Degenerations and representations of twisted Shibukawa-Ueno R-operators

We study degenerations of the Belavin R-matrices via the infinite dimensional operators defined by Shibukawa-Ueno. We define a two-parameter family of generalizations of the Shibukawa-Ueno R-operators. These operators have finite dimensional representations which include Belavin's R-matrices in the elliptic case, a two-parameter family of twisted affinized Cremmer-Gervais R-matrices in the trigonometric case, and a two-parameter family of twisted (affinized) generalized Jordanian R-matrices in the rational case. We find finite dimensional representations which are compatible with the elliptic to trigonometric and rational degeneration. We further show that certain members of the elliptic family of operators have no finite dimensional representations. These R-operators unify and generalize earlier constructions of Felder and Pasquier, Ding and Hodges, and the authors, and illuminate the extent to which the Cremmer-Gervais R-matrices (and their rational forms) are degenerations of Belavin's R-matrix

Triangular Poisson structures on Lie groups and symplectic reduction

We show that each triangular Poisson Lie group can be decomposed into Poisson submanifolds each of which is a quotient of a symplectic manifold. The Marsden-Weinstein-Meyer symplectic reduction technique is then used to give a complete description of the symplectic foliation of all triangular Poisson structures on Lie groups. The results are illustrated in detail for the generalized Jordanian Poisson structures on SL(n).Comment: 12 pages, AMS-Late

Algebraic structure of multi-parameter quantum groups

Multi-parameter versions U_p(g) and C_p[G] of the standard quantum groups U_q(g) and C_q[G] are considered where G is a semi-simple connected complex algebraic group and g is the Lie algebra of G. The primitive spectrum of C_p[G] is calculated, generalizing a result of Joseph for the standard quantum groups. This classification is compared with the classification of symplectic leaves for the associated Poisson structure on G.Comment: AMS Latex, 37 pages, June 1994; to appear in Advances in Mat

Motor Adaptations to Pain during a Bilateral Plantarflexion Task: Does the Cost of Using the Non-Painful Limb Matter?

During a force-matched bilateral task, when pain is induced in one limb, a shift of load to the non-painful leg is classically observed. This study aimed to test the hypothesis that this adaptation to pain depends on the mechanical efficiency of the non-painful leg. We studied a bilateral plantarflexion task that allowed flexibility in the relative force produced with each leg, but constrained the sum of forces from both legs to match a target. We manipulated the mechanical efficiency of the non-painful leg by imposing scaling factors: 1, 0.75, or 0.25 to decrease mechanical efficiency (Decreased efficiency experiment: 18 participants); and 1, 1.33 or 4 to increase mechanical efficiency (Increased efficiency experiment: 17 participants). Participants performed multiple sets of three submaximal bilateral isometric plantarflexions with each scaling factor during two conditions (Baseline and Pain). Pain was induced by injection of hypertonic saline into the soleus. Force was equally distributed between legs during the Baseline contractions (laterality index was close to 1; Decreased efficiency experiment: 1.16±0.33; Increased efficiency experiment: 1.11±0.32), with no significant effect of Scaling factor. The laterality index was affected by Pain such that the painful leg contributed less than the non-painful leg to the total force (Decreased efficiency experiment: 0.90±0.41, P<0.001; Increased efficiency experiment: 0.75±0.32, P<0.001), regardless of the efficiency (scaling factor) of the non-painful leg. When compared to the force produced during Baseline of the corresponding scaling condition, a decrease in force produced by the painful leg was observed for all conditions, except for scaling 0.25. This decrease in force was correlated with a decrease in drive to the soleus muscle. These data highlight that regardless of the overall mechanical cost, the nervous system appears to prefer to alter force sharing between limbs such that force produced by the painful leg is reduced relative to the non-painful leg

Homological Characterization of bounded F2F_2-regularity

Semi-regular sequences over $\mathbb{F}_2$ are sequences of homogeneous elements of the algebra $B^{(n)}=\mathbb{F}_2[X_1,...,X_n]/(X_1^2,...,X_n^2)$, which have as few relations between them as possible. It is believed that most such systems are semi-regular and this property has important consequences for understanding the complexity of Grobner basis algorithms such as F4 and F5 for solving such systems. In fact even in one of the simplest and most important cases, that of quadratic sequences of length $n$ in $n$ variables, the question of the existence of semi-regular sequences for all $n$ remains open. In this paper we present a new framework for the concept of semiregularity which we hope will allow the use of ideas and machinery from homological algebra to be applied to this interesting and important open question. First we introduce an analog of the Koszul complex and show that $\mathbb{F}_2$-semi-regularity can be characterized by the exactness of this complex. We show how the well known formula for the Hilbert series of a semiregular sequence can be deduced from the Koszul complex. Finally we show that the concept of first fall degree also has a natural description in terms of the Koszul complex