The quantum divided power algebra of a finite-dimensional Nichols algebra of diagonal type

We prove (i) Nichols algebra 𝔅(V) of vector space V is finite dimensional if and only if Nichols braided Lie algebra 𝔏(V) is finite dimensional; (ii) if the rank of connected V is 2 and 𝔅(V) is an arithmetic root system, then 𝔅(V) = F ⊕ 𝔏(V); and (iii) if Δ(𝔅(V)) is an arithmetic root system and there does not exist any m-infinity element with puu ≠ 1 for any u ∈ D(V), then dim (𝔅(V)) = ∞ if and only if there exists V′, which is twisting equivalent to V, such that dim (𝔏-(V′)) = ∞. Furthermore, we give an estimation of dimensions of Nichols Lie algebras and two examples of Lie algebras which do not have maximal solvable ideals.

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On the restricted Jordan plane in odd characteristic

Journal of Algebra and Its Applications ◽

10.1142/s0219498821400120 ◽

2020 ◽

pp. 2140012

Author(s):

Nicolás Andruskiewitsch ◽

Héctor Peña Pollastri

Keyword(s):

Hopf Algebra ◽

Positive Characteristic ◽

Drinfeld Double ◽

Nichols Algebra ◽

Simple Modules ◽

Restricted Version ◽

Defining Relations ◽

Finite Dimensional ◽

Nichols Algebras

In positive characteristic the Jordan plane covers a finite-dimensional Nichols algebra that was described by Cibils et al. and we call the restricted Jordan plane. In this paper, the characteristic is odd. The defining relations of the Drinfeld double of the restricted Jordan plane are presented and its simple modules are determined. A Hopf algebra that deserves the name of double of the Jordan plane is introduced and various quantum Frobenius maps are described. The finite-dimensional pre-Nichols algebras intermediate between the Jordan plane and its restricted version are classified. The defining relations of the graded dual of the Jordan plane are given.

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Non-Linear Dynamics of Cardiovascular Variability Signals

Methods of Information in Medicine ◽

10.1055/s-0038-1634981 ◽

1994 ◽

Vol 33 (01) ◽

pp. 81-84 ◽

Cited By ~ 14

Author(s):

S. Cerutti ◽

S. Guzzetti ◽

R. Parola ◽

M.G. Signorini

Keyword(s):

Dimensional Space ◽

Severe Heart Failure ◽

Parametric Models ◽

Deterministic Systems ◽

Finite Dimensional ◽

Return Maps ◽

Pathological Conditions ◽

Non Linear ◽

Space State

Abstract:Long-term regulation of beat-to-beat variability involves several different kinds of controls. A linear approach performed by parametric models enhances the short-term regulation of the autonomic nervous system. Some non-linear long-term regulation can be assessed by the chaotic deterministic approach applied to the beat-to-beat variability of the discrete RR-interval series, extracted from the ECG. For chaotic deterministic systems, trajectories of the state vector describe a strange attractor characterized by a fractal of dimension D. Signals are supposed to be generated by a deterministic and finite dimensional but non-linear dynamic system with trajectories in a multi-dimensional space-state. We estimated the fractal dimension through the Grassberger and Procaccia algorithm and Self-Similarity approaches of the 24-h heart-rate variability (HRV) signal in different physiological and pathological conditions such as severe heart failure, or after heart transplantation. State-space representations through Return Maps are also obtained. Differences between physiological and pathological cases have been assessed and generally a decrease in the system complexity is correlated to pathological conditions.

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Finite-Dimensional Modeling of Network-Induced Delays for Real-Time Control Systems

10.23919/acc.1988.4789881 ◽

1988 ◽

Cited By ~ 1

Author(s):

Asok Ray ◽

Yoram Halevi

Keyword(s):

Control Systems ◽

Real Time ◽

Real Time Control ◽

Time Control ◽

Finite Dimensional ◽

Dimensional Modeling

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A closer look at the stable completion

10.23943/princeton/9780691161686.003.0005 ◽

2017 ◽

Author(s):

Ehud Hrushovski ◽

François Loeser

Keyword(s):

Inverse Limit ◽

Unit Vector ◽

Vector Spaces ◽

Dimensional Vector ◽

Compact Sets ◽

Linear Topology ◽

Topological Embedding ◽

Definable Set ◽

Finite Dimensional ◽

The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

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