1. The statement This brief note is devoted to a simple (and well-known) result in noncommutative algebra, which is not deep but nevertheless subtler than it appears. It concerns the so-called quaternion algebras: Definition 1.1. Let k be a commutative ring1. Let a ∈ k and b ∈ k. The quaternion algebra Ha,b is defined to be the k-algebra with generators i and j and relations i2 = a, j2 = b, ij ...

Let H be a semisimple Hopf algebra over an algebraically closed field k. It is assumed that either char k = 0 or char k > dimH . Semisimplicity of H means that H is a semisimple left H-module. In that case H has finite dimension. Throughout the paper we shall keep to notations from [M]. For example H stands for the dual Hopf algebra with the natural pairing 〈−,−〉 : H ⊗ H → k. The algebra H is a...

Let k be an algebraically closed field of characteristic 0 and let D be a division algebra whose center F contains k. We shall say that D can be reduced to r parameters if we can write D ≃ D0 ⊗F 0 F , where D0 is a division algebra, the center F0 of D0 contains k and trdeg k (F0) = r. We show that every division algebra of odd degree n ≥ 5 can be reduced to ≤ 1 2 (n − 1)(n − 2) parameters. More...

1. The statement This brief note is devoted to a simple (and well-known) result in noncommutative algebra, which is not deep but nevertheless subtler than it appears. It concerns the so-called quaternion algebras: Definition 1.1. Let k be a commutative ring1. Let a ∈ k and b ∈ k. The quaternion algebra Ha,b is defined to be the k-algebra with generators i and j and relations i2 = a, j2 = b, ij ...

A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an embedding of universal C∗-algebras. We show how to build a (k + 1)-graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of k-graph algebras under embeddings induced from coverings as the universal algebra of a (k + 1)-graph. Our main focus is on computing the K-theory of ...

We observe a correspondence between the zero modes of superconformal algebras S(2, 1) and W (4) ([8]) and the Lie superalgebras formed by classical operators appearing in the Kähler and hyper-Kähler geometry. 1 Lie superalgebras of the Hodge theory 1.1 Kähler manifolds Let M = (M2n, g, I, ω) be a compact Kähler manifold of real dimension 2n, where g is a Riemannian (Kähler) metric, I is a compl...

In this paper we prove the Schur-Weyl duality between the symplectic group and the Brauer algebra over an arbitrary infinite field K. We show that the natural homomorphism from the Brauer algebra Bn(−2m) to the endomorphism algebra of the tensor space (K2m)⊗n as a module over the symplectic similitude group GSp2m(K) (or equivalently, as a module over the symplectic group Sp2m(K)) is always surj...

A free bimodule resolution is constructed for the integral group ring of the dihedral group of order 4m. This resolution is applied for a description, in terms of generators and defining relations, of the Hochschild cohomology algebra of this group ring. Introduction Let K be a commutative ring with unity, let R be an associative K-algebra that is a finitely generated projective K-module, let Λ...

Let K be a field of characteristic zero and suppose that D is a K-fv.lslpn algebra; i.e. a finite dimensional division algebra over K with center K. In Mollin [i] we proved that if K contains no non-trivial odd order roots of unity, then every finite odd order subgroup of D the multiplicative group of D, is cyclic. The first main result of this paper is to generalize (and simplify the proof of)...

The Clebsch-Gordan and Racah-Wigner coefficients for the positive (or negative) discrete series of irreducible representations for the noncompact form Uq(SU(1, 1)) of the algebra Uq(sl(2)) are computed. n0. Introduction Now it is well known the great significance which the Clebsch-Gordan and RacahWigner coefficients for the algebra Uq(SU(2)) has in the conformal field theory, topological field ...