A construction of the quantum affine algebra Uq(ĝ) is given in two steps. We explain how to obtain the algebra from its positive Borel subalgebra Uq(b +), using a construction similar to Drinfeld’s quantum double. Then we show how the positive Borel subalgebra can be constructed with quantum shuffles.

Based on the shifted Schensted correspondence and the shifted Knuth equivalence, a shifted analog of the Poirier-Reutenauer algebra as a higher lift of Schur’s P-functions and a right coideal subalgebra of the Poirier-Reutenauer algebra is constructed. Its close relations with the peak subalgebra and the Stembridge algebra of peak functions are also uncovered.

Let G be a complex reductive algebraic group, g its Lie algebra and h a reductive subalgebra of g, n a positive integer. Consider the diagonal actions G : g, NG(h) : h . We study a relation between the algebra C[h]G and its subalgebra consisting of restrictions to h of elements of C[g].

In this paper we study the relation between nonhomogeneous and homogeneous Sagbi bases. As a consequence, we present a general principle of computing Sagbi bases of a subalgebra and its homogenized subalgebra, which is based on passing over to homogenized generators.

Any non-pure quantum state admits an innnity of non-trivial decom-positions. A recent proposal how to measure the information content of a quantum state with reference to a given subalgebra of operators, singles out some of them, called optimal decompositions, which depend both on the state and on the subalgebra. In this paper we start exploring their main features.

In this paper, we use the theory of natural duality to study subalgebra lattices in the finitely generated varieties of MV-algebras. With this tool, we obtain the dual atomicity of these lattices, and characterize the members of these varieties in which every subalgebra is an intersection of maximal subalgebras. Then, we determine the algebras that have a modular or distributive lattice of suba...

Any non-pure quantum state admits an infinity of non-trivial decompositions. A recent proposal how to measure the information content of a quantum state with reference to a given subalgebra of operators, singles out some of them, called optimal decompositions, which depend both on the state and on the subalgebra. In this paper we start exploring their main features.

We show that every logmodular subalgebra of Mn(C) is unitary equivalent to an algebra of block upper triangular matrices, which was conjectured in [5]. In particular, this shows that every unital contractive representation of a logmodular subalgebra of Mn(C) is automatically completely contractive.

We consider the decoherence free subalgebra which satisfies the minimal condition introduced by Alicki [1]. We show the manifest form of it and relate the subalgebra with the Kraus representation. The arguments also provides a new proof for generalized Lüders theorem [5].

A triangular quantum deformation of osp(2/1) from the classical r-matrix including an odd generator is presented with its full Hopf algebra structure. The deformation maps, twisting element and tensor operators are considered for the deformed osp(2/1). It is also shown that its subalgebra generated by the Borel subalgebra is self-dual.