The unrolled quantum group inside Lusztig’s quantum group of divided powers

For quantum symmetric pairs $(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct $\imath$ -canonical bases for the highest weight integrable $\textbf U$ -modules and their tensor products regarded as $\textbf {U}^\imath$ -modules, as well as an $\imath$ -canonical basis for the modified form of the $\imath$ -quantum group $\textbf {U}^\imath$ . A key new ingredient is a family of explicit elements called $\imath$ -divided powers, which are shown to generate the integral form of $\dot {\textbf {U}}^\imath$ . We prove a conjecture of Balagovic–Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi- $K$ -matrix and the constructions of $\imath$ -canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.

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Climate Policy in a System of Divided Powers: Dealing with Carbon Leakage and Regulatory Linkage

SSRN Electronic Journal ◽

10.2139/ssrn.2174024 ◽

2012 ◽

Author(s):

Daniel A. Farber

Keyword(s):

Climate Policy ◽

Carbon Leakage ◽

Divided Powers

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Dangerous Liaisons

10.1093/oso/9780198732174.003.0005 ◽

2017 ◽

Author(s):

András Sajó ◽

Renáta Uitz

Keyword(s):

Executive Branch ◽

Individual Liberty ◽

Checks And Balances ◽

Functional Distribution ◽

Governmental Functions ◽

Private Actors ◽

Divided Powers

This chapter examines the idea of separating distinct governmental functions into at least three branches (horizontal separation) as a means to safeguard individual liberty. The three branches of government have different functions: the legislature legislates, the executive branch executes the laws, and the judiciary administers justice. This corresponds to the functional distribution of essential governmental tasks and competences. The chapter explores how governments based on separated (or at least divided) powers work, in a perpetual balancing exercise as a result of the operation of checks and balances. Finally, it discusses independent agencies that are now routinely added to the old constitutional mix of powers and the problem of outsourcing public powers to private actors.

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RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000165 ◽

2021 ◽

pp. 1-37

Author(s):

Martijn Caspers

Keyword(s):

Quantum Group ◽

Quantum Groups ◽

Riesz Transform ◽

Wreath Products ◽

Compact Quantum Group ◽

Riesz Transforms ◽

Markov Semigroup ◽

Markov Semigroups ◽

Free Products ◽

Compact Quantum Groups

Abstract One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient- ${\mathcal {S}}_2$ condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

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The Real Forms of the Fractional Supergroup SL(2,C)

Mathematics ◽

10.3390/math9090933 ◽

2021 ◽

Vol 9 (9) ◽

pp. 933

Author(s):

Yasemen Ucan ◽

Resat Kosker

Keyword(s):

Quantum Group ◽

Lie Group ◽

The Real ◽

And Mathematics ◽

Real Forms

The real forms of complex groups (or algebras) are important in physics and mathematics. The Lie group SL2,C is one of these important groups. There are real forms of the classical Lie group SL2,C and the quantum group SL2,C in the literature. Inspired by this, in our study, we obtain the real forms of the fractional supergroups shown with A3NSL2,C, for the non-trivial N = 1 and N = 2 cases, that is, the real forms of the fractional supergroups A31SL2,C and A32SL2,C.

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