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# Lie Algebra - Science topic

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Is the hypothesis (formulated in the introduction to the ) about the one-to-one correspondence between Abelian and non-Abelian finite groups correct?
There are more non-commutative groups, since any commutative group can be re-written in "non-commutative" terms, yet the converse does not apply really.
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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?
Best,
FM.
Moreover, the independence of the whole BCH (formal) series with respect to the chosen basis only happens in the event that the series converge (otherwise the question does not have a meaning), and this depends on the "smallness" of the Lie algebra elements involved. In general, what you can expect by truncating the series is to get an element of the Lie algebra Z such that exp(Z) is close enough to log(exp(X)exp(Y)).
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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.
Infinitesimal theory of representations of semisimple Lie groups by V. S. Varadarajan
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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 sl2(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 sl2(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.
In fact, in the previous post, the path from Dirac's quantum geometry to Einstein's geometry was indicated, and the mathematical apparatus for successfully passing this path must be found in the mechanism of local algebras of vector fields1.
1)
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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?
Thank you very much for the reference my very dear Professor Bobadilla. I will search the Wikipedia reference books and articles online. In fact, I want to attend a congress of quantum loop gravitaty, which is the main rival theory of string theory. Although there are some researchers who try to unify the quantum loop gravity with string theory, and I think they have written articles about it. But there are also many detractors who say that they cannot be unified, that they are two different and incompatible theories and that only one alternative is possibly correct. I do not close and contemplate both possibilities. In fact I have many string theory books that I plan to read later, and an introductory book on quantum loop gravitaty for beginners, to do so in future research on both theories and see if I can unify in search of my ultimate goal of a theory of everything version 1.0. Since the true scientist must have many girlfriends and not marry any, contrary to the moral norm. Soon I will send you a bibliography about all it, for now I send you a video of the Bhagavad Gita, which is something the Dr. mentioned studies.
¡Saludos estimado profesor y amigo maestro!
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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?
Dear Sven Wirsing ,
It is well known that:
Every finite-dimensional Engel algebra is nilpotent.(Theorem by Engel)
So, if it is bounded ( due to finite dimension), then the nilpotent elements inherit the same property.
Best regards
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The quaternion algebra in the matrix representation is generated by the modified Pauli matrices iσ1, iσ2, iσ3, which in the representation of linear vector fields
1=x4∂x1-x3∂x2+x2∂x3-x1∂x4
2=x3∂x1+x4∂x2-x1∂x3-x2∂x4
3=x2∂x1-x1∂x2-x4∂x3+x3∂x4
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
x1dx1+x2dx2+x3dx3+x4dx4
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 sl2( 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 sl4( ℂ ).
What other mathematical structures based on the representation of spinor by the vector field, tangent to the Villarceau circles, it would be possible to offer a note to physicists.
Because of our closed trajectory of the vector field (the Villarceau circles) lie on a 4-torus in the 8-dimensional space, then our spinor there are extra degrees of freedom.
If the additional degree of freedom provides a linear motion along the 3-torus lying on the 3-sphere with the poles knocked out, the symmetries of this motion are described by the group U(3), since it is generated by the group O(3) of rotation of the 3-sphere around the axis passing through its poles, and the group of motion 3-torus, which is described by the complex diagonal matrices D(3).
Thus, due to additional degrees of freedom, closed trajectories of the vector field can form arbitrary closed curves on the torus, for example, on the 2-torus it can be a trefoil knot.
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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)
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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 R4 × S3. 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 S3 × S3.
See, for example,
Principles of a Unified Theory of Spacetime and Physical Interactions
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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?
Hi Wolfgang
I'm not sure I can give you a perfect answer to this, but I'll do the best I can.
Consider a manifold with two different coordinate systems on it, x and x'. If the coordinates in the x' system are analytic functions of those in x system, we can by definition expand the x' coordinates in a power series in the x coordinates. By considering two neighbouring points, it's possible to find the relation between a small variation in the x coordinates and a small variation in the x' coordinates. The first order term in this relation is the Jacobian matrix.
The Riemannian approach is to look at the spaces tangent to the manifold at each point. These are by definition flat spaces. At a given point, one can define a vector tangent to the curve of increasing x^1, x^2 and so on for each coordinate in the x system. These collectively are the coordinate basis for this system. An arbitrary infinitesimal displacement can then be written as a linear sum of the basis vectors corresponding to each coordinate. It can similarly be written as a linear sum of the basis vectors corresponding to each x' coordinate. Thus we have resolved the infinitesimal displacement vector into two different coordinate bases. It is easy to show that the components of the vector in the two coordinate bases are again related by the Jacobian matrix. This is used as the template for transforming components for any vector at that point.
The Riemannian approach is therefore based on the assumption that all coordinate systems can be related analytically, but in general they are non-linear functions of each other. (A linear change of coordinates is a combination of a rotation and a translation of the coordinate system.)
I'm therefore not sure what you mean by your first question. (If the paragraph above doesn't help sufficiently, could you please explain a) what you mean by "non-linear terms" - terms of what? and b) how they could cancel out locally but lead to long distance effects?)
This just deals with the tangent space at a given point. To compare tangent spaces at different points, you need a sense of what is "parallel" to what. This is where a connection is necessary. There are infinitely many ways of defining a connection for a given manifold, but the most common is to use the Levi-Civita connection. It is the only symmetric metric-compatible connection and its components are the Christoffel symbols. (I'm happy to give you a definition of "metric-compatible" if this would be useful.) The Riemann curvature tensor, which describes the curvature of the manifold, can be written entirely in terms of the Levi-Civita connection, in such a way that if the Christoffel symbols are zero, the curvature is zero - the manifold is flat space.
I'm not familiar with a "secant metric", so I'll go on what you've put by way of explanation. The metric in spherical polar coordinates on a the surface of a sphere (i.e. for theta and phi) contains sin theta, but it doesn't have zero Christoffel symbols. As explained above, zero Christoffel symbols means that the space is flat. So I can only assume that you're talking about normal flat space but with an unusual coordinate system on it, one that is reached from Cartesian coordinates by using the sin of at least one of those coordinates. Is this what you mean, and if so, what are you saying about it?
It is also worth being aware that the metric *at a given point* tells you nothing about the global shape of the manifold. However, the coordinate system that the metric corresponds to may be valid over a large region of the manifold. If the manifold has the same curvature properties over the whole manifold that it does over the region, this will determine the global shape of the manifold. I haven't managed to find much by way of mathematical theory relating local geometry to global topology, but there are some odds and ends. For example, if the Ricci tensor is proportional to the metric, the manifold is a "surface of constant curvature". Also, the concept of holonomy (see the Wikipedia article) can be useful, especially if you can apply the de Rham decomposition theorem. (Sorry if I've gone a little off-topic here and it doesn't sound familiar - it's quite possible that you wouldn't need to make use of any of this theory. I'm using it at the moment, so it's fresh in my mind!)
In summary, the non-linear approach I used is entirely in keeping with Riemannian geometry and all of the manifolds I looked at are Riemannian manifolds. (This is why I'm able to define inner products all the way through.) It's based on theory established in the 1960s to handle non-linear sigma models and spontaneous symmetry breaking, developed to bring out the group theory aspects. My paper just shows how to use these methods in an extremely simple case that is easy to visualise.
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Is there some decomposition theorem for chain complexes in terms of simpler complexes?
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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.
I haven't seen the term partial field show up at all, other than the definition you have already presented.
If you are familiar with the notion of an abstract Witt ring, which was introduced in the early 80's, then you have seen a way that a partial field appears naturally. An ordered triple (G,Q,q), called a quaternionic structure, is a group which additively generates a ring, q is a map from the cartesian product of G with itself subjectively onto Q. This, with some effort, produces a ring with G as its multiplicative group of units. If you are interested in such things, from a historical standpoint, then you should read "Abstract Witt Rings" by Murray Marshall. If you'd like to see some combinatorial results involving quaternionic structures and the Witt ring, then you should consider reading my thesis.
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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?
Thank you so much Yuri, I will take account on the notes and I will read it. Best regards Alejandro
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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.
Actually I used Borsuk-Ulam to prove an improved version of the Caratheodory theorem that is related to the kind of problems that I am studying.
For a compact convex set the image of an idempotent and a fix-point set for a selfmap is the same. Therefore some papers call such sets fixpoint sets. To call it a fix-point set only makes sense for certain topological spaces but the concept is of a more general nature since it can be defined in any category.
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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 $\sqrt{-1}u(n)$?
Hi Pan,
$u(n)$ is a real Lie algebra, and in particular a real vector space. Using a non-degenerate symmetric bilinear form on $u(n)$, you can identify $u(n)$ with its dual vector space. The $\sqrt{-1}$ is not that important in a sense, and probably comes from using a non-degenerate pairing between skew-hermitian and hermitian matrices (instead of a non-degenerate symmetric bilinear form). $u(n)$ is the space of skew-hermitian $n$ by $n$ matrices, and $\sqrt{-1} u(n)$ is the space of hermitian $n$ by $n$ matrices by the way.
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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) → TeG  be the differential of φ , where Tx 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 TeG × TeG ?
Dear Hanifa,
2) if M and N are two manifolds then T(x,y)(M \times N) =Tx(M)\times Ty(N). It is evident in local coordinates.
1) Let L=Te(G) be the Lie algebra of G. It is not hard to prove that
d(e,e)(\phi): L\times L ->L is just the (bi)linear mapping:
I  think that the part 1.5 of
or (see especially Chapter 3)
will provide more details.
Best regards, Yuri
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I became worried as most of the study in Einstein derivation and pre-Einstein derivation are treated in tandem with Lie groups.
The geometric significance, and motivation, comes from looking at Lie groups with left-invariant metrics, however the definition is purely algebraic, at the Lie algebra level.  If you look at algebras over R or C, then its algebra of derivations is the Lie algebra of an algebraic group.  As such, it has a Levi decomposition.  This seems to be enough to carry over the ideas of Nikolayevsky...give it a try!
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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?
i want some simple books to learn Lie algebra and topology basically . for example Frankel book about topology is very good case . could you please introduce me some others?
thanks again
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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.
This is part of the Weyl Integration Formula. The best understandable source I know is Wu-Yi Hsiang: Lectures on Lie Groups, Chapter "Oribital geometry of the adjoint action". There is an online edition (almost ...): https://books.google.de/books/about/Lectures_on_Lie_groups.html?id=gmI1zrW6gHAC
Here is a short answer. Adjoint and coadjoint orbits of compact Lie groups G can be identified, using any Ad(G)-invariant inner product on the Lie algebra \g. To find the volume of an adjoint orbit Ad(G)\lambda we have to compute the Jacobi determinant (volume factor) of the embedding  f : G/H \to Ad(G)\lambda  where  H = C_\lambda is the centralizer of \lambda, the stabilizer subgroup of \lambda under the adjoint representation on \g. In the easiest case G = SU(2), the orbit is a round sphere S^2, and the volume factor is given by its radius. In the general case, M_\lambda = Ad(G)\lambda contains several perpendicular round spheres through the point \lambda, and each one has its own volume factor. These spheres come from the different root spaces (with respect to a maximal abelian subspace containing \lambda), and the volume factors depend on the values of the roots on \lambda, the numbers <\alpha,\lambda>. The division by <\lambda,rho> is the common normalization.
Does this help? If not, I can provide more details. But look as Hsiang's book first.
Best regards
Jost
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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
Many thanks --will read with interest and cite in my upcoming paper on non-commutative space time and group cohomology of the poincare group in the context of non-commutative geometry (needs a snappier title :))
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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?
I have found a 4-dimensional example over GF(2).
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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.
My idea is that contradiction may be the impetus that forces decision during quantum measurement.  Again referring to the donkey dilemma  -- the donkey that starved to death after being placed equidistant between two bails of hay.  The donkey cannot feed because no preference is possible.
If the donkey was forced to move forward by some strict contradiction behind him, forcing a decision on him, the contradiction would be imperative while the non-preference is no imperative at all.  What do you think about this kind of idea?
Steve.
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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.
Thank you Kenji. I'll have a look at that.
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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?
Generally speaking few subgroups survives quantization. As an example the usual embedding o(n)->o(n+1) does not survive in quantization. A quantum group has few quantum subgroups (there are a couple of different notions of quantum subgroups though). In the strong sens of a Hopf subalgebra of the original Hopf algebra I have very strong doubts that U_q(so(2))xU_q(so(3) will sit inside U_q(so(2,3)). It has to be carefully checked with explicit formula of generators and relations.
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Let α be a regular curve in three dimensional Euclidean space E3 . T, N and B are Frenet vectors of α. We want to calculate the Lie brackets [T,N], [T,B] and [N,B] as a linear combination of T, N and B.
I agree with Low's answer above.
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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.
The answer is NO. Take the a trivial Lie algebra L, that is just a vector space L (over an arbitrary field) with the product [x,y]=0 for all x,y in L. Let a  f:L-->L a non-linear mapping satisfying f(0)=0.
Then [f(x),f(y)]=0=f(0)=f([x,y]) for all x,y in L. No "aritmetic" helps there.
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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?
I'm grateful to Alexander Premet for providing an explanation in the attached link.
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properties of the radical and nil-radical of a finite dimensional Lie Algebra L
The book of Jacobson "Lie algebras" contains  classical results on the nilradical. The classical book "Schafer R.D. Introduction to Nonassociative Algebras. Academic Press. 1966" contains also results of nilradical for nonassociative algebras
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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 iff
f(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'?
@Mohamed  Thank you so much Mohamed. I will most definitely take a look at the book you suggested.
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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?
Quaternions again, and again ... Ok, let as form Clifford basis, take scalar part (grade 0) and bivector part (grade 2) and we have even multivectors basis for quaternions (just check multiplication table) . With my Mathematica code for Clifford algebra I can do any calculation on quaternions. Any! Bivectors have natural geometric interpretation and they are consequence of vector multiplication. Quaternions are  ingenious step forward from 19 century, they are very suitable for rotations in 3D, but the same formalism doesn't apply to higher dimensions, there is no 5D quaternions. Clifford algebra deals easily with rotations in any dimension, using the same formalism as in 3D.
For complex quaternions just use even part of 3D Clifford algebra over complex field (but Clifford 3D algebra possess complex and hyper-complex structure over real field).
"Four dimensional dynamics" (van Leunen) follows naturally and easily  from 3D Clifford algebra (Baylis, Sobczyk, Chappell, ...).  There is no need for time dimension.
Question here, as I understand it, was about Pauli matrices and vectors in 3D, and, again, quaternions are not vectors in 3D, they are squared to -1.
Geometric (Clifford) algebra gives amassing possibility to express great extent of mathematical physics in one unique language and even simplify it, with many surprising consequences. Rotations are extremely simple, spinors are there, geometry is intuitive and powerful,  one can forget about matrices, tensors ... Fundamental theorem of calculus in geometric algebra contains all well known theorems from vector calculus in one formula and is richer then all them together, etc.
1) there is a vector product and it is not commutative
2) vectors are squared to reals.
This is legacy from 19-th cent., from Hamilton, Grassman and Clifford. Clifford died young, Gibbs published his famous book about vectors and mathematica started to grow as wild tree without proper care: many branches with too many languages. Geometric algebra contains all that branches and much more, but within just one powerful language. It is very easy to implement  it on computer. Forget about nonintuitive matrices, forget about coordinates, they are easy to include after your calculations.
So, dear physicists, especially young students, we all should to think seriously about our choice: to follow  complicated mathematics with so many languages, or just rethink about vectors multiplication and just one powerful language. Hestenes and his followers  are done so many nice work since 1960s.
Here is may challenge to readers: give me any "branch" from mathematical physics and I will give You an answer within geometric algebra, or at  least suggest You were to find answer, as a rule, simpler one and with unexpected consequences.
For example, what is origin of a spin in quantum mechanic? Well, look at rotations in 3D geometric algebra. Spin formulas appear as geometrical consequence of properties of rotations. Famous physicist Pauli introduced his matrices to describe spin in quantum mechanics. But there is no need for matrices, just learn how to properly multiply vectors and use them: ordinary well known unit vectors are suited here much better than Pauli's matrices. This is proven mathematical fact!
Multiply all base unit vectors from 3D and find interesting object: commutes with all elements of algebra, squares to -1, so, it is nice imaginary unit, but this time with geometrical interpretation. Now enjoy in quantum mechanic formalism, every appearance of "imaginary unit" will give You something interesting and   geometrically clear.
My prophecy for future is use of Clifford product of vectors (and Gibbs's book will be known  as sideway in history of physics).
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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.
Dear Ali:
The answer to your first question is Yes: only abelian Lie algebras can have O_n (similar: U_n) as group of automorphisms. Let us consider the real case. The Lie bracket on L = R^n  is a linear map  F :  L \wedge L \to L (where L\wedge L denotes the antisymmetric tensors or endomorphims on L), but on the other hand, L\wedge L is the Lie algebra of O_n (antisymmetric matrices). If  O_n  acts by automorphisms, then  F  is equivariant with respect to O_n, i.e.  F transforms the O_n-representation on L\wedge L into the one on L. But the first one is the adjoint representation on the Lie algebra  L\wege L = o_n which is irreducible. Thus F is either zero or it has zero kernel (the kernel would be subrepresentation, but there is none in an irreducible representation). Hence L\wedge L must be a subrepresentation of L. Counting dimensions, this is possible only if dim L = 3. Then there is in fact one such map F, the cross product (vector product)  on L = R^3 (= the Lie algebra of SO_3). But O_3 is not in the automorphism group of the cross product since reflections do not preserve orientation, note that  (a,b,a x b) is an oriented orthonormal basis if a,b is an orthonormal 2-frame. Thus  F = 0.
Best regards
Jost
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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?
You can use Engel's theorem: g is nilpotent iff ad_x is nilpotent for all x in g. A linear endomorphism (a matrix) is nilpotent iff 0 is its unique eigenvalue, i.e. its characteristic polynomial is (-\lambda)^n
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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.
The idea you suggest is classic. It has been used e.g. in the paper of P. Planche
Structures de Finsler invariantes sur les espaces symétriques (MR1366100).
The Riemannian metric obtained from the John ellipsoid gives you an analytic metric in the case of a Lie group (or more generally a homogenous space). But on a generic Finsler manifold, this construction is not smooth and it is better to work with the Binet-Legendre metric; see  Matveev-Troyanov "The Binet-Legendre metric in Finsler geometry" (MR3033515). See also teh appendix in http://arxiv.org/abs/1408.6401
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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.
The Lie algebra of smooth vector fields on a manifold has no finite dimensional ideals.
Proof. Assume that g is a finite dimensional linear subspace in the Lie algebra of vector fields over the manifold M. Then M has a finite number of points p_1,\dots,p_k such that for any X\in g, X\neq 0, at least one of the vectors X(p_1),\dots,X(p_k) is not zero. (This follows from a compactness argument on the unit sphere of g with respect to an arbitrary positive definite inner product.) Choose a vector field X\in g, X\neq 0 and a point q, such that q is different form  p_1,...,p_k and  X(q)\neq 0. Then there is a vector field Y on M such that [Y,X](q)\neq 0. Choose a smooth function h such that h vanishes on an open subset containing the points p_1,...,p_k and h is constant 1 on an open neighborhood  of q. Put Z=hY. Then the Lie bracket [Z,X] is not zero, since
[Z,X](q)=[Y,X](q) \neq 0, on the other hand, [Z,X](p_i)=0 for i=1,...,k, thus, [Z,X] cannot be in g by the choice of the points p_1,...,p_k. This means that g cannot be an ideal.
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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 have uploaded a revised version of this ( which is still not complete, but has a slightly different approach and eliminates some errors).
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I became worried about such possibilities since most studies on Einstein and pre-Einstein derivation are treated in tandem with Lie groups.
If you have a k- algebra A, where "algebra" means that A is a k-vector space equpiped with a k-bilinear map m:A\times A \to  A (denote m(a,b)=a.b, A may be Lie, associative, or just nothing), then a derivation makes sense: a k-linear map D:A\to A such that D(a.b)=D(a).b+a.D(b) (and it is always a subalgebra of the Lie algebra of endomorphism of A). If A is also finite dimensional, then you have the usual trace of endomorphisms. In this case, you have (almost) all the ingredients for the definition: call pre-Einstein a derivation D such that
- D is semisimple (as endomorphism)
- tr(DE)=tr(E) for all  E derivation of the algebra E
and if your field contains R, then you have all ingredients: you ask that the eigenvalues of D are all reals.
So, the notion MAKES SENSE for a general k-algebra (A,m). I think the reason that people only consider the Lie algebra case is because they use these derivations for constructing manifolds with special geometrical properties, that they know they come from homogeneous spaces.
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