We study parallel algorithms for addition of numbers having finite representation in a positional numeration system defined by a base β in C and a finite digit set A of contiguous integers containing 0. For a fixed base β, we focus on the question of the size of the alphabet allowing to perform addition in constant time independently of the length of representation of the summands. We produce lower bounds on the size of such alphabet A. For several types of well studied bases (negative integer, complex numbers −1 + ı, 2ı, and ı √ 2, quadratic Pisot unit, and the non-integer rational base), we give explicit parallel algorithms performing addition in constant time. Moreover we show that digit sets used by these algorithms are the smallest possible.

There are three fine gradings of the simple Lie algebra of type B 2 over the complex number field. They provide a basic information about the structure of the algebra. In the paper an explicit description of all fine gradings is given in terms of the four-dimensional symplectic ͓sp͑4, C͔͒ and five-dimensional orthogonal ͓o͑5, C͔͒ representations of the algebra. In addition, the real forms of B 2 are considered. It is shown which of the fine gradings survive the restriction to each of the real forms. These results should be useful in defining various sets of additive quantum numbers for systems with such symmetries, for systematic study of grading preserving contractions of this Lie algebra, and generally for choosing bases which reflect structural properties of the Lie algebra.

A grading of a Lie algebra is called fine if it cannot be further refined. Fine gradings provide basic information about the structure of the algebra. There are eight fine gradings of the simple Lie algebra of type A 3 over the complex number field. One of them ͑root decomposition͒ is the main tool of the theory and applications in working with A 3 and with its representations; one other has also been used in the literature, and the rest have apparently not been recognized so far. An explicit description of all the fine gradings of A 3 is given in terms of the four-dimensional ͓sl͑4, C͔͒ and six-dimensional orthogonal ͓o͑6, C͔͒ representations of the algebra. These results should be useful generally for choosing bases which reflect structural properties of the Lie algebra, for defining various sets of additive quantum numbers for systems with such symmetries, and for systematic study of grading preserving contractions of this Lie algebra.

A grading of a Lie algebra is called fine if it cannot be further refined. Fine gradings provide basic information about the structure of the algebra. There are six fine gradings of the semisimple Lie algebra of type A1×A1 over the complex number field. An explicit description of all the fine gradings of A1×A1 is given in terms of the four-dimensional representation o(4,C) of the algebra.

Study of the normalizer of the MAD-group corresponding to a fine grading offers the most important tool for describing symmetries in the system of non-linear equations connected with contraction of a Lie algebra. One fine grading that is always present in any Lie algebra sl(n, C) is the Pauli grading. The MAD-group corresponding to it is generated by generalized Pauli matrices. For such MAD-group, we already know its normalizer; its quotient group is isomorphic to the Lie group SL(2, Zn) × Z2.In this paper, we deal with a more complicated situation, namely that the fine grading of sl(p 2 , C) is given by a tensor product of the Pauli matrices of the same order p, p being a prime. We describe the normalizer of the corresponding MAD-group and we show that its quotient group is isomorphic to Sp(4, Fp) × Z2, where Fp is the finite field with p elements.