A Lie algebra gQ over Q is said to be R-universal if every homomorphism from gQ to gl(n,R) is conjugate to a homomorphism into gl(n,Q) (for every n). By using Galois cohomology, we provide a short proof of the known fact that every real semisimple Lie algebra has an R-universal Q-form. We also provide a classification of the R-universal Lie algebras that are semisimple.

Non-commutative Poisson algebras are the algebras having an associative algebra structure and a Lie algebra structure together with the Leibniz law. For finite-dimensional ones we show that if they are semisimple as associative algebras then they are standard, on the other hand, if they are semisimple as Lie algebras then their associative products are trivial. We also give the descriptions of ...

In this paper, we determine the character space of vector-valued Lipschitz algebra Lipd(X, E), where (X, d) is any metric and E a certain unital semisimple commutative *-Banach algebra.

We continue the study of the distribution of closed geodesics on nilmanifolds Γ\N , constructed from a simply connected 2-step nilpotent Lie group N with a left invariant metric and a lattice Γ in N . We consider a Lie group N with associated 2-step nilpotent Lie algebra N constructed from an irreducible representation of a compact semisimple Lie algebra g0 on a real finite dimensional vector s...

We prove that a depth two Hopf subalgebra K of a semisimple Hopf algebra H is normal (where the ground field k is algebraically closed of characteristic zero). This means on the one hand that a Hopf subalgebra is normal when inducing (then restricting) modules several times as opposed to one time creates no new simple constituents. This point of view was taken in the paper [13] which establishe...

We study the induction and restriction functor from a Hopf subalgebra of a semisimple Hopf algebra. The image of the induction functor is described when the Hopf subalgebra is normal. In this situation, at the level of characters this image is isomorphic to the image of the restriction functor. A criterion for subcoalgebras to be invariant under the adjoint action is given generalizing Masuoka’...

For each natural number n, poset T , and |T |–tuple of scalars Q, we introduce the ramified partition algebra P (T) n (Q), which is a physically motivated and natural generalization of the partition algebra [24, 25] (the partition algebra coincides with case |T | = 1). For fixed n and T these algebras, like the partition algebra, have a basis independent of Q. We investigate their representatio...

Let A be the function algebra on a semisimple orbit, M , in the coadjoint representation of a simple Lie group, G, with the Lie algebra g. We study one and two parameter quantizations of A, Ah, At,h, such that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group, Uh(g). In particular, the algebra At,h specializes at h = 0 to a U(g), or G, inv...

We study two variations of the Brauer algebra Bn(x). The first is the algebra An(x), which generalizes the Brauer algebra by considering loops. The second is the algebra Ln(x), the An(x)-subalgebra generated by diagrams without horizontal arcs. An(x) and Ln(x) exhibit for x 6= 0 an hereditary-chain indexed by all integers. Following the ideas of Martin [10] in the context of the partition algeb...

Let H be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field k of characteristic 0. If H has no nontrivial skew-primitive elements, we find some bounds for the dimension of H 1 , the second term in the coradical filtration of H. Using these results, we are able to show that every Hopf algebra of dimension 14 is semisimple and thus isomorphic to a group algebra or...