scholarly journals Quantum symmetries on noncommutative complex spheres with partial commutation relations

2018 ◽  
Vol 21 (04) ◽  
pp. 1850028
Author(s):  
Simeng Wang
Keyword(s):  
Quantum Groups ◽  
Unitary Groups ◽  
Symmetry Groups ◽  
Orthogonal Groups ◽  
Commutation Relations ◽  
Quantum Symmetry ◽  
Quantum Symmetries ◽  
Free Independence

We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial commutation relations. We also discuss some geometric aspects of the quantum orthogonal groups associated with the mixture of classical and free independence discovered by Speicher and Weber. We show that these quantum groups are quantum symmetry groups on some quantum spaces of spherical vectors with partial commutation relations.

LIE ALGEBRA OF THE q-POINCARÉ GROUP AND q-HEISENBERG COMMUTATION RELATIONS

1999 ◽  
Vol 13 (24n25) ◽  
pp. 2895-2902
Author(s):  
PAOLO ASCHIERI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Heisenberg Algebra ◽  
Poincaré Group ◽  
Orthogonal Groups ◽  
Poincare Group ◽  
Commutation Relations ◽  
Real Forms

We discuss quantum orthogonal groups and their real forms. We review the construction of inhomogeneous orthogonal q-groups and their q-Lie algebras. The geometry of the q-Poincaré group naturally induces a well defined q-deformed Heisenberg algebra of hermitian q-Minkowski coordinates xaand momenta pa.

scholarly journals Uq(N) GAUGE THEORIES

1994 ◽  
Vol 09 (30) ◽  
pp. 2835-2847 ◽  
Author(s):  
LEONARDO CASTELLANI
Keyword(s):  
Quantum Groups ◽  
Differential Calculus ◽  
Gauge Theories ◽  
Space Time ◽  
Yang Mills ◽  
Commutation Relations ◽  
Transformation Rules ◽  
Field Strengths

Improving on an earlier proposal, we construct the gauge theories of the quantum groups U q(N). We find that these theories are also consistent with an ordinary (commuting) space-time. The bicovariance conditions of the quantum differential calculus are essential in our construction. The gauge potentials and the field strengths are q-commuting "fields," and satisfy q-commutation relations with the gauge parameters. The transformation rules of the potentials generalize the ordinary infinitesimal gauge variations. For particular deformations of U (N) ("minimal deformations"), the algebra of quantum gauge variations is shown to close, provided the gauge parameters satisfy appropriate q-commutations. The q-Lagrangian invariant under the U q(N) variations has the Yang–Mills form [Formula: see text], the "quantum metric" gij being a generalization of the Killing metric.

scholarly journals Two-parameter families of quantum symmetry groups

2011 ◽  
Vol 260 (11) ◽  
pp. 3252-3282 ◽  
Author(s):  
Teodor Banica ◽  
Adam Skalski
Keyword(s):  
Symmetry Groups ◽  
Quantum Symmetry ◽  
Two Parameter

scholarly journals Socles of Verma modules in quantum groups

1993 ◽  
Vol 47 (2) ◽  
pp. 221-231
Author(s):  
A.V. Jeyakumar ◽  
P.B. Sarasija
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Verma Modules ◽  
Commutation Relations

In this paper the Verma modules Me(λ) over the quantum group vε(sl(n + 1), ℂ), where ε is a primitive lth root of 1 are studied. Some commutation relations among the generators of Ue are obtained. Using these relations, it is proved that the socle of Mε(λ) is non-zero.

scholarly journals Quantum Symmetries of Graph C*-algebras

2018 ◽  
Vol 61 (4) ◽  
pp. 848-864 ◽  
Author(s):  
Simon Schmidt ◽  
Moritz Weber
Keyword(s):  
Automorphism Group ◽  
Directed Graph ◽  
Operator Algebras ◽  
Automorphism Groups ◽  
Undirected Graphs ◽  
C Algebras ◽  
Quantum Symmetry ◽  
Quantum Symmetries ◽  
Multiple Edges ◽  
Definition Of

AbstractThe study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

scholarly journals Sur l'homologie des groupes unitaires à coefficients polynomiaux

2012 ◽  
Vol 10 (1) ◽  
pp. 87-139 ◽  
Author(s):  
Aurélien Djament
Keyword(s):  
Unitary Groups ◽  
Symmetric Power ◽  
Orthogonal Groups ◽  
Tensor Power ◽  
Full Generality ◽  
Standard Methods ◽  
Homology Groups ◽  
Special Cases ◽  
Functor Homology ◽  
Kunneth Formula

AbstractLet A be a ring with anti-involution and F a nice functor (tensor or symmetric power, for example) from finitely-generated projective A-modules to abelian groups. We show that the homology of the hyperbolic unitary groups Un,n(A) with coefficients in F(A2n) can be expressed stably (i.e. after taking the colimit over n) by the homology of these groups with untwisted coefficients and functor homology groups that we can compute in suitable cases (for example, when A is a field of characteristic 0 or a ring without ℤ-torsion and F a tensor power). This extends the result where A is a finite field, which was dealt with previously by C. Vespa and the author (Ann. Sci. ENS, 2010).The proof begins by relating, without any assumption on F, our homology groups to the homology of a category of hermitian spaces with coefficients twisted by F. Then, when F is polynomial, we establish — following a method due to Scorichenko — an isomorphism between this homology and the homology of another category of (possibly degenerate) hermitian spaces, which is computable (in good cases) by standard methods of homological algebra in functor categories (using adjunctions, Künneth formula…). We give some examples.Finally, we deal with the analogous problem for non-hyperbolic unitary groups in some special cases, for example euclidean orthogonal groups On (A) (the ring A being here commutative). The isomorphism between functor homology and group homology with twisted coefficients does not hold in full generality; nevertheless we succeed to get it when A is a field or, for example, a subring of ℚ containing ℤ[1/2]. The method, which is similar to that in the previous case, uses a general result of symmetrisation in functor homology proved at the beginning of the article.

Sandwich classification for O2n+1(R) and U2n+1(R,Δ) revisited

2018 ◽  
Vol 21 (4) ◽  
pp. 539-571 ◽  
Author(s):  
Raimund Preusser
Keyword(s):  
Natural Number ◽  
Commutative Ring ◽  
Unitary Groups ◽  
Orthogonal Groups

AbstractIn a recent paper, the author proved that if {n\geq 3} is a natural number, R a commutative ring and {\sigma\in GL_{n}(R)}, then {t_{kl}(\sigma_{ij})} where {i\neq j} and {k\neq l} can be expressed as a product of 8 matrices of the form {{}^{\varepsilon}\sigma^{\pm 1}} where {\varepsilon\in E_{n}(R)}. In this article we prove similar results for the odd-dimensional orthogonal groups {O_{2n+1}(R)} and the odd-dimensional unitary groups {U_{2n+1}(R,\Delta)} under the assumption that R is commutative and {n\geq 3}. This yields new, short proofs of the Sandwich Classification Theorems for the groups {O_{2n+1}(R)} and {U_{2n+1}(R,\Delta)}.

scholarly journals Odd-dimensional orthogonal groups as amalgams of unitary groups.

2007 ◽  
Vol 316 (2) ◽  
pp. 591-607 ◽  
Author(s):  
R. Gramlich ◽  
M. Horn ◽  
W. Nickel
Keyword(s):  
Unitary Groups ◽  
Orthogonal Groups
Sign in / Sign up

Export Citation Format

Share Document