Casimir invariants and characteristic identities for gl(∞)gl(\infty )

A full set of (higher order) Casimir invariants for the Lie algebra $gl(\infty )$ is constructed and shown to be well defined in the category $O_{FS}$ generated by the highest weight (unitarizable) irreducible representations with only a finite number of non-zero weight components. Moreover the eigenvalues of these Casimir invariants are determined explicitly in terms of the highest weight. Characteristic identities satisfied by certain (infinite) matrices with entries from $gl(\infty )$ are also determined and generalize those previously obtained for $gl(n)$ by Bracken and Green.$^{1,2}$Comment: 10 pages, PlainTe

Finite-Dimensional Representations of the Quantum Superalgebra Uq_{q}[gl(2/2)]: II. Nontypical representations at generic qq

The construction approach proposed in the previous paper Ref. 1 allows us there and in the present paper to construct at generic deformation parameter $q$ all finite--dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$. The finite--dimensional $U_{q}[gl(2/2)]$-modules $W^{q}$ constructed in Ref. 1 are either irreducible or indecomposible. If a module $W^{q}$ is indecomposible, i.e. when the condition (4.41) in Ref. 1 does not hold, there exists an invariant maximal submodule of $W^{q}$, to say $I_{k}^{q}$, such that the factor-representation in the factor-module $W^{q}/I_{k}^{q}$ is irreducible and called nontypical. Here, in this paper, indecomposible representations and nontypical finite--dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$ are considered and classified as their module structures are analized and the matrix elements of all nontypical representations are written down explicitly.Comment: Latex file, 49 page

Eigenvalues of Casimir operators for gl(m/∞)gl(m/\infty)

A full set of Casimir operators for the Lie superalgebra $gl(m/\infty)$ is constructed and shown to be well defined in the category $O_{FS}$ generated by the highest weight irreducible representations with only a finite number of non-zero weight components. The eigenvalues of these Casimir operators are determined explicitly in terms of the highest weight. Characteristic identities satisfied by certain (infinite) matrices with entries from $gl(m/\infty)$ are also determined.Comment: 10 pages, Te

Highest weight irreducible representations of the Lie superalgebra gl(1/∞)gl(1/\infty)

Two classes of irreducible highest weight modules of the general linear Lie superalgebra $gl(1/\infty)$ are constructed. Within each module a basis is introduced and the transformation relations of the basis under the action of the algebra generators are written down.Comment: 24 pages, TeX; Journ. Math. Phys. (to be published

Parafermions, parabosons and representations of so(\infty) and osp(1|\infty)

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so(\infty) and of the Lie superalgebra osp(1|\infty). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labelled by certain infinite but stable Gelfand-Zetlin patterns, and the transformation of the basis is given explicitly. We also present expressions for the character of the Fock space representations

Mangiferin: A Promising Anticancer Bioactive

Of late, several biologically active antioxidants from natural products have been investigated by the researchers in order to combat the root cause of carcinogenesis, i.e., oxidative stress. Mangiferin, a therapeutically active C-glucosylated xanthone, is extracted from pulp, peel, seed, bark and leaf of Mangifera indica. These polyphenols of mangiferin exhibit antioxidant properties and tend to decrease the oxygen-free radicals, thereby reducing the DNA damage. Indeed, its capability to modulate several key inflammatory pathways undoubtedly helps in stalling the progression of carcinogenesis. The current review article emphasizes an updated account on the patents published on the chemopreventive action of Mangiferin, apoptosis induction made on various cancer cells, along with proposed antioxidative activities and patent mapping of other important therapeutic properties. Considering it as promising polyphenol, this paper would also summarize the diverse molecular targets of Mangiferin

Jacobson generators of the quantum superalgebra Uq[sl(n+1∣m)]U_q[sl(n+1|m)] and Fock representations

As an alternative to Chevalley generators, we introduce Jacobson generators for the quantum superalgebra $U_q[sl(n+1|m)]$. The expressions of all Cartan-Weyl elements of $U_q[sl(n+1|m)]$ in terms of these Jacobson generators become very simple. We determine and prove certain triple relations between the Jacobson generators, necessary for a complete set of supercommutation relations between the Cartan-Weyl elements. Fock representations are defined, and a substantial part of this paper is devoted to the computation of the action of Jacobson generators on basis vectors of these Fock spaces. It is also determined when these Fock representations are unitary. Finally, Dyson and Holstein-Primakoff realizations are given, not only for the Jacobson generators, but for all Cartan-Weyl elements of $U_q[sl(n+1|m)]$.Comment: 27 pages, LaTeX; to be published in J. Math. Phy