We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group Q4. Our main tool is a new presentation for the algebra C(Q4), corresponding to an isomorphism of type Q4 ≃ SO−1(3). The quantum subgroups of Q4 are subject to a McKay type correspondence, that we describe at the level of algebraic invariants.

Let K be any field, let L n denote the Leavitt algebra of type (1, n − 1) having coefficients in K, and let M d (L n) denote the ring of d × d matrices over L n. In our main result, we show that M d (L n) ∼ = L n if and only if d and n − 1 are coprime. We use this isomorphism to answer a question posed in [14] regarding isomorphisms between various C*-algebras. Furthermore, our result demonstra...

In this paper we present an example of elaborate categorical structures hidden in very simple algebraic objects. We look at the algebra of polynomial differential operators in one variable x, also known as the Weyl algebra, and its irreducible representation in the ring of polynomials Q[x]. We construct an abelian category C whose Grothendieck group can be naturally identified with the ring of ...

Let K be any field, let L n denote the Leavitt algebra of type (1, n − 1) having coefficients in K, and let M d (L n) denote the ring of d × d matrices over L n. In our main result, we show that M d (L n) ∼ = L n if and only if d and n − 1 are coprime. We use this isomorphism to answer a question posed in [14] regarding isomorphisms between various C*-algebras. Furthermore, our result demonstra...

We show that the Hochschild–Kostant–Rosenberg map from the space of multivector fields on a graded manifold N (endowed with a Berezinian volume) to the cohomology of the algebra of multidifferential operators on N (as a subalgebra of the Hochschild complex of C∞(N)) is an isomorphism of Batalin–Vilkovisky algebras. These results generalize to differential graded manifolds.

We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group Q4. Our main tool is a new presentation for the algebra C(Q4), corresponding to an isomorphism of type Q4 ≃ SO−1(3). The quantum subgroups of Q4 are subject to a McKay type correspondence, that we describe at the level of algebraic invariants.

The sub algebra functor Sub A is faithful for Boolean algebras (Sub A •Sub B implies A • B, see D. Sachs [7] ), but it is not faithful for bounded distributive lattices or unbounded distributive lattices. The automorphism functor Aut A is highly unfaithful even for Boolean algebras. The endomorphism functor End A is the most faithful of all three. B. M. Schein [8] and K. D.' Magill [5] establis...

An isomorphism is established between the plethystic Hopf algebra Pleth(Super[L]) and the algebra of vector symmetric functions. The Hall inner product of symmetric function theory is extended to the Hopf algebra Pleth(Super[L]).

In this work, the Z3-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algebra and the dual algebra is given. E-mail: [email protected]

Let R be an irreducible root system with the Coxeter number h. Let l > h be an odd integer (we assume that l is not divisible by 3 if R is of type G2). Let U be the quantum group of type 1 with divided powers associated to these data, see [10] (of type 1 means that the elements K l i are equal to 1). Let u ⊂ U be the Frobenius kernel, see loc. cit. Let 1 be the trivial U−module. The cohomology ...