Let g=g0¯⊕g1¯ be a finite-dimensional complex Lie superalgebra such that g0¯ is reductive and the adjoint representation of g0¯ in g1¯ is completely reducible. A Lie supergroup associated to g is defined by fixing the Hopf superalgebra of regular functions on the supergroup. Then it is shown that on this Hopf superalgebra there exists a non-zero left integral. According to an earlier work by the authors, this integral is unique up to scalar multiples.

The structure of the Hopf superalgebra B of regular functions on the general linear supergroup is developed. and applied to study the representation theory of the supergroup. It is shown that the general linear supergroup can he reconstructed from B in a way reminiscent of the Tannaka-Krein theory. A Boret-Weil type realization is also obtained for the irreducible integrable representations. As a side result on the structure of B, Schur superalgebras are introduced and are shown to be semi-simple over the complex field, with the simple ideals determined explicitly. (C) 2002 Elsevier Science (USA). All rights reserved.

Invariant integrals on Hopf superalgebras, in particular, the classical and quantum Lie supergroups, are studied. The uniqueness (up to scalar multiples) of a left integral is proved, and a Z2-graded version of Maschke’s theorem is discussed. A construction of left integrals is developed for classical and quantum Lie supergroups. Applied to several classes of examples the construction yields the left integrals in explicit form.

The R-matrix of the symplecto-orthogonal quantum superalgebra U_q(spo(2n|2m)) in the vector representation is calculated, and its basic properties are derived.

Recently, the R-matrix of the symplecto-orthogonal quantum superalgebra U_q(spo(2n|2m)) in the vector representation has been calculated. In the present work, this R-matrix is used to introduce the corresponding quantum supergroup SPO_q(2n|2m) and to construct an SPO_q(2n|2m)-covariant quantum Weyl superalgebra.

For a large class of finite-dimensional Lie superalgebras (including the classical simple ones) a Lie supergroup associated to the algebra is defined by fixing the Hopf superalgebra of functions on the supergroup. Then it is shown that on this Hopf superalgebra there exists a non-zero left integral. According to a recent work by the authors, this integral is unique up to scalar multiples.

Using techniques developed in a recent article by the authors, it is proved that the 2-cohomology of the Lie superalgebra sl ( m | 1 ) m ≥ 2, with coefficients in its enveloping algebra is trivial. The obstacles in solving the analogous problem for sl ( 3 | 2 ) are also discussed.

The cohomology groups of Lie superalgebras and, more generally, of ε Lie algebras, are introduced and investigated. The main emphasis is on the case where the module of coefficients is nontrivial. Two general propositions are proved, which help to calculate the cohomology groups. Several examples are included to show the peculiarities of the super case. For L=sl(1|2), the cohomology groups H1(L,V) and H2(L,V), with V a finite-dimensional simple graded L-module, are determined, and the result is used to show that H2(L,U(L)) [with U(L) the enveloping algebra of L] is trivial. This implies that the superalgebra U(L) does not admit any nontrivial formal deformations (in the sense of Gerstenhaber). Garland’s theory of universal central extensions of Lie algebras is generalized to the case of ε Lie algebras.

All finite dimensional irreducible unitary representations of the quantum supergroup Uq[gl(m‖n)] are classified in terms of their highest weights, for generic q≳0.

The universal R-matrices and, dually, the coquasitriangular structures of the group Hopf algebra of a finite Abelian group (resp. of an arbitrary Abelian group) are determined. This is used to formulate graded multilinear algebra in terms of triangular or cotriangular Hopf algebras. For the convenience of the reader, in a separate section the definitions and basic properties of quasitriangular and coquasitriangular Hopf algebras are recalled.

We consider an extension of the t-U Hubbard model taking into account new interactions between the numbers of up and down electrons. We confine ourselves to a one-dimensional open chain with L sites (4L states) and derive the effective Hamiltonian in the strong repulsion (large U) regime. This Hamiltonian acts on 3L states. We show that the spectrum of the latter Hamiltonian (not the degeneracies) coincides with the spectrum of the anisotropic Heisenberg chain (XX Z model) in the presence of a Z field (2L states). The wave functions of the 3L-state system are obtained explicitly from those of the 2L-state system, and the degeneracies can be understood in terms of irreducible representations of [Formula: see text].

The present work is a direct sequel to a recent article by the author, in which he has analysed the tensor product of tensor operators over quantum algebras. Here the results obtained there are summarized and then specialized and extended to prepare possible applications to quantum spin chains. In particular, certain invariant two-point operators are introduced (whose expectation values yield the invariant two-point correlation functions) and their multiplicative properties are derived.

Using U(q)[SU(2)] tensor calculus we compute the two-point scalar operators (TPSO), their averages on the ground state give the two-point correlation functions. The TPSOs are identified as elements of the Temperley-Lieb algebra and a recurrence relation is given for them. We have not tempted to derive analytic expressions for the correlation functions in the general case but got some partial results. For q = e(ipi)/3, all correlation functions are (trivially) zero, for q = e(ipi)/4, they are related in the continuum to the correlation functions of left-handed and right-handed Majorana fields in the half-plane coupled by the boundary condition. In the case q = e(ipi)/6, one gets the correlation functions of Mittag's and Stephen's parafermions for the three-state Potts model. A diagrammatic approach to compute correlation functions is also presented.

The generalized Lie algebras, which have recently been introduced under the name of color (super) algebras, are investigated. The generalized Poincaré–Birkhoff–Witt and Ado theorems hold true. We discuss the so-called commutation factors which enter into the defining identities of these algebras. Moreover, we establish a close relationship between the generalized Lie algebras and ordinary Lie (super) algebras.

It is pointed out that a star operation in a Hopf algebra, i.e., an involutive semilinear mapping of the Hopf algebra into itself which is (anti)multiplicative and (anti)comultiplicative, automatically satisfies a certain compatibility relation with the antipode.

It is shown that the naive Serre presentation corresponding to the simplest Cartan matrix of the special linear Lie superalgebra sl(m,n) yields, for m,n≥2, a larger Lie superalgebra s(m,n) of which sl(m,n) is a nontrivial quotient. The missing relations are found and the q deformation of the universal enveloping algebra of sl(m,n) is constructed. Some auxiliary results related to associative algebras and ε Lie algebras defined in terms of generators and relations are also included.

It is pointed out that, for m, n ≥2, the naive Serre presentation corresponding to the simplest Cartan matrix of sl(m, n) does not define the Lie superalgebra sl(m, n) but a larger algebra s(m, n) of which sl(m, n) is a nontrivial quotient. The supplementary relations for the generators are found and the definition of the q-deformed universal enveloping algebra of sl(m, n) is modified accordingly.

Tensor operators are defined for Hopf algebras. If the Hopf algebra is quasitriangular, the properly defined tensor product of tensor operators is a tensor operator. In one application, the problem of defining invariant correlation functions for the U(q) (sl(2))-symmetric spin 1/2 Heisenberg chain is considered. In another application, it is shown that suitably modified bosonic and fermionic q-creation and annihilation operators are tensor operators of U(q) (sl(n)).

It is shown that the braid generator σ associated with the universalR-matrix is diagonalizable on all unitary representations of quantum supergroups. An example is considered using U q (gl(2|1)) and a family of eight-dimensional typical representations.