Milena Svobodová

Fine gradings of o(4,C)

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.

Fine grading of sl(p2,C) generated by tensor product of generalized Pauli matrices and its symmetries

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 nonlinear 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(p2,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.

The eight fine gradings of sl(4, C) and o(6, C)

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 A3 over the complex number field. One of them (root decomposition) is the main tool of the theory and applications in working with A3 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 A3 is given in terms of the four-dimensional [sl(4, C)] and six-dimensional orthogonal [o(6, C)] 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.

Fine gradings of o(5, C), sp(4, C) and of their real forms

There are three fine gradings of the simple Lie algebra of type B2 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 [sp(4, C)] and five-dimensional orthogonal [o(5, C)] representations of the algebra. In addition, the real forms of B2 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.

k-Block parallel addition versus 1-block parallel addition in non-standard numeration systems

Parallel addition in integer base is used for speeding up multiplication and division algorithms. k-block parallel addition has been introduced by Kornerup in [14]: instead of manipulating single digits, one works with blocks of xed length k. The aim of this paper is to investigate how such notion inuences the relationship between the base and the cardinality of the alphabet allowing block parallel addition. In this paper, we mainly focus on a certain class of real bases | the so-called Parry numbers. We give lower bounds on the cardinality of alphabets of non-negative integer digits allowing block parallel addition. By considering quadratic Pisot bases, we are able to show that these bounds cannot be improved in general and we give explicit parallel algorithms for addition in these cases. We also consider the d-bonacci base, which satises the equation X d = X d 1 +X d 2 + +X + 1. If in a base being a d-bonacci number 1-block parallel addition is possible on an alphabetA, then #A > d + 1; on the other hand, there exists a k2 N such that k-block parallel addition in this base is possible on the alphabetf0; 1; 2g, which cannot be reduced. In particular, addition in the Tribonacci base is 14-block parallel on alphabet f0; 1; 2g.

Fine Group Gradings of the Real Forms of sl(4,C), sp(4,C), and o(4,C)

We present an explicit description of the “fine group gradings” (i.e., group gradings which cannot be further refined) of the real forms of the semisimple Lie algebras sl(4,C), sp(4,C), and o(4,C). All together 12 real Lie algebras are considered, and a total of 44 of their fine group gradings are listed. The inclusions sl(4,C)⊃sp(4,C)⊃o(4,C) are an important tool in our presentation. Systematic use is made of the faithful representations of the three Lie algebras by 4×4 matrices.

On-line Multiplication and Division in Real and Complex Bases

A positional numeration system is given by a base and by a set of digits. The base is a real or complex number β such that |β| > 1, and the digit set A is a finite set of real or complex digits (including 0). In this paper, we first formulate a generalized version of the on-line algorithms for multiplication and division of Trivedi and Ercegovac for the cases that β is any real or complex number, and digits are real or complex. We show that if (β, A) satisfies the so-called (OL) Property, then on-line multiplication and division are feasible by the Trivedi-Ercegovac algorithms. For a real base β and alphabet A of contiguous integers, the system (β, A) has the (OL) Property if #A > |β| . Provided that addition and subtraction are realizable in parallel in the system (β, A), our on-line algorithms for multiplication and division have linear time complexity. Three examples are presented in detail: base β = 3+√5/2 with alphabet A = {-1, 0, 1}; base β = 2i with alphabet A = {-2, -1, 0, 1, 2} (redundant Knuth numeration system); and base β = -3/2 + z√3/2 = -1 + ω, where ω = exp 2iπ/3 , with alphabet A = {0, ±1, ±ω, ±ω²} (redundant Eisenstein numeration system).

Characterization of Cut-and-Project Sets Using a Binary Operation

AbstractCut-and-project sets with convex acceptance windows, based on irrationalities τ=

s-Convexity and cut-and-project sets

\frac{1}{2}