Casimir Invariants and Characteristic Identities for Gl(∞)
نویسندگان
چکیده
A full set of (higher order) Casimir invariants for the Lie algebra gl(∞) is constructed and shown to be well defined in the category O F S 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(∞) are also determined and generalize those previously obtained for gl(n) by Bracken and Green.
منابع مشابه
On characteristic equations, trace identities and Casimir operators of simple Lie algebras
Two approaches are developed to exploit, for simple complex or compact real Lie algebras g, the information that stems from the characteristic equations of representation matrices and Casimir operators. These approaches are selected so as to be viable not only for ‘small’ Lie algebras and suitable for treatment by computer algebra. A very large body of new results emerges in the forms, a) of id...
متن کاملThe Procesi–Razmyslov theorem for O(n)-invariants in prime characteristic
A linear group G < GL(n) acts on d-tuples of n×n matrices by simultaneous conjugation. In [Adv. Math. 19 (1976), 306–381] Procesi established generators and relations between them for G-invariants, where G is GL(n), O(n), and Sp(n) and the characteristic of base field is zero. We continue generalization of the mentioned results to the case of positive characteristic originated by Donkin in [Inv...
متن کاملRelabeling Symmetries in Hydrodynamics and Magnetohydrodynamics
Lagrangian symmetries and concomitant generalized Bianchi identities associated with the relabeling of fluid elements are found for hydrodynamics and magnetohydrodynamics (MHD). In hydrodynamics, relabeling results in Ertel's theorem of conservation of potential vorticity, while in MHD it yields the conservation of cross helicity. The symmetries of the reduction from Lagrangian (material) to Eu...
متن کاملEigenvalues of Casimir Invariants for U Q [osp(m|n)]
For each quantum superalgebra U q [osp(m|n)] with m > 2, an infinite family of Casimir invariants is constructed. This is achieved by using an explicit form for the Lax operator. The eigenvalue of each Casimir invariant on an arbitrary irreducible highest weight module is also calculated.
متن کاملBraided central elements
We present and study two families of polynomials with coefficients in the center of the universal enveloping algebra. These polynomials are analogues of a determinant and a characteristic polynomial of a certain non-commutative matrix, labeled by irreducible representations of gl n (C). The matrix is an image of the universal R-matrix of a Yangian of gl n (C) under certain representation. We co...
متن کامل