Search

We present classical groups SL⋆(2) and SU⋆(2) as well as classical Lie algebra sl2⋆(C) associated with a new associative multiplication on 2 × 2 matrices. The idea of the new multiplication is generalized to the action of a 2 × 2 square matrix on a 2 × 1 column one. The coordinate Hopf algebra O(SLq⋆(2)) is introduced as a q-generalization of Hopf algebra O(SL⋆(2)), and it is shown that the coordinate algebra corresponding to the quantum plane Cq2 is a SLq⋆(2)-left-covariant algebra. Furthermore, the quantized universal enveloping algebra Uq(sl2⋆) with the Hopf structure as a dual of O(SLq⋆(2)) is introduced. For each of the Hopf algebras O(SLq⋆(2)) and Uq(sl2⋆), we associate two different real forms with two inequivalent families of *-involutions, with O(SUq⋆(2)) and Uq(su2⋆) as one of the real forms. It is shown that the Hopf algebra pairing is a dual pairing of two Hopf *-algebras O(SUq⋆(2)) and Uq(su2⋆).

Fisher-Hartwig asymptotics refers to the large $n$ form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin-spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the original Fisher-Hartwig formula to the asymptotic decay of the Ising correlations above $T_c$, while the study of the Bose gas density matrix leads us to generalize the Fisher-Hartwig formula to the asymptotic form of random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our generalizations is that they extend to Hankel determinants the Fisher-Hartwig asymptotic form known for Toeplitz determinants., 25 pages

A graphical formalism is introduced for describing subgroup type coordinates on n-dimensional Lorentzian hyperboloids imbedded into n+1 dimensional Minkowski spaces. The O(n,1) group element is parametrized according to different subgroup chains, involving Lorentz, rotation, and Euclidean subgroups. The coordinates are then induced by the corresponding group action. Eigenfunctions of the Laplace–Beltrami operator are obtained as products of Jacobi functions, associated Legendre functions, and modified Bessel functions.

We present closed forms for the exponential of some infinitesimal generators of Lie groups which play an important role in physics and engineering applications. These explicit forms are based on the Putzer’s method. We also compare this methodology and results with related work by other authors.

After reviewing the complete Lie group for the full Boltzmann equation, it is shown that projective transformations play a special role in the general case, and the class of invariant solutions giving rise to homoenergetic affine flows is presented. Homoenergetic affine flows in the two‐dimensional case [potential U(r)∝r−2] are considered in detail: It is shown that the general solution of this problem can be essentially simplified by projective transformations.

The unitary irreducible representations of the central extension of the Poincare group in 1+1 dimensions are constructed by an application of the Kirillov theory. These are then lifted to projective unitary irreducible representations of the Poincare group. The 1+1 Galilean group is treated separately in an appendix.

It is shown that the complete symmetry group for the Kepler problem, as introduced by Krause, can be derived by Lie group analysis. The same result is true for any autonomous system.

In this paper, a set of De Rham complexes of quantum group GLq(N) determined by one parameter r is given, and it is proven that the differential calculi on the quantum group GLq(N) given in this paper are bicovariant. The noncommutative differential calculi on the quantum groups SLq(N) and SUq(N) are also discussed.

Nontrivial elements of homotopy groups for unitary, orthogonal, and symplectic groups are given explicitly. In particular, (a) representatives of generators of nontrivial homotopy groups of stable special unitary, orthogonal, and symplectic groups are constructed using Clifford algebras; (b) the values for ‘‘winding numbers’’ for stable SU, SO, and Sp are calculated for generators of homotopy groups; and (c) representatives of generators of homotopy groups Πn−2(O(n−1)), Π2n−2(U(n−1)), Π4n−2(Sp(n−1)) are given.

The unitary Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the special unitary groups. The Jordan–Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.

The symplectic Cayley–Klein groups are defined as the groups that are obtained by the contractions and analytical continuations of the classical symplectic groups. The Jordan–Schwinger representations of the groups under consideration are discussed based on the mixed sets of creation and annihilation operators of boson or fermion type. The matrix elements of finite group transformations are obtained in the bases of coherent states.

Some features of Manin’s construction of quantum groups are developed and extended to supergroups.

The ordinary formalism for classical field theory is applied to dynamical group field theories. Focusing first on a local group field theory over one copy of SU(2) and, then, on more involved nonlocal theories (colored and non colored) defined over a tensor product of the same group, we address the issue of translation and dilatation symmetries and the corresponding Noether theorem. The energy momentum tensor and dilatation current are derived and their properties identified for each case.

The root systems of SO(8), SO(9) and F4 are constructed by quaternions. Triality manifests itself as permutations of pure quaternion units e1, e2 and e3. It is shown that the automorphism groups of the associated root systems are the finite subgroups of O(4) generated by left-right actions of unit quaternions on the root systems. The relevant finite groups of quaternions, the binary tetrahedral and binary octahedral groups, play essential roles in the construction of the Weyl groups and their conjugacy classes. The relations between the Dynkin indices, standard orthogonal vector and the quaternionic weights are obtained.

The Casimir operators of the following groups are explicitly constructed :Rn⧠Gl(n,R), Rn⧠Sl(n,R), Rn⧠O(p,q),Cn⧠U(p,q),R2n⧠Sp(n,R), Cn⧠Gl(n,C),Cn⧠Sl(n,C), Cn⧠O(n,C), C2n⧠Sp(n,C), Hn⧠U*(2n), Hn⧠SU*(2n), Hn⧠O*(2n), Hn⧠Sp( p,q), and R⧠(R2n⧠Sp(n,R)). The method is based on a particular fiber bundle structure of the generic orbits generated by the co‐adjoint representation of a semidirect product.

In general, if the parameters of a real or complex algebraic matrix group are replaced by Grassmann parameters, without changing the algebraic constraints, the resulting set fails to form a group. It is shown how to remedy this defect of naive Grassmannification by generalizing the constraint relations. In particular, it is shown how to define Grassmann analogs of the orthogonal, unitary, and symplectic groups.

The relation between two methods of finding the unitary projective representations of a connected Lie group is studied. It is shown that the one of the projective covering group G is singled out as the simplest one, but one could also get a connected and simply connected Lie splitting group of minimal dimension by using a similar procedure.

We extend global particle symmetries from the traditional group framework to that of generalized groups. The nature of these latter are presented, and various invariants constructed for them. The problem of gauging generalized groups is discussed and a no–go theorem proved under reasonable conditions on the generalized group structure.

By constructing the appropriate generating functions, the eigenvalues of the Casimir operators for the orthogonal and the symplectic groups are expressed in terms of power sums which are formally the same for the O(2n), Sp(2n), O(2n+1) groups as for the U(n) groups. The results for the O(2n), Sp(2n), and the O(2n+1) groups are written as the corresponding results for the U(n) groups plus very simple correction terms. This approach unifies the treatment of the problem for the semisimple Lie groups. Explicit evaluation of the eigenvalues of the Casimir operators becomes very simple.

We show how the definitions of the classical Lie groups have to be modified in the case where Grassmann variables are present. In particular we construct the general linear, the special linear and the orthosymplectic graded Lie groups. Special attention is paid to the question of how to formulate an adequate ’’unitarity condition.’’

A formalism for obtaining shift operators for all classical groups in the form of tensor operators is described in application to the orthogonal groups. This formalism is also applicable to the irreps in the discrete series of the noncompact versions of the classical groups.

Eigenvalues of the Casimir operators of the orthogonal and the symplectic groups are obtained in closed and simple form by diagonalizing directly the matrices introduced by Perelomov and Popov. This method unifies the treatment of the problem for the semisimple Lie groups.

Normalization coefficients are derived for the shift operators of O(n) introduced previously. The resultant normalized lowering operator for O(n) is presented in a form analogous to the U(n) case.

We study the graded Lie groups corresponding to the graded Lie algebras SU(2,2/1) and OSp(1/4). General finite group transformations are parametrized, and nonlinear representations are obtained on coset spaces. Jordan and traceless algebras are constructed which admit these groups as automorphism groups.

Eigenvalues of the invariant operators of the orthogonal and symplectic groups have been obtained in closed form. All semisimple Lie groups, the unitary, orthogonal, and symplectic groups, are treated in a systematic way by modifying Perelomov and Popov’s method. The eigenvalues of the invariant operators for the orthogonal and symplectic groups are then calculated with reference to the unitary group.

Diagrammatic methods for decomposing Kronecker products of arbitrary representations of any of the classical groups are presented. For convenience, efficient ways of computing the dimensions and quadratic Casimir’s C2(R) are also given. These methods seem more useful for hand calculations than the method of Schur functions (or characteristic polynomials). An appendix presents the Kronecker products for any two representations of dimension ⩽100.

Matrix elements of the generators of the symplectic group Sp(2n) have been obtained for the most degenerate irreducible representations (m2n,?) and (?2n). These are the only two cases for n≳2 where the state labels according to the branching laws of Hegerfeldt give rise to an orthogonal set of basic vectors.

The general formulas found in a preceding paper for the characters of irreducible representationsof simple Lie groups are developed in the case of the classical groups: the four series Al, Bl, Cl, Dl,plus the exceptional case G2. Groups of rank 1 and 2, plug A3 ≃ D3 are studied in detail.

The main purpose of this paper is to show the conditions under which a finite dimensional representation of a group, irreducible over the complex field, is reducible over quaternions. The answer is simply stated in terms of the Frobenius‐Shur classification of group representations.

For the classical compact Lie groups G, a formula for the multiplicity of weights (called inner multiplicity) is given. This formula relates the inner multiplicity of a group G to the inner multiplicity of a naturally embedded subgroup G′. For the SU(n) groups the formula can be brought into a particularly simple form—namely, a sum over Kronecker symbols—by choosing the group SU(2) for G′. The multiplicity of irreducible representations of a subgroup G′ into which an irreducible representation of a group G decomposes if G is restricted to G′—called restriction multiplicity of G with respect to G′—is related to the inner multiplicity of the group G.

A diagrammatic technique is presented for computing the matrix elements of the generators of the unitary, orthogonal, symplectic, and symmetric groups (An, Dn, Bn, Cn, and Sn) within any of their unitary irreducible representations. Examples are worked out.

A comparison is made between the method of constructing finite‐dimensional representations of the classical groups using the boson calculus in its standard or modified form and the method of Zhelobenko, which uses polynomials over a homogeneous space defined by a certain triangular subgroup. It is shown that the two methods can be directly related, so that one construction can be mapped into the other.

A method has been found whereby the inner multiplicity of all classical groups [all irreducible representations of SU(n) and SO(2k + 1) and some simple irreducible representations of SO(2k) and Sp(2k)] can be obtained easily from the branching laws of Weyl (unitary), Boerner (orthogonal), and Hegerfeldt (symplectic). Once the inner multiplicity is known, the same formula can be used again to obtain the decomposition of a classical group into its subgroups without any restriction. Finally, since the inner multiplicity is connected to the outer multiplicity through the Racah‐Speiser lemma, this method enables us to obtain the Clebsch‐Gordan series for the direct product of all classical groups.

The generating functions of classical groups are used to set up recursion relations for their partition functions. These are then used to find the internal multiplicity structure of the weights using Kostant's formula.

We suggest a very simple method for handling relativistic n‐body kinematics. In fact the method works well for studying the n‐fold Kronecker products of the representations of many groups, including all compact Lie groups as a simple case. The method is explicitly covariant, and treats all particles symmetrically.

We discuss here in detail the validity of the branching‐law method suggested in a previous paper [M. K. F. Wong, J. Math. Phys. 11, 1489 (1970)] for the calculation of the inner multiplicity of weights in an irreducible representation of a classical group. It is found that the method works for all the irreducible representations of SU(n) and SO(2n + 1), but that in the case of SO(2n) and Sp(2n) the method does not always give complete solutions except in some simple cases. It is then suggested that Racah's recurrence relation be used in these cases so that complete solutions may be obtained. It is also noted that Racah's recurrence relation alone is sufficient to obtain the inner multiplicity of all weights. This fact is utilized in the calculation of inner multiplicities in another paper [B. Gruber, J. Math. Phys. 11, 3077 (1970)]. The method suggested in this paper is illustrated through the calculation of some typical examples of the inner multiplicity of weights in the two classical groups SO(2n) and ...

A tensor symmetrization procedure obtained in a recent publication [Phys. Rev. Letters 16, 1058 (1966)] is shown to support rather than disprove Weyl's tensor symmetrization theorem. This ``extended'' symmetrization procedure differs from Weyl's approach in that to construct a subspace irreducible under GL(n, c) one starts with a set of formal states (symmetrized tensors with formal index values) spanning an irreducible representation of the permutation group rather than starting with a single formal state. Extended symmetrization is often more useful than Weyl's approach because the states obtained are highly organized and because it also yields an efficient independent state selection method for the symmetrization procedures using modified Young symmetrizers and Wigner projection operators. The state organization obtained makes it possible to show that the nonorthogonality which is present for bases obtained with Young symmetrizers can be easily removed. The state organization also makes it possible to simplify the task of recoupling symmetrized tensor representations to gain a simply‐coupled form. This form enlarges the class of Clebsch‐Gordan and recoupling coefficients which can be evaluated by tensor methods. Group matrices and Lie group generator matrix elements are also obtained by tensor methods. Extended symmetrization using unitary representation Wigner projection operators based on unitary representations is shown to result in orthogonal states although usually not the orthogonal states desired. The usual Young symmetrizers are shown to often be more useful than modified Young symmetrizers or Wigner projection operators.

In a previous paper, a method has been developed which allows a calculation in the same manner of all multiplicities occurring in the theory of linear representations of the classical compact Lie groups. This method is applied to the group SO(9) and some of its subgroups which are of interest in physics. It is demonstrated with the group SO(9) as example that this method of calculating the multiplicities allows us to accumulate in a condensed way a large amount of information and, moreover, that this method can be easily applied to any other (classical, compact) rank‐4 group and its semisimple subgroups.

Recurrence relations for the ``outer multiplicity'' (multiplicity of irreducible representations occurring in the reduction of the direct product of two irreducible representations) as well as the ``branching multiplicity'' (multiplicity of irreducible representations into which an irreducible representation of a group decomposes, if this group is restricted to a subgroup) are obtained. These recurrence relations are shown to be closely related to each other. Racah has obtained a recurrence relation for the ``inner multiplicity'' (multiplicity of weights). It turns out that the recurrence relation for the inner multiplicity is a special case of the recurrence relation for the outer multiplicity which, in turn, can be looked upon as a special case of recurrence relation for the branching multiplicity. Moreover, Kostant's recurrence relation for the partition function as well as Racah's recurrence relation for the inner multiplicity have been generalized. This generalization introduces a parameter which can be used to simplify actual calculations. In an appendix, finally, a formula is given which allows one to calculate the branching multiplicity from known inner multiplicities of the subgroup to which it is restricted.

The Adler–Bell–Jackiw (ABJ) anomalies for all representations of the classical groups are calculated with the aid of a technique proposed recently for the representation theory of these groups. The method presented here is useful also for the calculations of the eigenvalues of Casimir invariants with rank higher than 3.

An irreducible unitary representation of any group belongs to one of the three Wigner classes, potentially real, pseudo‐real, or complex. The irreducible unitary representations of all the compact simple Lie groups except those of the type E are hereby classified. The similar classification for the simple groups E6, E7, and E8 is completed in the next paper.

The explicit form of a vector constant of the motion for an arbitrary relativistic spherically symmetric time independent classical Hamiltonian is obtained by showing that its construction is achieved on solving a linear second order ordinary differential equation. The solution of this equation is presented and the vector, in conjunction with the angular momentum to which it is normal, is used to generate the algebras of the Euclidean group E(3), the orthogonal rotation group O(4) and the special unitary group SU(3). The mass is assumed to be structureless and to move in an externally prescribed scalar potential field.