Representations of quantum *-algebras 𝑠𝑙_{𝑡}(𝑁+1,ℝ)

1992 ◽  
pp. 57-66
Author(s):  
E. Vaysleb
Keyword(s):  
Quantum Algebras

scholarly journals Residuated Structures and Orthomodular Lattices

Studia Logica ◽  
2021 ◽  
Author(s):  
D. Fazio ◽  
A. Ledda ◽  
F. Paoli
Keyword(s):  
Algebraic Logic ◽  
Residuated Lattices ◽  
Quantum Structures ◽  
Heyting Algebras ◽  
Quantum Algebras ◽  
Orthomodular Lattices ◽  
Lattice Of Subvarieties ◽  
Residuated Structures ◽  
Common Framework ◽  
Mv Algebras

AbstractThe variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$ ℓ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$ ℓ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$ ℓ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.

scholarly journals Degenerate Sklyanin algebras

Open Physics ◽  
2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Andrey Smirnov
Keyword(s):  
Difference Operators ◽  
Baxter Equation ◽  
Quantum Algebras ◽  
Rational Solutions ◽  
R Matrix ◽  
Gauge Transformations ◽  
Sklyanin Algebras ◽  
Sklyanin Algebra ◽  
The Difference

AbstractNew trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.

scholarly journals Normalité de certains anneaux déterminantiels quantiques

1999 ◽  
Vol 42 (3) ◽  
pp. 621-640 ◽  
Author(s):  
Laurent Rigal
Keyword(s):  
Residue Class ◽  
Quantum Algebras ◽  
Regular Functions ◽  
Maximal Orders ◽  
Ring Of Fractions ◽  
Prime Factors ◽  
The Matrix ◽  
Residue Class Ring ◽  
Class Ring ◽  
The Ideal

Let Kq[X] = Oq(M(m, n)) be the quantization of the ring of regular functions on m × n matrices and Iq (X) be the ideal generated by the 2 × 2 quantum minors of the matrix X=(Xij)l≤i≤m, I≤j≤n of generators of Kq[X]. The residue class ring Rq(X) = Kq[X]/Iq(X) (a quantum analogue of determinantal rings) is shown to be an integral domain and a maximal order in its divisionring of fractions. For the proof we use a general lemma concerning maximalorders that we first establish. This lemma actually applies widely to prime factors of quantum algebras. We also prove that, if the parameter isnot a root of unity, all the prime factors of the uniparameter quantum space are maximal orders in their division ring of fractions.

Analogues of quantum Schubert cell algebras in PBW-deformations of quantum groups

2016 ◽  
Vol 15 (10) ◽  
pp. 1650179 ◽  
Author(s):  
Yongjun Xu ◽  
Dingguo Wang ◽  
Jialei Chen
Keyword(s):  
Weyl Group ◽  
Quantum Groups ◽  
Braid Group ◽  
Group Actions ◽  
Necessary Condition ◽  
Quantum Algebras ◽  
Ore Extension ◽  
Schubert Cell ◽  
Sufficient And Necessary Condition

We focus on a class of filtered quantum algebras [Formula: see text] which are both coideal subalgebras of quantum groups and Poincaré–Birkhoff–Witt (PBW)-deformations of their negative parts. In [Y. Xu and S. Yang, PBW-deformations of quantum groups, J. Algebra 408 (2014) 222–249], Xu and Yang proved that braid group actions on [Formula: see text] introduced by Kolb and Pellegrini can be used to define root vectors and construct PBW bases for [Formula: see text]. In this present paper, for each element [Formula: see text] in the Weyl group of [Formula: see text] we first introduce a subspace [Formula: see text] and a subalgebra [Formula: see text] of [Formula: see text], where [Formula: see text] can be considered as an analogue of quantum Schubert cell algebra. Then a sufficient and necessary condition on [Formula: see text] is given for [Formula: see text]. Moreover, we prove that [Formula: see text] if and only if [Formula: see text] and [Formula: see text] can be generated by the same simple reflections. Finally, we characterize the algebra [Formula: see text] which can be obtained via an iterated Ore extension. Our results show that quantum groups and their PBW-deformations really have some different properties.

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