Search results

...ant bilinear form is unique up to scaling. A Lie algebra is [[semi-simple Lie algebra|semi-simple]] if and only if its Killing form is [[non-degenerate]] ...ath> where x,y,z are elements in the algebra and the brackets [] are the [[Lie brackets]] ...

...of [[../TrivialGroupoid/|Abelian groups]], category of sheaves of abelian groups, category of reversible automata <math>\mathcal{A}_R</math> ...ps]] #Category of [[../CubicalHigherHomotopyGroupoid/|crossed modules]] of groups ...

===Lie algebras in quantum theories=== ...inite--Lie algebra that completely specifies locally the properties of the Lie group everywhere. ...

===Topic on Lie algebroids=== This is a topic entry on Lie algebroids that focuses on their quantum applications and extensions of cur ...

...nce, let A and B be square matrices, and let α(s) and β(t) be paths in the Lie group of nonsingular matrices which satisfy ...

...he Cartesian closed category of finite Abelian [[../QuantumGroup4/|quantum groups]] with exponential elements (including [[../BilinearMap/|Lie groups]]) and quantum group [[../TrivialGroupoid/|homomorphisms]] is a <math>C_3</ ...

The physics of [[../QuarkAntiquarkPair/|spins]] and mathematics of spin groups are both important subjects respectively in Physics and [[../PhysicalMathem In Physics, the term ''spin 'groups' '' is often used with the broad meaning of a collection of coupled, or int ...

of [[../TrivialGroupoid/|groups]], \textit{there is a correspondence between the unitary R. Gilmore: ''Lie Groups, Lie Algebras and Some of Their Applications.'' , ...

*[[../PolishGroup/|topological groups]] *[[../BilinearMap/|Lie groups]] and [[../BilinearMap/|Lie algebras]] ...

...bras as "building blocks"; and in the case of finite [[../TrivialGroupoid/|groups]] they are obtained from the latter by Fourier transformation of the group Let us consider next, alternative definitions of quantum groups that indeed possess [[../TopologicalOrder2/|extended quantum symmetries]] a ...

...urier transformation are finite [[../CompactQuantumGroups/|compact quantum groups]] that are Abelian. Let us consider next, alternative definitions of quantum groups that indeed possess [[../ExtendedQuantumSymmetries/|extended quantum symmet ...

Lie, S.,1975. Sophus Lie's 1880 Transformation Group Paper, in ''Lie Groups : History, Frontiers, and Applications.'' , Volume 1, translated by M. Acke Lie, S., 1976. Sophus Lie's 1884 Differential Invariant Paper, in ''Lie Groups : History, Frontiers, and Applications.'' , Volume 3, translated by M. Acke ...

M. Chaician and A. Demichev: ''Introduction to Quantum Groups'' , World Scientific (1996). V. G. Drinfel'd: Quantum groups, In \emph{Proc. Int. Cong. of ...

M. Chaician and A. Demichev: ''Introduction to Quantum Groups'' , World Scientific (1996). V. G. Drinfel'd: Quantum groups, In \emph{Proc. Int. Congress of ...

...[[Lie groupoid]], which has more complex mathematical properties than the Lie group. ...

...e/|homotopy]], homology and [[../CohomologyTheoryOnCWComplexes/|cohomology groups]]. There are close connections between algebraic topology, ...HomotopyGroupoid/|homotopy theory]] and [[../HomotopyCategory/|fundamental groups]] #Topology and [[../QuantumOperatorAlgebra5/|groupoids]]; van Kampen theor ...

...nics, Poisson algebras are also used in the study of [[../BilinearMap/|Lie groups]]. ...

...the BC-conjecture, but it is nevertheless `suspected' to be true for such groups. Moreover, the BC-conjecture remains so far unproven in general for discrete groups. ...

\subsubsection{Lie groupoids, their dual algebroids and representations .../TrivialGroupoid/|group]] <math>\mathsf{G}_x</math> is a [[../BilinearMap/|Lie group]], and for a (left or right) Haar ...

===Hilbert spaces, Von Neumann algebras and Quantum Groups=== \htmladdnormallink{locally compact quantum groups <math>CQG_{lc}</math>}{http://planetphysics.us/encyclopedia/LocallyCompactQ ...