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# Lie Algebra - Science topic

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Questions related to Lie Algebra

Is the hypothesis (formulated in the introduction to the ) about the one-to-one correspondence between Abelian and non-Abelian finite groups correct?

I have a Lie Algebra with a basis:

- A_1 = [[0 1 0], [0 0 0], [-1 0 0]]
- A_2 = [[0 0 1], [-1 0 0], [0 0 0]]
- A_3 = [[0 0 0], [0 1 0], [0 0 -1]]

The Lie brackets are [A_1,A_2] = A_3 ; [A_1,A_3] = A_1 and [A_3,A_2] = A_2.

I want to compute log(exp(A_1)exp(A_2)). I tried using the well-know Baker-Campbell-Hausdorff formula until order 5 (with the terms already computed as on wikipedia) and also the Tables 1 and 2 of the paper

*An efficient algorithm for computing...*from Casas and Murua published in 2009. The three terms of order 5 that I get are different.Could someone confirm that or explain the mistakes that I made in the computation?

Thank you in advance for your help.

Best,

FM.

A detailed consideration of the First and Second Universal Enveloping Algebras of a semi-simple Lie algebra and their contributions to infinite-dimensional representations of the group are recently undertaken. Hopefully, the second Universal Enveloping Algebra of a semi-simple Lie algebra would make the classification of these representations complete.

Let us have Minkowski space-time, which must be curved so that its metric does not change, and the coordinates cease to be straight lines. How can I do that? In this matter, a hint can be found in the mathematical apparatus of quantum mechanics. Indeed, if we take the Pauli matrices and the Pauli matrices multiplied by the imaginary unit as the basis of the Lie algebra sl

_{2}(**C**), then the four generators of this algebra can be associated with the coordinates of Minkowski space-time not only algebraically, but also geometrically through the correspondence of the elements of the algebra sl_{2}(**C**) and linear vector fields of the 4-dimensional space. Then the current lines of the vector fields of space-time become entangled in a ball, which, when untangled, surprisingly turns into Minkowski space-time. What other formalisms exist to solve the general relativity equations and their observables (such as energy, momentum and angular momentum of gravitational waves), in addition to the Cartan structure equations formalism, Killing equations, Lie algebras, Newman-Penrose formalism, and spinorial formalism, and what are their characteristics, advantages and disadvantages, including the advantages and disadvantages of these five formalisms mentioned (since I know their characteristics more or less)? And where can I consult them?

Let $(L_n)_{n\in \mathbb{N}}$ a monoton increasing sequence of finite-dimensional nilpotent Lie algebras over a field of characteristic $p>0$.

Question: If the Engel length of the sequence is bounded, then the class of nilpotency is also bounded for it?

The quaternion algebra in the matrix representation is generated by the modified Pauli matrices iσ

_{1}, iσ_{2}, iσ_{3,}which in the representation of linear vector fields iσ

_{1}=x_{4}∂x_{1}-x_{3}∂x_{2}+x_{2}∂x_{3}-x_{1}∂x_{4} iσ

_{2}=x_{3}∂x_{1}+x_{4}∂x_{2}-x_{1}∂x_{3}-x_{2}∂x_{4} iσ

_{3}=x_{2}∂x_{1}-x_{1}∂x_{2}-x_{4}∂x_{3}+x_{3}∂x_{4}define pair rotations in 4-dimensional space. In turn, these linear vector fields are tangent vector fields to the Villarceau circles of classic 2-tori, lying on 3-spheres, because they specify the simultaneous rotation of the defining circles (lying in the orthogonal planes) of the torus and because they are orthogonal to the differential 1-form

x

_{1}dx_{1}+x_{2}dx_{2}+x_{3}dx_{3}+x_{4}dx_{4}However, if the paired rotations to replace the pair pseudo-rotations, which are a composition of pair rotations and equiaffine transformations of the classical 2-torus, the lie algebra of pseudo-rotations of the Villarceau circles will correspond to the lie algebra su(2).

Moreover, if we allow rotations and pseudo-rotations of the Villarceau circles, we get the lie algebras sl

_{2}( ℂ ).Similarly, the algebra of rotations (pseudo-rotations) of the Willarceau circles of 4-tori lying on 7-spheres (7-hyperspheres of the spaces with a neutral metric) corresponds to the lie algebra sl

_{4}( ℂ ).for linear control systems x_dot=Ax+Bu the reachability set can be calculated using the Image of the controllability matrix, i.e

R=([B AB A^2B,....,]) and reachability set=Im(R)

when rank(R)=n, and we do not have any control constraint the reachibilaty set of linear system is R^n (n is the dimension of states)

if we have a non-linear affine-control system

x_dot=f(x)+g(x)*u

R can be calculated using Lie algebra

R=[g1,g2,[f,g1],[f,g2],...]

my question is, in this case reachability set is again Im(R)?

and if reachability set=Im(R), how can we compute the rachability set, because here R will be a matrix with function arrays (function of states)

On the one hand, it is known that a real hypersphere of an 8-dimensional neutral space, that is, a space with signature of the metric (+ 4, -4), is homeomorphic to the space R

^{4}× S^{3}. On the other hand, the algebra of linear tangent vector fields of hyperspheres of an 8-dimensional neutral space and hyperspheres of an 8-dimensional Euclidean space is isomorphic to the Lie algebra of the Dirac matrix algebra. At the same time, at the intersection of the hypersphere of the neutral and Euclidean space is the product of the spheres S^{3}× S^{3}.A three dimensional rotation operation is identical to a displacement operation in a three dimensional space of orientations. The rotation operation is defined by the displacement vector and a rotation axes. If we define that the rotation axes is perpendicular to the starting orientation and to the displacement vector, the definition of the rotation operation is complete.

But according to that definition, the space of orientations with this displacement operator has a strange algebra with a mixture of local and global properties.

Any nonzero displacement operation with the displacement vector V is different from a sequence of n displacement operations with the displacement vector V/n. Cause of the difference is the (local) property of the axes to be perpendicular to the displacement vector and to the current orientation. The (differential) relation between an infinitesimal displacement operation and a finite displacement operation in that space is therefore fundamentally (non-linearly) different from the (linear) relation in a Euclidian space. This raises the following questions:

What kind of algebra describes the properties of that space of orientations with a displacement operation defined by those rotations? (It seems that it is not a Lie Algebra! Do we even need a new kind of algebra due to nonlinear relations? Is this even beyond known mathematics?) Has anyone already studied the properties of such a space?

Background:

Potentially this algebra provides a static, homogenous, and isotropic model for the geometry of our universe. Geodesic lines generated with those displacement operations diverge and therefore allow explaining the red shift.

Some simple maths:

Given is:

Point p in Cartesian coordinates: p=(x,y,z); Radius R of the universe;

Orientation O in Euler angles: O=(x/R,y/R,z/R);

Displacement d in cartesian coordinates: d=(dx,dy,dz);

Rotation Q in Euler angles: Q=(dx/R,dy/R,dz/R);

Result p’ of the displacement in cartesian coordinates: p’=p+d ;

Orientation O’ after the displacement: O’=((x+dx)/R,(y+dy)/R,(z+dz)/R);

Calculation of the unit quaternion q representing the rotation Q:

Pure unit quaternion e perpendicular to O and O’; e=O x O’ / | O x O’| in (i,j,k) components

Unit Quaternion q given as : q=0.5|Q| +e*sqrt(1-0.25Q²)

Transpose quaternion qt: qt=transpose(q)

Quaternion o representing the Orientation O: o= Euler_angles_to_unit_Quaternion (O);

Quaternion o’, the result of the rotation: o’=qoqt

Shifted/rotated Orientation in Euler angles O’=Unit_Quaternion_to_Euler_angles (o’)

To be verified: p’ = R*O’

To be calculated: What happens to neighbour points of p after an Operation equivalent to the q-rotation?

Is there some decomposition theorem for chain complexes in terms of simpler complexes?

As far as I could find they (Partial Field) were first introduced in 1996, but still playing an important role when it comes to Matroids Representation. Then I've lately read a paper about skew partial fields and matroids representation over it, that become the generalization of representability of matroids over any skew field. I'd like to know if there are any other theory beside Matroids' that are related to partial fields. If there's somebody to give me any clue.

It's known that if we have an irreducible representation of a semisimple Lie algebra, there exist one highest weight associated to some root system, and for every dominant weight there exist a irreducible representation of the semisimple Lie algebra.

There exist some equation or condition about weights that give us as an equivalence the irreducibility of a representation?

Let K be a convex. Some subsets are the image of an affine map g of K into itself. What should such subsets be called? The affine map g may be considered to be idempotent.

I guess that there exist a concept from category theory that describe objects L with the property that there exists morphisms f:L->K and g:K->L such that gf is the identity on L.

Suppose $u(n)$ is the Lie algebra of the unitary group $U(n)$, why the dual vector space of $u(n)$ can be identified with $\sqrt{-1}u(n)$?

Let G be a Lie group and φ : G × G → G, (g, h) → ghg−1.

Let d

_{(e,e)}φ : T_{(e,e)}(G × G) → T_{e}G be the differential of φ , where T_{x}M designate the tangent space of a differential variety M.My questions are:

1) how to calculate d

_{(e,e)}φ? (So we can see if it is linear or bilinear map)2) How to prove that T

_{(e,e)}(G × G) is isomorphic to T_{e}G × T_{e}G ?I became worried as most of the study in Einstein derivation and pre-Einstein derivation are treated in tandem with Lie groups.

i saw in a paper that u(3,1) has one s-boson and three p-boson. i want to get a general reception about all lie algebras. could any one help me please?

For a compact Lie group G every coadjoint orbit is of the form M_\lambda = G.\lambda with \lambda\in \mathfrak{t}^*, the Lie algebra of the maximal torus T of G. When \lambda is regular, we can compute the volume vol(M_\lambda) of the symplectic manifold M_\lambda.

Proposition: Let \lambda be regular and \Lambda=\sqrt{-1}\lambda. Then

vol(M_\lambda) = vol(G.\lambda)=\prod_{\alpha>0}\frac{<\alpha,\Lambda>}{<\alpha,\rho>}, where \rho is half the sum of all positive roots of G.

Can anybody give a detailed interpretation about this formula? In particular, what happens for G=U(n)?

In General Relativity the Riemannian curvature tensor at a point in space-time is measured in terms of parallel transport of a vector around a closed loop locally at that point. In quantum terms (Penrose, 'the road to reality' published 2004) proposes to represent local space-time curvature in the space as involving the Lie bracket of the Lie Algebra generators of local translations in space-time

It is well-known that the nilradical of a finite-dimensional Lie algebra over a field of characteristic p is not necessarily characteristic (that is, not invariant under all derivations of the algebra). But is there a

**solvable**Lie algebra whose nilradical is not characteristic?Could contradiction play a role in quantum systems, as part of the mechanism of measurement, forcing a single random outcome from the spectrum of possibilities?

All ideas are welcome, including outrageous ones.

Does anyone know of a reference for the existence of projective covers for modules over a finite-dimensional simple Lie algebra S over a field of characteristic zero? Specifically I am really interested just in the trivial irreducible S-module.

Let's say I take the Lie group SO(2,3). It's maximal compact subgroup is SO(2)xSO(3). What I would like to know is what happens to this maximal compact subgroup if I deform to the quantum group SOq(2,3) [By this I mean a deformation of the universal enveloping algebra of SO(2,3) into the quantum group]. Is it simply SOq(2)xSOq(3) or is something more complicated going on?

Let

**be a regular curve in three dimensional Euclidean space***α***E**^{3}^{ }.**T, N**and**B**are Frenet vectors of**. We want to calculate the Lie brackets [***α***T,N**], [**T,B**] and [**N,B**] as a linear combination of**T, N**and**B.**Is a bracket-preserving map between Lie algebras necessarily Linear? Usually, it is assumed to be linear in advance and the second condition of bracket-preservation males it a Lie homomorphism, by definition.

When applying Block's Theorem on the structure of differentiably simple rings to Lie algebras most authors require an algebraically closed field, but I can see no reference to algebraic closure in Block's paper. What extra does the algebraic closure give?

properties of the radical and nil-radical of a finite dimensional Lie Algebra L

Let gl(n, R) be the Lie algebra of matrices with real entries and GL(n, R) its associated Lie group. Recall that a linear subgroup G ⊂ GL(n,R) acts by conjugation on gl(n, R), that is, for g ∈ G its action on A ∈ gl(n, R), is defined by

g(A) = g^{-1}Ag.

Definition: Let G ⊂ GL(n,

**R**) be a subgroup. A polynomial f ∈**R**[(Xij)1≤i,j≤n] is called invariant on gl(n, R) with respect to conjugation by elements in G ifff(g^{-1}Ag) = f(A) (4) ∀g ∈ G, ∀A ∈ gl(n,

**R**).We will denote by G' the set of all invariant polynomials.

**Question: What is known about G'?**Unit vectors i, j, k are defined as the basis for orthogonal 3-space. My view is that these unit vectors are man-made and that Nature does not know about them. That is to say, the information they contain is in the minds of mathematicians, not in the machinery of Nature. I am therefore looking for a self consistent mathematical system, that asserts the orthogonal information of 3-space.

Is it possible that this man-defined orthogonality is false representation that enters General Relativity which then obstructs progress in quantising gravity?

In these two questions we try to find a relation between vector bundle theory and Lie algebras:(Note that we identify a finite dimensional Lie algebra L with R^n or C^n)

1)Assume that $L$ is a n dimensional real (complex) Lie algebra such that O(n) (U(n)) is a subgroup of Aut(L). Is L necessarily an Abelian Lie algebra? Of course every Lie algebra with this property satisfies the following:

The structure group of every vector bundle E over compact space X can be reduced to Aut(L).

2) Assume that E is a n-dimensional vector bundle over a compact space X such that for every n- dimensional Lie algebra L, the structure group of E can be reduced to Aut(L). Is E a trivial vector bundle? I confess that perhaps this second question is very broad and general, so we can consider this second part in low dimensional case.

Through cartan's criterion,we can easily know when a lie algebra is solvable.But to nilpotent lie algebra, we don't have such a criterion.Is it worth to find a criterion for nilpotent lie algebra?

Can someone tell me something about Finsler structures on Lie groups? The idea is to pick a convex norm on Lie algebra. Does this define a convex ellipsoid and extend this convex body with left-translations through all over the Lie group? It should work but I never tried myself. The other interesting question is, what if the norm is not convex? These ideas came to me while reading a paper about homogeneous (in a rather wide sense) spaces.

Assume that a Lie group G acts on a manifold M, effectively. So the Lie algebra g of G is embedded in $\chi^{\infty}(M)$ in a natural way. (effective action: if x.g=x for all x then g=e)

Under what dynamical conditions this embedding is an "Ideal embedding"?

That is : The image of g is an ideal in the Lie algebra of smooth vector fields on M.

By dynamical conditions I mean the dynamical properties arising from the action of G on M.

This is work in progress. If N is the nilradical of a solvable Lie algebra then its centraliser, C_L(N), is contained in N. The problem is to find a generalised version of the nilradical, N*, such that C_L(N*), is contained in N* for any Lie algebra. Following the group theory approach seems to work for characteristic zero, but the characteristic p case is more interesting. I've tried a number of modifications but all flounder at some point. Of course, the lack of Levi's Theorem and decomposition of semisimples into simples may be insuperable hurdles; Block's Theorem doesn't seem to give enough. Also the nilradical isn't necessarily characteristic in characteristic p.

I became worried about such possibilities since most studies on Einstein and pre-Einstein derivation are treated in tandem with Lie groups.

I want to know how can I arrive at commutative relations (16), by starting from killing vector (14) and its constraint (15) ?