### Article

## Rings of Fractions of Reduction Algebras

We establish the absence of zero divisors in the reduction algebra of a Lie algebra {Mathematical expression} with respect to its reductive Lie subalgebra {Mathematical expression}. We identify the field of fractions of the diagonal reduction algebra of {Mathematical expression} with the standard skew field; as a by-product we obtain a two-parametric family of realizations of this diagonal reduction algebra by differential operators. We also present a new proof of the Poincaré-Birkhoff-Witt theorem for reduction algebras.

Let K be a ﬁeld and A be a commutative associative K-algebra which is an integral domain. The Lie algebra DerA of all K-derivations of A is an A-module in a natural way and if R is the quotient ﬁeld of A then RDerA is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDerA of rank k over R (i.e. such that dimR RL = k), then the derived length of L is at most k and L is ﬁnite dimensional over its ﬁeld of constants. In case of solvable Lie algebras over a ﬁeld of characteristic zero their derived length does not exceed 2k. Nilpotent and solvable Lie algebras of rank 1 and 2 (over R) from the Lie algebra RDerA are characterized. As a consequence we obtain the same estimations for nilpotent and solvable Lie algebras of vector ﬁelds with polynomial, rational, or formal coeﬃcients.

We introduce a unital associative algebra associated with degenerate CP1. We show that is a commutative algebra and whose Poincare' series is given by the number of partitions. Thereby, we can regard as a smooth degeneration limit of the elliptic algebra introduced by Feigin and Odesskii [Int. Math. Res. Notices 11, 531 (1997)]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using and the Heisenberg representation of the commutative family studied by Shiraishi [ Commun. Math. Phys. 263, 439 (2006)]. It is found that the Ding-Iohara algebra [Lett. Math. Phys. 41, 183 (1997)] provides us with an algebraic framework for the free field construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting [Leningrad Math. J. 1, 1419 (1990)] in the sence of Babelon-Bernard-Billey [Phys. Lett. B. 375, 89 (1996)], the Ruijsenaars difference operator [Commun. Math. Phys. 110, 191 (1987)], and the operator M(q,t1,t2) of Okounkov-Pandharipande [e-print arXiv:math-ph/0411210].

We discuss some well-known facts about Clifford algebras: matrix representations, Cartan’s periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in <span data-mathml="nn dimensions, etc. We also present our point of view on some problems. Namely, we discuss the generalization of the Pauli theorem, the basic ideas of the method of averaging in Clifford algebras, the notion of quaternion type of Clifford algebra elements, the classification of Lie subalgebras of specific type in Clifford algebra, etc.

Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials in the limit t→∞ and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q-Whittaker function. These coefficients are pure q-powers and their degrees are expected to coincide in the untwisted setting with the extremal degrees of the so-called PBW-filtration in the corresponding finite-dimensional irreducible representations of the simple Lie algebras for any root systems. This is a particular case of a general conjecture in terms of the level-one Demazure modules. We prove this coincidence for all Lie algebras of classical type and for G2, and also establish the relations of our extremal degrees to minimal q-degrees of the extremal terms of the Kostant q-partition function; they coincide with the latter only for some root systems. © 2015 Elsevier Inc.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.