of a lattice subgroup Dn of a (n + 2)-dimensional connected simply connected nilpotent Lie group Fn whose corresponding Lie algebra is the generic filiform Lie algebra fn. Other simple infinite dimensional quotients of C∗(Dn) are also...

This paper studies the representations of semisimple Lie algebras, with care given to the case of sln(C). We develop and utilize various tools, including the adjoint representation, the Killing form, root space decomposition, and the Weyl group to classify the irreducible representations of semisimple Lie algebras.

We determine the identities of degree ≤ 6 satisfied by the symmetric (Jordan) product a◦ b = ab+ ba and the associator [a, b, c] = (ab)c−a(bc) in every nonassociative algebra. In addition to the commutative identity a◦b = b◦a we obtain one new identity in degree 4 and another new identity in degree 5. We demonstrate the existence of further new identities in degree 6. These identities define a ...

We begin to study the Lie theoretical analogs of symplectic reflection algebras for Γ a finite cyclic group, which we call “cyclic double affine Lie algebra”. We focus on type A : in the finite (resp. affine, double affine) case, we prove that these structures are finite (resp. affine, toroidal) type Lie algebras, but the gradings differ. The case which is essentially new is sln(C[u, v]⋊Γ). We ...

The K-groups, the range of trace on K0, and isomorphism classifications are obtained for simple infinite dimensional quotient C*-algebras of the group C*-algebras of six lattice subgroups, corresponding to each of the six non-isomorphic 5-dimensional connected, simply connected, nilpotent Lie groups. Connes’ non-commutative geometry involving cyclic cocycles and the Chern character play a key r...

In this article we initiate a systematic study of irreducible weight modules over direct limits of reductive Lie algebras, and in particular over the simple Lie algebras A(∞), B(∞), C(∞) and D(∞). Our main tool is the shadow method introduced recently in [DMP]. The integrable irreducible modules are an important particular class and we give an explicit parametrization of the finite integrable m...

We first introduce the Wigner-Weyl-Moyal formalism for a theory whose phase-space is an arbitrary Lie algebra. We also generalize to quantum Lie algebras and to supersymmetric theories. It turns out that the non-commutativity leads to a deformation of the classical phase-space: instead of being a vector space it becomes a manifold, the topology of which is given by the commutator relations. It ...