The embedding structure and the shift operator of the U(1) lattice current algebra

The structure of block-spin embeddings of the U(1) lattice current algebra is described and the inner realizations of the shift automorphism are classified for an odd number of lattice sites. We present a particular inner shift operator which admits a factorization involving quantum dilogarithms analogous to the results of Faddeev and Volkov.On leave of absence from Steklov Mathematical Institute, St. Petersburg, Russia.     

The Schur function realization of vertex operators

We present a realization of untwisted vertex operators in terms of operations on Schur functions. Calculations of matrix elements and traces of products of vertex operators are performed using results from the classical theory of symmetric functions. The concepts of compound, composite and supersymmetric Schur functions naturally appear in this context. Furthermore, a trace formula for a product of vertex operators turns out to be a generalization of a MacDonald identity reformulated in terms of Schur functions.     

Vertex operator representations of the quantum affine algebra U q(B r (1) )

We construct the level one vertex operator representations of the q-deformation U _q(B _r ⁽¹⁾ ) of the affine Kac-Moody algebra B _r ⁽¹⁾ . Beside the q-deformed vertex operators introduced by Frenkel and Jing, this construction involves a q-deformation of free fermionic fields.     

Heisenberg realization for U q (sl n ) on the flag manifold

We give the Heisenberg realization for the quantum algebra U _q (sl _n ), which is written by theq-difference operator on the flag manifold. We construct it from the action of U _q (sl _n ) on theq-symmetric algebraA _q (Mat _n ) by the Borel-Weil-like approach. Our realization is applicable to the construction of the free field realization for U _q [2].     

Finite-dimensional representations of U q (osp(1/2n)) and its connection with quantum so(2n+1)

It is shown that the quantum supergroup U _q (osp(1/2_n)) is essentially isomorphic to the quantum group U _-q (so(2n+1)) restricted to tensorial representations. This renders it straightforward to classify all the finite-dimensional irreducible representations of U _q (osp(1/2n)) at generic q. In particular, it is proved that at generic q, every-dimensional irrep of this quantum supergroup is a deformation of an osp(1/2n) irrep, and all the finite-dimensional representations are completely reducible.     

The homogeneous realization of the basic representation of A inf1 sup(1) and the toda lattice

It is well-known that the principal realization of the basic module L(₀) over A _inf1 ^sup(1) gives rise to the KdV hierarchy of partial differential equations. Here we use the homogeneous realization of the same module to construct a hierarchy of differential-difference equations, the first member of which turns out to be the equation for the Toda lattice.     

Hypergeometric solutions of Knizhnik-Zamolodchikov equations

Explicit integral formulas are presented for the solutions of Knizhnik-Zamolodchikov equations associated with an arbitrary Kac-Moody Lie algebra.     

Vertex operators of level-one U q (B n (1) )-modules

We construct explicitly the level-one vertex operators for the fundamental modulesV(₁) (i=0, 1,n) of the quantum affine algebra of typeB using free boson and fermion fields.     

On a possible algebra morphism of U q [osp(1/2n)] onto the deformed oscillator algebra W q (n)

We formulate a conjecture stating that the algebra ofn pairs of deformed Bose creation and annihilation operators is a factor algebra of U _q [osp(1/2n)], considered as a Hopf algebra, and prove it for then = 2 case. To this end, we show that for any value ofq, U _q [osp(1/4)] can be viewed as a superalgebra freely generated by two pairsB ₁ ^± ,B ₂ ^± of deformed para-Bose operators. We write down all Hopf algebra relations, an analogue of the Cartan-Weyl basis, the commutation relations between the generators and a basis in U _q [osp(1/2n)] entirely in terms ofB ₁ ^± ,B ₂ ^± .     

The construction of a representation of loop algebras

By considering the cohomology of the loop algebraL , a representation ofL is constructed. the construction is based on a derivation ofL and a two-dimensional closed cochain ofl with coefficients in real numbersR ¹. In the case of =0, the differential of the energy representation of the corresponding loop groupLG is derived.This work was supported in part by the National Natural Science Foundation of China.     

Fusion algebra at a rational level and cohomology of nilpotent subalgebras of\widehat{\mathfrak{s}\mathfrak{l}}(2)

We define and calculate the fusion algebra of a WZW model at a rational level using cohomological methods. As a byproduct, we obtain a cohomological characterization of admissible representations of ₂.     

Remarks on the structure constants of the Verlinde algebra associated to sl3

We present a new formula for the structure constants of the Verlinde algebra associated to sl₃. We show that after an affine change of variables the structure constants, considered as a function of highest weights, become the weight function of a suitable sl₃ highest weight representation.Supported in part by NFS grants DMS-9400841 and DMS-9203929.     

New boson representations of the sl q (2) with multiplicity two and new solutions to the Yang-Baxter equation atq p =1

Whenq is a root of unity, the representations of the quantum universal enveloping algebra sl _q (2) with multiplicity two are constructed from theq-deformed boson realization with an arbitrary parameter which is in a very general form and is first presented in this Letter. The new solutions to the Yang-Baxter equation are obtained from these representations through the universalR-matrix.This work is supported in part by the National Foundation of Natural Science of China.     

The quantum Weyl group and the universal quantumR-matrix for affine lie algebraA1(1)

The universal quantumR-matrix is obtained in the case of the affine Kac-Moody Lie algebra sl(2).     

Left regular representation and contraction of sl q (2) to e q (2)

The left regular representation of the quantum algebras sl _q (2) and e _q (2) are discussed and shown to be related by contraction. The reducibility is studied andq-difference intertwining operators are constructed.     

Contraction of graded su(2) algebra

The Inönu-Wigner contraction scheme is extended to Lie superalgebras. The structure and representations of extended BRS algebra are obtained from contraction of the graded su(2) algebra. From cohomological consideration, we demonstrate that the graded su(2) algebra is the only superalgebra which, on contraction, yields the full BRS algebra.     

Free-field realization of theWKP(N) algebra

TheW _KP ^(N) algebra has been identified with the second Hamiltonian structure in theNth Hamiltonian pair of the KP hierarchy. In this Letter, by constructing the Miura map that decomposes the second Hamiltonian structure in theNth pair of the KP hierarchy, we show thatW _KP ^(N) can also be decomposed toN independent copies ofW _KP ⁽¹⁾ algebras, therefore its free-field realization can be worked out by constructing free fields for each copy ofW _KP ⁽¹⁾ . In this way, the free fields may consist ofN + 2n number of bosons, among them, 2n are in pairs, wheren is an arbitrary integer between 1 andN. We also express the currents ofW _KP ^(N) in terms of the currents ofN —n copies of U(1) andn copies of SL(2,R) _k algebras with levelk = 1. By reductions, we give similar results forW ^(N) andW ₃ ⁽²⁾ algebra.     

Quantization of U q [so(2n + 1)] with deformed para-Fermi operators

The observation thatn pairs of para-Fermi (pF) operators generate the universal enveloping algebra of the orthogonal Lie algebra so(2n + 1) is used in order to define deformed pF operators. It is shown that these operators are an alternative to the Chevalley generators. With this background U _q [so(2n + 1)] and its Cartan-Weyl generators are written down entirely in terms of deformed para-Fermi operators.     

量子力学表示理论的一种实现

量子力学创立伊始,狄拉克就关注到了一般表示的问题,其后量子力学的发展又引入了福克态、相干态等表示。好的表示应能提供正交归一的完备基,同时又能给出问题的严格解析解或者允许方便地得到近似解,但这常常是做不到的。我们意识到此前得到的相干态正交化方法恰恰满足表示理论的一般性要求,且因为包含自由参数为构造归一化的完备正交基实际上提供了无限的选择,这样甚至在解决问题的过程中都可以灵活地选择不同的表示,从而带来计算量的大幅减小。通过对不同耦合强度下的近共振态Rabi模型最初10个能级的计算,并同关联的JC模型的结果相比较,验证了相干态正交化方法的有效性。     

Spontaneous symmetry breaking of affine field theory

It is possible to construct non-Abelian field theories by gauging Kac-Moody algebras. Here we discuss the spontaneous symmetry breaking of such theories via the Higgs mechanism. If the Higgs particle lies in the Cartan subalgebra of the Kac-Moody algebra, the previously massless vectors acquire a mass spectrum that is linear in the Kac-Moody index and has additional fine structure depending on the associated Lie algebra.