scholarly journals Parametrizations of degenerate density matrices

2017 ◽  
Vol 29 (08) ◽  
pp. 1750026
Author(s):  
E. Brüning ◽  
S. Nagamachi
Keyword(s):  
Lie Algebra ◽  
Quotient Space ◽  
Homogeneous Spaces ◽  
Density Matrices

It turns out that a parametrization of degenerate density matrices requires a parametrization of [Formula: see text], [Formula: see text] where [Formula: see text] denotes the set of all unitary [Formula: see text]-matrices with complex entries. Unfortunately, the parametrization of this quotient space is quite involved. Our solution does not rely on Lie algebra methods directly, but succeeds through the construction of suitable sections for natural projections, by using techniques from the theory of homogeneous spaces. We mention the relation to the Lie algebra background and conclude with two concrete examples.

scholarly journals On Classes of Local Unitary Transformations

2012 ◽  
Vol 19 (02) ◽  
pp. 283-292
Author(s):  
Naihuan Jing
Keyword(s):  
Homogeneous Spaces ◽  
Unitary Groups ◽  
Mixed States ◽  
Density Matrices ◽  
Double Cosets ◽  
Unitary Transformations ◽  
One To One ◽  
Local Unitary Invariance ◽  
Unitary Invariance

We give a one-to-one correspondence between classes of density matrices under local unitary invariance and the double cosets of unitary groups. We show that the interrelationship among classes of local unitary equivalent multi-partite mixed states is independent from the actual values of the eigenvalues and only depends on the multiplicities of the eigenvalues. The interpretation in terms of homogeneous spaces of unitary groups is also discussed.

Symbols in Berezin quantization for representation operators

2021 ◽  
pp. 296-304
Author(s):  
Vladimir F. Molchanov ◽  
Svetlana V. Tsykina
Keyword(s):  
Homogeneous Space ◽  
Lie Algebra ◽  
Grassmann Manifold ◽  
Homogeneous Spaces ◽  
Finite Multiplicity ◽  
Basic Notion ◽  
Berezin Quantization ◽  
Algebraic Version ◽  
Enveloping Algebra ◽  
Algebra Of Operators

The basic notion of the Berezin quantization on a manifold M is a correspondence which to an operator A from a class assigns the pair of functions F and F^♮ defined on M. These functions are called covariant and contravariant symbols of A. We are interested in homogeneous space M=G/H and classes of operators related to the representation theory. The most algebraic version of quantization — we call it the polynomial quantization — is obtained when operators belong to the algebra of operators corresponding in a representation T of G to elements X of the universal enveloping algebra Env g of the Lie algebra g of G. In this case symbols turn out to be polynomials on the Lie algebra g. In this paper we offer a new theme in the Berezin quantization on G/H: as an initial class of operators we take operators corresponding to elements of the group G itself in a representation T of this group. In the paper we consider two examples, here homogeneous spaces are para-Hermitian spaces of rank 1 and 2: a) G=SL(2;R), H — the subgroup of diagonal matrices, G/H — a hyperboloid of one sheet in R^3; b) G — the pseudoorthogonal group SO_0 (p; q), the subgroup H covers with finite multiplicity the group SO_0 (p-1,q -1)×SO_0 (1;1); the space G/H (a pseudo-Grassmann manifold) is an orbit in the Lie algebra g of the group G.

scholarly journals On Formality of Some Homogeneous Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1011
Author(s):  
Aleksy Tralle
Keyword(s):  
Homogeneous Space ◽  
Root System ◽  
Lie Algebra ◽  
Lie Group ◽  
Maximal Torus ◽  
Homogeneous Spaces ◽  
Direct Proof ◽  
Classical Type ◽  
Sasakian Manifolds ◽  
Full Analysis

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

scholarly journals A Note on Simple Anti-Commutative Algebras Obtained from Reductive Homogeneous Spaces

1968 ◽  
Vol 31 ◽  
pp. 105-124 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Homogeneous Space ◽  
Lie Algebra ◽  
Lie Group ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Commutative Algebras ◽  
Direct Sum ◽  
Reductive Homogeneous Spaces ◽  
Connected Lie Group

LetGbe a connected Lie group andHa closed subgroup, then the homogeneous spaceM = G/His calledreductiveif there exists a decomposition(subspace direct sum) withwhereg(resp.) is the Lie algebra ofG(resp.H); in this case the pair (g,) is called areductive pair.

scholarly journals Homogeneous spaces of Hilbert type

2015 ◽  
Vol 11 (02) ◽  
pp. 397-405 ◽  
Author(s):  
Mikhail Borovoi
Keyword(s):  
Number Field ◽  
Function Field ◽  
Quotient Space ◽  
Homogeneous Spaces ◽  
Global Field ◽  
Simply Connected ◽  
Hilbert Type

Let k be a global field. Let G be a connected linear algebraic k-group, assumed reductive when k is a function field. It follows from a result of a paper by Bary-Soroker, Fehm and Petersen that when H is a smooth connected k-subgroup of G, the quotient space G/H is of Hilbert type. We prove a similar result for certain non-connected k-subgroups H of G. In particular, we prove that if G is a simply connected k-group over a number field k, and H is an abelian k-subgroup of G, not necessarily connected, then G/H is of Hilbert type.

scholarly journals Further results on q-Lie groups, q-Lie algebras and q-homogeneous spaces

2021 ◽  
Vol 9 (1) ◽  
pp. 119-148
Author(s):  
Thomas Ernst
Keyword(s):  
Weyl Group ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Cartan Subalgebra ◽  
Homogeneous Spaces ◽  
Iwasawa Decomposition ◽  
Base Field ◽  
Killing Form ◽  
Exact Sequences

Abstract We introduce most of the concepts for q-Lie algebras in a way independent of the base field K. Again it turns out that we can keep the same Lie algebra with a small modification. We use very similar definitions for all quantities, which means that the proofs are similar. In particular, the quantities solvable, nilpotent, semisimple q-Lie algebra, Weyl group and Weyl chamber are identical with the ordinary case q = 1. The computations of sample q-roots for certain well-known q-Lie groups contain an extra q-addition, and consequently, for most of the quantities which are q-deformed, we add a prefix q in the respective name. Important examples are the q-Cartan subalgebra and the q-Cartan Killing form. We introduce the concept q-homogeneous spaces in a formal way exemplified by the examples S U q ( 1 , 1 ) S O q ( 2 ) {{S{U_q}\left( {1,1} \right)} \over {S{O_q}\left( 2 \right)}} and S O q ( 3 ) S O q ( 2 ) {{S{O_q}\left( 3 \right)} \over {S{O_q}\left( 2 \right)}} with corresponding q-Lie groups and q-geodesics. By introducing a q-deformed semidirect product, we can define exact sequences of q-Lie groups and some other interesting q-homogeneous spaces. We give an example of the corresponding q-Iwasawa decomposition for SLq(2).

scholarly journals Contractions of Lie Algebras and Separation of Variables.

1997 ◽  
Vol 12 (01) ◽  
pp. 53-61 ◽  
Author(s):  
A. A. Izmest'ev ◽  
G. S. Pogosyan ◽  
A. N. Sissakian ◽  
P. Winternitz
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lorentz Group ◽  
Homogeneous Spaces ◽  
Separation Of Variables ◽  
Second Order ◽  
Coordinate Systems ◽  
Euclidean Group ◽  
Enveloping Algebra ◽  
Four Levels

The Inönü-Wigner contraction from the Lorentz group O(2,1) to the Euclidean group E(2) is used to relate the separation of variables in the Laplace-Beltrami operators on the two corresponding homogeneous spaces. We consider the contractions on four levels: the Lie algebra, the commuting sets of second order operators in the enveloping algebra o(2,1), the coordinate systems and some eigenfunctions of the Laplace-Beltrami operators.

scholarly journals Some Homogeneous Einstein Manifolds

1970 ◽  
Vol 39 ◽  
pp. 81-106 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Homogeneous Space ◽  
Lie Algebra ◽  
Lie Group ◽  
Closed Subgroup ◽  
Einstein Manifold ◽  
Homogeneous Spaces ◽  
Einstein Manifolds ◽  
Direct Sum ◽  
Reductive Homogeneous Spaces ◽  
Connected Lie Group

Let G be a connected Lie group and H a closed subgroup with Lie algebra such that in the Lie algebra g of G there exists a subspace m with (subspace direct sum) and In this case the corresponding manifold M = G/H is called a reductive homogeneous space and (g,) (or (G,H)) a reductive pair. In this paper we shall show how to construct invariant pseudo-Riemannian connections on suitable reductive homogeneous spaces M which make M into an Einstein manifold.

scholarly journals On the Dimension of Certain Graded Lie Algebras Arising in Geometric Integration of Differential Equations

2000 ◽  
Vol 3 ◽  
pp. 44-75
Author(s):  
Arieh Iserles ◽  
Antonella Zanna
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Algebra ◽  
Homogeneous Spaces ◽  
Practical Interest ◽  
Exponential Map ◽  
Graded Lie Algebras ◽  
Graded Algebras ◽  
Algebraic Setting ◽  
Cayley Transformation

AbstractMany discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is the calculation of commutators, a task that can be made significantly cheaper by the introduction of appropriate bases of function values and by the exploitation of redundancies inherent in a Lie-algebraic structure by means of graded spaces. In many Lie groups of practical interest a convenient alternative to the exponential map is a Cayley transformation, and the subject of this paper is the investigation of graded algebras that occur in this context. To this end we introduce a new concept, a hierarchical algebra, a Lie algebra equipped with a countable number of m-nary multilinear operations which display alternating symmetry and a ‘hierarchy condition’. We present explicit formulae for the dimension of graded subspaces of free hierarchical algebras and an algorithm for the construction of their basis. The paper is concluded by reviewing a number of applications of our results to numerical methods in a Lie-algebraic setting.

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