His latest book, an update and expansion of his well-known Lie Groups, Lie Algebras, and Some of Their Applications Wiley , is targeted to mathematical by: views Lie Groups, Physics, and Geometry by Robert Gilmore - Drexel University, The book emphasizes the most useful aspects of Lie groups, in a way that is easy for students to acquire and to assimilate.
It includes a chapter dedicated to the applications of Lie group theory to solving differential equations. Veltman B. This is quite a useful introduction to some of the basics of Lie algebras and Lie groups, written by a physicist for physicists. It is a bit idiosyncratic in its coverage, but what it does cover is explained reasonably well. Size: KB. Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological of the key ideas in the theory of Lie groups is to replace the global object, the group, with its local or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra.
Last edited by Kakora. Want to Read. Written in English Subjects: Group theory, Lie groups About the Edition Introduction to Lie groups for graduate and undergraduate students in physics, mathematics and electrical engineering. Edition Notes Description based on print version record.
Share this book. Smart urban growth in China. The origin of the family, private property, and the state. Not Necessarily the New Age. The intimate male. Lie Groups, Lie Algebras, and Representations. This book addresses Lie groups, Lie algebras, and representation theory. The author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all of the most interesting examples.
From the reviews:"Sure to. Lie Groups Beyond an Introduction. The structure of a Lie algebra can be determined by inspecting its regular representation, once this has been brought to suitable form by a similarity transformation. To facilitate constructing this transformation, we have shown how to use the Cartan—Killing inner product to determine the linear vector subspaces in the Lie algebra that are maximal nilpotent invariant subalgebras, the maximal solvable invariant subalgebra, the semisimple subalgebra, and its maximal compact subalgebra.
Compute the decomposition 8. The photon algebra n, b. The algebra so 3, 1. The algebra for the Galilei group Eq. Commutators involving bilinear trilinear,.
What is the relation between the Cartan—Killing inner product computed using the defining matrix representation of a matrix Lie algebra and using the regular matrix representation of the Lie algebra? Compute the matrix infinitesimal generators for each. Decompose each Lie algebra into the standard form 8. If there is one quadratic Casimir operator, it must therefore be proportional to C 2.
Hint: use the Jacobi identity. Determine the structure of the Lie algebra defined by the following operators cf.
These two chapters focus on determining the structure of a Lie algebra and putting it into some canonical form. In the previous chapter we determined the types of subalgebras that every Lie algebra is constructed from. In this chapter we put the commutation relations into a standard form.
This can be done for any Lie algebra. For semisimple Lie algebras this standard form has a very rigid structure whose usefulness is surpassed only by its beauty. This study was carried out using as a tool the Cartan—Killing inner product. As far as possible, this was the only method used. In the present chapter we introduce a second powerful tool from the theory of linear vector spaces. This is the eigenvalue decomposition.
This tool is introduced in an attempt to find standard forms for the commutation relations. If a standard form is available then the properties of a Lie algebra, as well as its identification classification , can be determined at sight.
The eigenoperator decomposition is effected by computing and studying a secular equation determined from the matrix of the regular or any other matrix representation of the Lie algebra. To get the most information from this study we seek the maximum number of independent roots of this equation.
The decomposition of the Lie algebra into eigenoperators according to the roots of the secular equation, and the properties of these roots, can also be discussed for any Lie algebra. However, for Lie algebras with a nonsingular Cartan—Killing inner product — semisimple and simple Lie algebras — the properties of the roots are very rigidly prescribed. This leads to a very elegant set of canonical commutation relations. Without this extension it is not always possible to find roots of the secular equation.
This field extension has the drawback that several different Lie algebras e. We return to this question in Chapter 11, where the problem is resolved. The converse is true. Therefore, there is a permutation transformation of the basis vectors that brings the regular representation of this Lie algebra to strictly upper triangular form, and the algebra is nilpotent by inspection.
This same rotation of coordinates maps the Lie algebra su 2 to the Lie algebra su 1, 1. The secular equation was written down for the regular representation, since it can always be constructed from the Lie algebra. A secular equation could just as easily be written down for any matrix representation of the Lie algebra.
We are by and large interested in studying matrix Lie algebras, so secular equations can be written directly for the defining matrix algebras. There is a great deal of utility in this approach. First, the matrices in a matrix algebra are almost always smaller — much smaller — than the matrices of its regular representation. Second, a matrix Lie algebra contains at least as much information certainly not less as its regular representation. Example From the secular equation 9. Coefficients in the secular equation are expressed in terms of the fully antisymmetric Levi—Civita tensor on n symbols.
In this figure the vertical symbol is the Levi—Civita symbol for n dimensions e. Contracted dummy indices are connected by lines. The invariance of these operators depends only on the commutation relations of the Lie algebra. Therefore 9. The existence of two second-order Casimir operators for so 4 is another piece of evidence that this algebra is semisimple rather than simple. We do this by choosing a Z for which we: 1.
Such elements Z in the Lie algebra can always be found. The six operators in the two-photon algebra can be organized according to their roots, which are eigenvalues of a secular equation. Two operators have zero root. When this inner product is nonsingular, the decomposition of the algebra into its subspaces, one for each root of the secular equation, has additional properties.
We list these properties here, providing only an occasional proof. A more Structure theory for simple Lie algebras complete treatment of this, the most beautiful part of Lie algebra theory, can be found elsewhere Gilmore, b; Helgason As a result i.
Thus, nonzero roots occur in pairs of opposite sign. We use this expression because it shows clearly how the boundary conditions are imposed. All of the rank-two root space diagrams are shown in Fig. There the symmetries of root spaces under reflection and rotation may be seen.
Two-dimensional root space diagrams. Top: A2 , B2 , C2. Bottom: D2 , G 2. Root space C2. All commutators not explicitly shown in this table vanish.
For this ranktwo algebra two phases may be set arbitrarily. Using the Cartan—Killing inner product it is possible to determine the semisimple part of a Lie algebra and its complement, the maximal solvable invariant subalgebra. An eigenvalue decomposition can be used to put the commutation relations of the semisimple part into a standard form.
When the algebra is simple or semisimple the commutation relations are elegantly summarized by a root space diagram. This is a simple geometric structure in a Euclidean space of dimension l, where l is the rank of the Lie algebra. The rank is: i ii iii iv v the number of functionally independent coefficients in the secular equation; the number of independent roots of the secular equation; the number of Casimir invariant operators; the dimension of the root space diagram; the number of mutually commuting operators in the Lie algebra.
We have illustrated how to extract commutation relations from a root space diagram for C2. In doing so, we have introduced a situation in which different algebras have the same complex extension e. Root spaces classify commutation relations of these complex Lie algebras. However, determining the real subalgebras of a complex Lie algebra is a not entirely trivial task to which we return in Chapter Determine the secular equation for this matrix.
Determine the rank of this Lie algebra. Construct the secular equation. Both are quadratic. Construct these coefficients. Use these to construct the two quadratic invariant operators on this semisimple Lie algebra. Show that the secular equation of the regular representation has just three independent coefficients. Do this by showing that there is a relation between the secular equation for the regular representation and the secular equation for the defining matrix representation.
What is this relation? The three independent coefficients in the secular equation for the defining representation are of degree 2, 3, 4. Construct the invariant operators on su 4 of degree 2, 3, and 4. This is true also for so 2n , with one difference: the invariant operator of degree 2n is a perfect square. Explicitly write out C2 for so 4 and C3 for so 6. Compare your results with Fig. Both Lie algebras have invariant operators of degree 2, 3, 4.
Constuct the isomorphism between these Lie algebras and their invariant operators. This algebra is ten dimensional. Show that this root space diagram is isomorphic to C2 , shown in Fig. Repeat Problem 6 for the algebra of two fermion operators for two modes. This algebra is six dimensional. Show that the resulting root space diagram is D2 Fig. The Lie algebras su 2 and so 3 are isomorphic.
In fact, the latter is the regular representation for the former. Show that the two results are proportional. What is the proportionality constant? Choose two vectors X and Y in the Lie algebra su n. The two inner products are proportional.
This vector plays a major role in computing the spectrum of the quadratic Casimir operator for each of the irreducible representations of each of the simple Lie algebras.
The Weyl group of reflections for a simple Lie algebra is generated by reflections in planes orthogonal to all the nonzero roots.
Show that the product of the degrees of the functionally independent coefficients in the secular equation for each of these algebras is equal to the order of the Weyl group.
Show that the product of the degrees of the Casimir operators for each of these algebras is equal to the order of the Weyl group. Show A1 , A2 ,. What are the conditions on a i j for each matrix Lie algebra? Show that the even coefficients are all functionally independent. Conclude that each of these three matrix Lie algebras has rank n. Show that each coefficient f j is a function of the invariants of the matrix X. Show that the Cayley—Hamilton expansion simplifies considerably if the matrix X is chosen as generic diagonal.
Problem 9. Although roots were introduced to simplify the expression of commutation relations, they can be used to classify Lie algebras and to provide a complete list of simple Lie algebras. We achieve both aims in this chapter. However, we use two different methods to accomplish this.
We classify Lie algebras by specifying their root space diagrams. However, it is not easy to prove the completeness of root space diagrams by this method. Completeness is obtained by introducing Dynkin diagrams. These specify the inner products among a fundamental set of basis roots in the root space diagram. In this approach completeness is relatively simple to prove, while enumeration of the remaining roots within a root space diagram is less so. The rank of an algebra is, among other things: i ii iii iv the number of independent functions in the secular equation; the number of independent roots of the secular equation; the number of mutually commuting operators in the Lie algebra; the number of invariant operators that commute with all elements in the Lie algebra Casimir operators ; v the dimension of the positive-definite root space that summarizes the commutation relations.
These two observations are all that is required to construct root space diagrams of any rank. The results are summarized in Table Begin with the rank-one root space. To construct rank-two root spaces, add a noncollinear vector to this root space in such a way that the constraints exhibited in Table Only a small number of rank-two root spaces can be constructed in this way. A noncoplanar vector is added to a rank-two root space diagram subject to the condition that all the requirements of Table The resultant set of roots is completed by reflection in hyperplanes orthogonal to all roots.
If any pair of roots in the completed diagram does not satisfy these conditions, the resulting diagram is not an allowed root space diagram. The allowed rank-three root space diagrams are shown in Fig. This procedure is inductive. Rank-three root space diagrams. Top: A3 , D3. Bottom: B3 , C3. Root spaces constructed by the building-up principle. There are four infinite series and five exceptional Lie algebras.
The root spaces are organized by rank. In this figure all root spaces are shown by rank. Arrows connect pairs related by the building-up principle. Remark 1. The root space D2 consists of two orthogonal root subspaces. Both describe the rank-one algebra A1. Several different Lie groups algebras are associated with each root space. This comes about because root spaces classify complex Lie algebras. Recall that extension of the field from real to complex numbers was required to guarantee that the secular equation could be solved.
Remark 2. The decomposition is shown in Fig. Orthogonal root spaces describe semisimple Lie algebras. Root subspaces that do not have an orthogonal decomposition describe simple Lie algebras.
Complete reducibility of the regular representation corresponds to decomposition of the root space into disjoint orthogonal root spaces and of the semisimple Lie algebras to simple invariant subalgebras. Remark 3. The root spaces B2 and C2 are equivalent, as is easily seen by rotation. Remark 4.
In the building-up construction the roots in each root space diagram are explicitly constructed. What is not immediately obvious is that there are no more simple root spaces than those listed. How are we sure that there are no more than five exceptional root spaces?
This question is not easy to resolve in the context of root space constructions alone. However, it is easily resolved by another algorithmic procedure.
This procedure yields a beautiful completeness argument. The price we pay is a somewhat greater difficulty in constructing the complete set of roots for However, since they have been constructed above, this poses no severe limitation. Among the positive roots the l nearest to this hyperplane in a rank-l root space are linearly independent. They can therefore be chosen as a basis set in this space.
Every positive root can be expressed in terms of this basis as a linear combination of these fundamental roots with integer coefficients. The integers are all positive or zero, because every shift operator defined by a positive root can be written as a multiple commutator of shift operators with fundamental positive roots. By symmetry, every negative root is a linear combination of fundamental roots with nonpositive integer coefficients.
The fundamental roots for G 2 are shown in Fig. Root space for G 2. Disconnected Dynkin diagrams describe semisimple Lie algebras. Each fundamental root is represented by a dot. Dots i and j are joined by n i j lines. Orthogonal roots are not connected. Such a diagram is called a Dynkin diagram. Orthogonal root spaces for semisimple Lie algebras are represented by disconnected Dynkin diagrams.
In these diagrams the relative squared lengths of the fundamental roots 3, 1 for G 2 are indicated over the root symbol, by an arrow pointing from the shorter to the longer, and by open and solid dots. The conventions are interchangeable: normally not more than one is adopted.
We will use only one at a time. Only a very limited number of distinct kinds of Dynkin diagrams can occur. The limitations derive from two observations.
Observation 1 The root space is positive-definite. A simple linear chain can be removed. If the original is an allowed Dynkin diagram, the shortened diagram is also an allowed Dynkin diagram.
In this case the original diagram is not an allowed Dynkin diagram. General forms of allowed root space diagrams after the process of contraction has been performed.
Property 1 There are no loops. A diagram containing a loop has at least as many lines as vertices. Property 2 The number of lines connected to any node is less than four. This results from Observation 2.
This table provides a complete list of simple root spaces. Each root space was constructed in Section The complete set of roots in each of the root spaces is listed in that section. The completeness of this construction is guaranteed by the correspondence between the root space diagrams constructed in Section Show that the following three statements for a semisimple Lie algebra are equivalent: a.
Do these statements extend to semisimple Lie algebras with three or more simple invariant subalgebras? Identify H1 and H2. Apply the Schwartz inequality to the two vectors in Eq. Use the projection inequality of Eq. Carry out a similar decomposition for any value of n. The Lie algebras so classified exist over the field of complex numbers. Each simple Lie algebra over C of complex dimension n has a number of inequivalent real subalgebras over R of real dimension n.
These are obtained by putting reality restrictions on the coordinates in the complex Lie algebra. The different real forms of a complex simple Lie algebra are obtained systematically by a simple eigenvalue decomposition. For the classical matrix Lie algebras, three different procedures suffice to construct all real forms. These are: block submatrix decomposition; subfield restriction; and field embedding.
This equation cannot be solved in general unless the field is extended from the real to the complex numbers. Allowing that extension, we were able to find a canonical form for the operators in semisimple Lie algebras. The commutation relations were classified in terms of a root space diagram. These diagrams were used to enumerate all the simple Lie algebras over the complex field. These decompositions are: block submatrix decomposition; subfield restriction; and field embeddings.
Example The noncompact Lie algebras sl 2; R and su 1, 1 have commutation relations described by the root space A1. The nonisomorphic Lie algebra su 2 has the same root space. This relation is shown in Fig. The complex extension Lie algebra has root space A1 describing canonical commutation relations for the diagonal and shift operators shown in Fig. The most general element in this Lie algebra is a complex linear combination of the three matrices shown.
The algebras sl 2; R and su 2 have real dimension 3 while their common complex extension has complex dimension 3 real dimension 6. In the following sections we present a systematic way for determining how to restrict the complex parameters to real parameters in order to construct all inequivalent real Lie algebras with the same dimension as the complex Lie algebra whose commutation relations are described by a root space diagram.
The canonical form for the diagonal and shift operators in their Lie algebras is also shown. H1 H2 H This is the trace of the normalized Cartan—Killing form. Inspection of Since the compact real form can always be constructed easily for a simple Lie algebra see Eq. The mapping, commutation relations, and A systematic method exists for finding Cartan decompositions Under T , one eigenspace of T is mapped into itself while the other its orthogonal complement is mapped into its negative.
These three mapping types are derived from block matrix decomposition, subfield restriction, and field embeddings. We discuss each in the next three subsections, indicating the real forms that are produced. In all instances we begin with the compact Lie algebras. Under the procedure described in the previous section the off-diagonal block is multiplied by i.
A Lie algebra over the complex numbers can be divided into two subsets: real matrices and the remainder, imaginary matrices. It is possible to choose coordinates p1 , q1 , p2 , q2 ,. Symplectic transformations leave invariant the canonical form of the hamiltonian equations of motion in classical mechanics. This table indicates the root space associated with each real form.
Some of the low-dimensional root spaces are equivalent. For example, A1 where the compact real form is su 2 , B1 so 3 , and C1 sp 1 are equivalent, as are B2 so 5 and C2 sp 2.
So also are A3 su 4 and D3 so 6. As a result, there are equivalences between the real forms of these Lie algebras. These equivalences are summarized in Table Table The subscript in parentheses after the rank is the character of the real form. By placing various reality restrictions on the coefficients of the complex algebra, a spectrum of real subalgebras is obtained, each of which has the same complex extension. To each root space there corresponds a unique real form that is compact.
For the simple classical matrix Lie algebras three types of mappings T suffice to construct all real forms: block submatrix decomposition; subfield restriction; and field embedding. These operators close under commutation. Identify the simple three-dimensional subalgebra as sl 2; R or su 1, 1.
Show that the compact real form is obtained by multiplying the two noncompact generators by i. Multiply the two noncompact operators by i. Real forms Spectrum of quadratic Casimir a. Use the metric Identify basis states in a Hilbert space by their eigenvalues under the operators Hi : n 1 , n 2 ,.
For the orthogonal groups S O n , impose suitable reality conditions i. In this problem we will review the construction of the UIR unitary irreducible representations of the compact group SU 2 and will use similar methods to construct all the UIR of its analytic continuation, the noncompact Lie group SU 1, 1. Since the algebras are related by analytic continuation, so also are the UIR. We will begin with the analytic hermitian matrix elements for su 2 and continue to hermitian matrix elements for the analytically continued algebra su 1, 1.
Show that all commutation relations are satisfied in the representation afforded by this set of basis states. The integer lattice in two dimensions carries representations of the algebras su 2 and su 1, 1 that exponentiate to unitary irreducible representations with careful choice of the basis set.
Now relax the condition that n 1 and n 2 are integers. In the latter case, it is not possible for the matrix element in Eq. This procedure is delicate: one must be careful of the complex unit i with quaternions. Show by more careful arguments that the result stated is correct. In both cases these subspaces exponentiate onto algebraic manifolds on which the invariant metric gi j is definite, either negative or positive.
Manifolds with a definite metric are Riemannian spaces. These spaces are also globally symmetric in the sense that every point looks like every other point — because each point in the space EXP p or EXP ip is the image of the origin under some group operation.
We briefly discuss the properties of these Riemannian globally symmetric spaces in this chapter. These spaces are shown in Fig.
On S 2 the Cartan—Killing metric is negative-definite. We may just as well take it as positive-definite.
Under this metric the sphere becomes a Riemannian manifold since there is a metric on it with which to measure distances. The upper sheet of the two-sheeted hyperboloid is topologically equivalent to the flat space R 2 but geometrically it is not: it has intrinsic curvature that can be computed, via its Cartan—Killing metric and the curvature tensor derived from it.
The first two are Riemannian symmetric spaces, the third is a pseudo-Riemannian symmetric space. The most interesting of these spaces is the single-sheeted hyperboloid H The Cartan—Killing inner product in this linear vector space is indefinite. The space is a pseudo-Riemannian manifold. In addition it is multiply connected. For a compact simple Lie algebra g i.
This space is globally symmetric. This is because they all look like the identity EXP 0 , since the identity and its neighborhood can be shifted to any other point in the space by multiplication by the appropriate group operation for example, by EXP p or EXP i p.
The exponential of a straight line through the origin in p returns periodically to the neighborhood of the identity. The space P is not topologically equivalent to any Euclidean space, in which a straight line geodesic through the origin never returns to the origin. The space P may be simply connected or multiply connected.
The exponential of a straight line through the origin in i p a geodesic through the identity in EXP i p simply goes away from this point without ever returning. Geometrically it is not Euclidean since it has nonzero curvature.
This space is simply connected. The rank of these spaces is min p, q. For Riemannian globally symmetric spaces the rank is cf. Section We will not elaborate on these points here. The number of nonisotropic directions is determined by computing the number of distinct eigenvalues of the Cartan—Killing metric on P, or equivalently and more easily, of the Cartan—Killing inner product on p same as the metric at the identity. In each of the spaces P there is a Euclidean subspace submanifold.
For S 2 , any great circle is Euclidean. These spaces are classical because they involve the classical series of Lie groups: the orthogonal, the unitary, and the symplectic. As before, to each there is a dual compact real form.
The metric at the identity is chosen as the Cartan—Killing inner product on i p, or its negative on p. If d x Id are infinitesimal displacements at the identity that are translated to infinitesimal displacements d x p at point p, then these two sets of infinitesimals are linearly related by a nonsingular linear transformation cf.
Since the intrinsic properties of the Riemannian symmetric space are entirely encoded in its metric tensor, we can begin to compute its important properties, for example, the curvature tensor. It is first useful to compute the Christoffel symbols as a way-station on the road to computing the full Riemannian curvature tensor. The Christoffel symbols not a tensor! This task is greatly simplified in a symmetric space, for all points look the same and we can compute the tensors wherever the computation is easiest.
This turns out to be at the origin. We illustrate by carrying out the computations in the neighborhood of the identity for the compact case, the sphere. At the origin the components of the Christoffel symbols all vanish, so it is sufficient to retain only the first two terms in the expression for the curvature tensor. The computation can be carried out just as easily for the noncompact space H Assume also that the inner product is indefinite on p also h.
The metric on this space is indefinite. The space is curved and typically multiply connected. All of the algebraic properties associated with a Riemannian symmetric space hold also for pseudo-Riemannian symmetric spaces. That is, rank can be defined, and carries most of the implications listed in Section There is a systematic method for constructing pseudo-Riemannian symmetric spaces.
Note that each mapping Ti has one positive and two negative eigenvalues, and chooses a different generator for the maximal compact subalgebra h of the noncompact real form g. All Riemannian globally symmetric spaces are constructed as quotients of a simple Lie group G by a maximal compact subgroup K. More specifically, they are exponentials of a subalgebra p of a Lie algebra g for which commutation relations and inner products are given by Pseudo-Riemannian globally symmetric spaces are similarly constructed.
For these spaces the rank can be defined. This determines a number of algebraic properties maximal number of independent mutually commuting generators and Laplace—Beltrami operators as well as geometric properties number of nonisotropic directions, dimension of maximal Euclidean subspaces. Metric and measure are determined on these spaces in an invariant way. These are the eigenvalues of these square, hermitian matrices.
Show that there is one Laplace—Beltrami on the two-sheeted hyperboloid H22 and compute it in the standard parameterization in terms of the coordinates on the plane R2. Show that these two Laplace—Beltrami operators are dual in some sense. What sense? Discuss dualities. There is one independent root. What else can be said about this Riemannian symmetric space?
The metric on a pseudo-Riemannian symmetric space is gi j x. The unit disk, with a suitable metric, provides a second representation of the hyperbolic plane.
Here z 0 is any point in the upper half-plane. Compute the inverse of this mapping, and show that it maps the interior of the unit disk unto the upper half of the complex plane and the boundary of the unit disk onto the real axis boundary of the upper half-plane. Show that this transformation maps the invariant metric and measure on the upper half-plane onto the invariant metric and measure on the unit disk.
In general, the properties of the original Lie group have well-defined limits in the contracted Lie group. For example, the parameter space for the contracted group is well defined and noncompact.
Other properties with well-defined limits include: Casimir operators; basis states of representations; matrix elements of operators; and Baker—Campbell— Hausdorff formulas. Contraction provides limiting relations among the special functions of mathematical physics.
As long as the change of basis transformation is nonsingular the Lie algebra is unchanged. If the transformation becomes singular, the structure constants Cr st may still converge to a well-defined limit.
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