Lie bialgebra contractions are introduced and classified. A non-degenerate coboundary bialgebra structure is implemented into all pseudo-orthogonal algebras so(p, q) starting from the one corresponding to so(N +1). It allows to introduce a set of Lie bialgebra contractions which leads to Lie bialgebras of quasi-orthogonal algebras. This construction is explicitly given for the cases N = 2, 3, 4...

The simple (or semisimple) classical Lie groups (algebras) may be transformed by contractions to the nonsemisimple ones of the different structure. We named the set of such groups (algebras) as Cayley-Klein (CK) groups (algebras). In our approach [2] the contractions of the groups (algebras) are described with the help of dual valued parameters, i.e. the nonsemisimple CK groups are regarded as ...

This paper continues the study of quantised function algebras Oǫ[G] of a semisimple group G at an lth root of unity ǫ. These algebras were introduced by De Concini and Lyubashenko in 1994, and studied further by De Concini and Procesi and by Gordon, amongst others. Our main purpose here is to increase understanding of the finite dimensional factor algebras Oǫ[G](g), for g ∈ G. We determine the ...

We replace the group of group-like elements of the quantized enveloping algebra Uq(g) of a finite dimensional semisimple Lie algebra g by some regular monoid and get the weak Hopf algebra w q (g). It is a new subclass of weak Hopf algebras but not Hopf algebras. Then we devote to constructing a basis of w q (g) and determine the group of weak Hopf algebra automorphisms of w q (g) when q is not ...

Weyl’s character formula is a useful tool in understanding the structure of irreducible representations of semisimple Lie algebras. Even more important than the character formula itself is a corollary, the Weyl dimension formula, which gives a fairly simple expression for the total dimension of the representation in terms of the root system of the Lie algebra. Although the results discussed are...

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, 55 and 77 is semisimple and thus isomorphic to a group...

This paper is concerned with partially defined derivations on Banach algebras and particularly on C*-algebras and H*-algebras, a subject whose study was motivated by the question of time evolution and spatial translation in quantum physics (see [2, 7] for a full account of the theory). It is well known [4] that everywhere defined derivations on semisimple Banach algebras are automatically conti...

In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfel’d double for a class of graded Hopf algebras is investigated. Various quantum algebras, in...