# Lie algebra generated by logarithm of differentiation and logarithm

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I encountered this problem while working on my research. Will it be an intertwiner for the group representations? The only reference I could find on this says without proof that an intertwiner of Lie algebra representations that can be integrated to representations of the groups is also an intertwiner for the group representations.

If this is is indeed true, how would one prove it? It's enough if the group is connected; a finite group is a Lie group, and this definitely won't work in that case. The Lie algebra condition implies that the map commutes with the exponential of any Lie algebra element, and thus with the subgroup generated by these. Since the group is connected, that's the whole group. EDIT: Even if you are working in a situation where there's no guarantee the exponential map exists, you can still make this work.

Being a Lie algebra representation is the fact that the above map is 0, so this function is constant. It's obviously 0 at the identity, so it's 0 everywhere.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 5 years, 3 months ago. Active 5 years, 3 months ago. Viewed times. Giuseppe Sellaroli Giuseppe Sellaroli 33 4 4 bronze badges.

Active Oldest Votes.In the theory of Lie groupsthe exponential map is a map from the Lie algebra g of a Lie group G into G. In case G is a matrix Lie groupthe exponential map reduces to the matrix exponential.

The formula for d exp was first proved by Friedrich Schur The formula is important both in pure and applied mathematics. It enters into proofs of theorems such as the Baker—Campbell—Hausdorff formulaand it is used frequently in physics  for example in quantum field theoryas in the Magnus expansion in perturbation theoryand in lattice gauge theory.

Throughout, the notations exp X and e X will be used interchangeably to denote the exponential given an argument, except when, where as noted, the notations have dedicated distinct meanings. The calculus-style notation is preferred here for better readability in equations. On the other hand, the exp -style is sometimes more convenient for inline equations, and is necessary on the rare occasions when there is a real distinction to be made. The derivative of the exponential map is given by . The proof given below assumes a matrix Lie group. This means that the exponential mapping from the Lie algebra to the matrix Lie group is given by the usual power series, i. The conclusion of the proof still holds in the general case, provided each occurrence of exp is correctly interpreted.

See comments on the general case below. The outline of proof makes use of the technique of differentiation with respect to s of the parametrized expression. Lemma Let Ad denote the adjoint action of the group on its Lie algebra. A frequently useful relationship between Ad and ad is given by  [nb 1]. Using the formal power series to expand the exponential, integrating term by term, and finally recognizing 2. The proof, as presented here, is essentially the one given in Rossmann A proof with a more algebraic touch can be found in Hall The formula in the general case is given by .

Here the exp -notation is used for the exponential mapping of the Lie algebra and the calculus-style notation in the fraction indicates the usual formal series expansion. For more information and two full proofs in the general case, see the freely available Sternberg reference. An immediate way to see what the answer must be, provided it exists is the following. Existence needs to be proved separately in each case. By direct differentiation of the standard limit definition of the exponential, and exchanging the order of differentiation and limit.

The inverse function theorem together with the derivative of the exponential map provides information about the local behavior of exp. From 3 it follows that this will happen precisely when.

This, in turn, happens when the eigenvalues of this operator are all nonzero. The eigenvalues of ad X are, in turn, related to those of X. Fix an ordered basis e i of the underlying vector space V such that X is lower triangular. Let E ij be the corresponding basis for matrix space, i. One checks that the action of ad X is given by. The conclusion is that d exp X is invertible, hence exp is a local bianalytical bijection around Xwhen the eigenvalues of X satisfy  [nb 4].

However, using the relationship between Ad and ad given by 4it is straightforward to further see that. A textbook proof along these lines can be found in Hall and Miller Dynkin's formula mentioned may also be derived analogously, starting from the parametric extension.

It is at this point evident that the qualitative statement of the BCH formula holds, namely Z lies in the Lie algebra generated by XY and is expressible as a series in repeated brackets A.

The resulting Dynkin's formula is then.In mathematicsthe commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. The commutator of two elements, g and hof a group Gis the element. The set of all commutators of a group is not in general closed under the group operation, but the subgroup of G generated by all commutators is closed and is called the derived group or the commutator subgroup of G.

Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group. The definition of the commutator above is used throughout this article, but many other group theorists define the commutator as. Commutator identities are an important tool in group theory. It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator see next section.

Similar identities hold for these conventions. Many identities are used that are true modulo certain subgroups. These can be particularly useful in the study of solvable groups and nilpotent groups. For instance, in any group, second powers behave well:. If the derived subgroup is central, then. The commutator of two elements a and b of a ring including any associative algebra is defined by.

It is zero if and only if a and b commute. In linear algebraif two endomorphisms of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis.

By using the commutator as a Lie bracketevery associative algebra can be turned into a Lie algebra. The anticommutator of two elements a and b of a ring or an associative algebra is defined by. The commutator of two operators acting on a Hilbert space is a central concept in quantum mechanicssince it quantifies how well the two observables described by these operators can be measured simultaneously. Relation 3 is called anticommutativitywhile 4 is the Jacobi identity.

In other words, the map ad A defines a derivation on the ring R.This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. Functions, tensor fields and forms can be differentiated with respect to a vector field. The Lie derivative commutes with contraction and the exterior derivative on differential forms.

Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in this case the word "Lie" is dropped, and one simply speaks of the derivative of a function.

The space of vector fields forms a Lie algebra with respect to this Lie bracket. The Lie derivative constitutes an infinite-dimensional Lie algebra representation of this Lie algebra, due to the identity. Considering vector fields as infinitesimal generators of flows i. Generalisations exist for spinor fields, fibre bundles with connection and vector-valued differential forms. However, this definition is undesirable because it is not invariant under changes of coordinate systeme.

On an abstract manifold such a definition is meaningless and ill defined. In differential geometrythere are three main coordinate independent notions of differentiation of tensor fields: Lie derivatives, derivatives with respect to connectionsand the exterior derivative of completely anti symmetric covariant tensors or differential forms.

The main difference between the Lie derivative and a derivative with respect to a connection is that the latter derivative of a tensor field with respect to a tangent vector is well-defined even if it is not specified how to extend that tangent vector to a vector field. However a connection requires the choice of an additional geometric structure e.

In contrast, when taking a Lie derivative, no additional structure on the manifold is needed, but it is impossible to talk about the Lie derivative of a tensor field with respect to a single tangent vector, since the value of the Lie derivative of a tensor field with respect to a vector field X at a point p depends on the value of X in a neighborhood of pnot just at p itself.

Finally, the exterior derivative of differential forms does not require any additional choices, but is only a well defined derivative of differential forms including functions.

The Lie derivative may be defined in several equivalent ways. To keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields, before moving on to the definition for general tensors.

The Log Derivative Song (How to Differentiate a Logarithmic Function with a Song)

The above system of differential equations is more explicitly written as a system. There are several approaches to defining the Lie bracket, all of which are equivalent. We list two definitions here, corresponding to the two definitions of a vector field given above:. The Lie derivative of T is defined at a point p by.

We now give an algebraic definition. The algebraic definition for the Lie derivative of a tensor field follows from the following four axioms:. The Lie derivative acting on a differential form is the anticommutator of the interior product with the exterior derivative.

This follows easily by checking that the expression commutes with exterior derivative, is a derivation being an anticommutator of graded derivations and does the right thing on functions.

### 5. Derivative of the Logarithmic Function

Explicitly, let T be a tensor field of type pq.This works for any positive value of x we cannot have the logarithm of a negative number, of course. Note 1: Actually, this result comes from first principles.

Note 2: We are using logarithms with base e. If you need a reminder about log functions, check out Log base e from before.

We will not prove this limit here. For some problems, we can use the logarithm laws to simplify our log expression before differentiating it.

The derivative will be simply 2 times the derivative of ln x. The above graph only shows the positive arm for simplicity. Unfortunately, we can only use the logarithm laws to help us in a limited number of logarithm differentiation question types. Most often, we need to find the derivative of a logarithm of some function of x. See change of base rule to see how to work out such constants on your calculator. Note 1: This formula is derived from first principles.

Note 2: If we choose e as the base, then the derivative of ln uwhere u is a function of xsimply gives us our formula above:. The first term, log 2 6is a constant, so its derivative is 0. The term on the top, log 2 eis a constant. If we need a decimal value, we can work it out using change of base as follows:. The value 3. Note: Where possible, always use the properties of logarithms to simplify the process of obtaining the derivatives.

NOTE: We need to be careful with the domain of this solution, as it is only correct for certain values of x. So when we find the differentiation of a logarithm using the shortcut given above, we need to be careful that the domain of the function and the domain of the derivative are stated. Derivative of square root of sine x by first principles.

Derivative graphs interactive. Differentiating tanh by Haida [Solved! Name optional. Derivatives of Sin, Cos and Tan Functions 2. Derivatives of Csc, Sec and Cot Functions Differentiation interactive applet - trigonometric functions 3.

Derivatives of Inverse Trigonometric Functions 4. Applications: Derivatives of Trigonometric Functions 5. Derivative of the Logarithmic Function 6.

Derivative of the Exponential Function 7. Derivative of the Logarithmic Function. For this proof, we'll need the following background mathematics. We need to recognise that this is an implicit function.

NOTE : This has an exponent which is variable.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Now, many Physics books treating Quantum Field Theory, immediately relate the representations of Lie Groups and Lie Algebras without citing the result being used nor explaining how is it used really.

Now since physicists don't clear this in the books, I'm asking here. What is actually the relation between representations of Lie Groups and Lie Algebras that allows one to find the representations of the Lie Group in terms of the representations of the Lie Algebra?

I think the answer on your question is given probably from a geometric point of view.

## How to Differentiate with Logarithmic Functions

There is a beautiful theorem from Lie himself and is usually referred as Lie's 3rd Theoremand states something which nowadays is rephrased as follows over the complex numbers. Theorem: There is an equivalence between the category of complex simply connected Lie groups and category of complex Lie algebras. The above theorem isn't difficult to get proved and what it says in fact is that any complex Lie Algebra can be thought as the Lie Algebra of some Lie group.

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Active Oldest Votes. There is a beautiful theorem from Lie himself and is usually referred as Lie's 3rd Theoremand states something which nowadays is rephrased as follows over the complex numbers Theorem: There is an equivalence between the category of complex simply connected Lie groups and category of complex Lie algebras. Hope the above helps! So a Lie group representation induces a Lie algebra representation, and we know how to construct it. But why the opposite is true?The derivative of a logarithmic function is the reciprocal of the argument.

As always, the chain rule tells us to also multiply by the derivative of the argument. Differentiate by taking the reciprocal of the argument. Don't forget the chain rule! Differentiate using the formula for derivatives of logarithmic functions. When the argument of the logarithmic function involves products or quotients we can use the properties of logarithms to make differentiating easier.

We can avoid the product rule by first re-writing the function using the properties of logarithms and then differentiating, as shown below. Use the properties of logarithms to expand the function. Differentiate using the quotient rule. Expand the function using the properties of logarithms. Free Algebra Solver Make a Graph Graphing Calculator. Error : Please Click on "Not a robot", then try downloading again. Using the properties of logarithms will sometimes make the differentiation process easier.

Step 1 Differentiate by taking the reciprocal of the argument. Step 1 Differentiate using the formula for derivatives of logarithmic functions.

Step 1 Use the properties of logarithms to expand the function. Step 1 Differentiate using the formula for derivatives of logarithms. Step 2 Simplify the derivative. Step 1 Differentiate using the derivatives of logarithms formula.

Step 2 Simplify. Step 1 Rewrite the function so the square-root is in exponent form. Step 2 Identify the factors used in the function. Step 3 Differentiate using the product rule. Step 4 Simplify. Step 1 Differentiate using the quotient rule. Step 1 Expand the function using the properties of logarithms.

Step 2 Differentiate each term. Popular pages mathwarehouse. Surface area of a Cylinder. Unit Circle Game. Pascal's Triangle demonstration. Create, save share charts. Interactive simulation the most controversial math riddle ever! Calculus Gifs. How to make an ellipse. Volume of a cone.