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We establish a duality of the non-periodic Gaudin model associated with superalgebra gl m | n and the non-periodic Gaudin model associated with algebra gl k. The Hamiltonians of the Gaudin models are given by expansions of a Berezinian of an (m + n) × (m + n) matrix in the case of gl m | n and of a column determinant of a k × k matrix in the case of gl k. We obtain our results by proving Capelli type identities for both cases and comparing the results. [ABSTRACT FROM AUTHOR]

Inspired by the Capelli-type identities for group determinants researched by Tôru Umeda, we give Capelli identities for irreducible representations of any finite group, and Capelli elements of the group algebra associated with these identities. These elements construct a basis of the centre of the group algebra. [ABSTRACT FROM AUTHOR]

A quantum Capelli identity is given on the multiparameter quantum general linear group based on the ( p , u)-condition. The multiparameter quantum Pfaffan of the ( p , u)-quantum group is also introduced and the transformation under the congruent action is given. Generalization to the multiparameter hyper-Pfaffan and relationship with the quantum minors are also investigated. [ABSTRACT FROM AUTHOR]

Let ( G , V ) be an irreducible multiplicity-free finite-dimensional representation of a connected reductive complex group G , as classified by V.G. Kac [17] , and G ′ its derived subgroup. Denote by g the Lie algebra of G , and U ( g ) its universal enveloping algebra. Assume that there exists a polynomial f generating the algebra of G ′ -invariant polynomials on V ( C [ V ] G ′ ≃ C [ f ] ) and such that f ∉ C [ V ] G . Such representations are said to be of Capelli type if the algebra of G -invariant differential operators is the image of the center of U ( g ) under the differential of the G -action. They fall into eight cases given by R. Howe and T. Umeda [14] : five infinite families and three “exceptional” examples. We prove that the category of regular holonomic D V -modules invariant under the action of G ′ is equivalent to the category of graded modules of finite type over a suitable algebra A , except for few special cases. Indeed the Levasseur's conjecture [28, Conjecture 5.17, p. 508] fails in these cases because of the disconnectedness of the stabilizers of some “smaller” orbits. [ABSTRACT FROM AUTHOR]

The column space of a real n × k matrix x of rank k is a k-plane. Thus we get a map from the space X of such matrices to the Grassmannian G of k-planes in Rn, and hence a GLn-equivariant isomorphism C8(G)"C8(X)GLk We consider the On × GLk-invariant differential operator C On X given by C = det(xtx)det(?t?), where x = (xij), ? = (?/?xij). By the above isomorphism, C defines an On-invariant operator on G. Since G is a symmetric space for On, the irreducible On-submodules of C8 (G) have multiplicity 1; thus, On-invariant operators act by scalars On these submodules. Our main result determines these scalars for a general class of such operators including C. This answers a question raised by Howe and Lee and also gives new Capelli-type identities for the orthogonal Lie algebra. [ABSTRACT FROM AUTHOR]

Abstract: The classic Cayley identity states that where is an matrix of indeterminates and is the corresponding matrix of partial derivatives. In this paper we present straightforward algebraic/combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (=exterior algebra) and Grassmann–Berezin integration. Among the new identities proven here are a pair of “diagonal-parametrized” Cayley identities, a pair of “Laplacian-parametrized” Cayley identities, and the “product-parametrized” and “border-parametrized” rectangular Cayley identities. [Copyright &y& Elsevier]

This article presents a natural extension of the tensor algebra. In addition to "left multiplications" by vectors, we can consider "derivations" hy co vector s as husic operators on this extended algebra. These two types of operators satisfy an analogue of the canonical commutation relations. This algebra and these operators have the following applications: (i) applications to invariant theory related to tensor products und (ii) applications to immanunts. The latter includes a new method to study the quantum immanants in the universal enveloping algebras of the general linear Lie algebras and their Capclli type Identities ( the higher Capelli identities). [ABSTRACT FROM AUTHOR]

Abstract: We give a classification of regular holonomic -modules on -skew-symmetric matrices related to the action of the general linear group . Actually we establish one more case of the conjecture by T. Levasseur (Levasseur, 2009 ). [Copyright &y& Elsevier]

Abstract: We study a class of matrices with noncommutative entries, which were first considered by Yu.I. Manin in 1988 in relation with quantum group theory. They are defined as “noncommutative endomorphisms” of a polynomial algebra. More explicitly their defining conditions read: (1) elements in the same column commute; (2) commutators of the cross terms are equal: (e.g. ). The basic claim is that despite noncommutativity many theorems of linear algebra hold true for Manin matrices in a form identical to that of the commutative case. Moreover in some examples the converse is also true, that is, Manin matrices are the most general class of matrices such that linear algebra holds true for them. The present paper gives a complete list and detailed proofs of algebraic properties of Manin matrices known up to the moment; many of them are new. In particular we provide complete proofs that an inverse to a Manin matrix is again a Manin matrix and for the Schur formula for the determinant of a block matrix; we generalize the noncommutative Cauchy–Binet formulas discovered recently arXiv:0809.3516, which includes the classical Capelli and related identities. We also discuss many other properties, such as the Cramer formula for the inverse matrix, the Cayley–Hamilton theorem, Newton and MacMahon–Wronski identities, Plücker relations, Sylvester''s theorem, the Lagrange–Desnanot–Lewis Carroll formula, the Weinstein–Aronszajn formula, some multiplicativity properties for the determinant, relations with quasideterminants, calculation of the determinant via Gauss decomposition, conjugation to the second normal (Frobenius) form, and so on and so forth. Finally several examples and open question are discussed. We refer to [A. Chervov, G. Falqui, Manin matrices and Talalaev''s formula, J. Phys. A 41 (2008) 194006; V. Rubtsov, A. Silantiev, D. Talalaev, Manin matrices, elliptic commuting families and characteristic polynomial of quantum elliptic Gaudin model, in press] for some applications in the realm of quantum integrable systems. [Copyright &y& Elsevier]

Abstract: Let be a prime ring with extended centroid and let be a reduced differential polynomial with coefficients in , the symmetric Martindale quotient ring of , and with zero constant term. Let and . We prove that the finiteness of and the finite-dimensionality of the -span of are equivalent to that of and that of the -span of , respectively. Hence some questions on differential polynomials are reduced to those on ordinary generalized polynomials. Let and be two derivations of a Lie ideal of and a right ideal of . As applications of our theorems, we obtain the necessary and sufficiency conditions for the finiteness of and and for the finite-dimensionality of the -spans of and . [Copyright &y& Elsevier]

This paper presents new generators for the center of the universal enveloping algebra of the symplectic Lie algebra. These generators are expressed in terms of the column-permanent and it is easy to calculate their eigenvalues on irreducible representations. We can regard these generators as the counterpart of central elements of the universal enveloping algebra of the orthogonal Lie algebra given in terms of the column-determinant by Wachi. The earliest prototype of all these central elements is the Capelli determinants in the universal enveloping algebra of the general linear Lie algebra. [ABSTRACT FROM AUTHOR]

Abstract: We give an algebraic description of regular holonomic -modules whose characteristic variety is the union of conormal bundles to the orbits of the action of the general linear group on square matrices. [Copyright &y& Elsevier]

Abstract: This paper gives a direct proof for the coincidence of the following two central elements in the universal enveloping algebra of the orthogonal Lie algebra: an element recently given by A. Wachi in terms of the column-determinant in a way similar to the Capelli determinant, and an element given by T. Umeda and the author in terms of the symmetrized determinant. The fact that these two elements actually coincide was shown by A. Wachi, but his observation was based on the following two non-trivial results: (i) the centrality of the first element, and (ii) the calculation of the eigenvalue of the second element. The purpose of this paper is to prove this coincidence of two central elements directly without using these (i) and (ii). Conversely this approach provides us new proofs of (i) and (ii). A similar discussion can be applied to the symplectic Lie algebras. [Copyright &y& Elsevier]

We describe a connection between discrete birth process and a certain family of multivariate interpolation polynomials. This enables us to compute all asymptotic moments of the birth process, generalizing previously known results for the mean and variance. [ABSTRACT FROM AUTHOR]

Abstract: Suppose that R is a semiprime ring which is either a PI-ring or a countable ring. Then there exists a collection of prime ideals , , such that each is invariant under all derivations of R and . [Copyright &y& Elsevier]

Abstract: Analogues of the Capelli identity are given for every irreducible reductive dual pair in the complex symplectic group and the complex orthogonal group. They describe a natural correspondence between the “centers” of the two universal enveloping algebras and an algebra of invariant operators. [Copyright &y& Elsevier]

We define and study a family of completely prime rank ideals in the universal enveloping algebra U(gln). A rank ideal is a noncommutative analogue of a determinantal ideal, the defining ideal for the closure of the set of n×n matrices of fixed rank. We introduce a notion of rank for gln-modules and determine the rank of simple highest weight modules and of simple finite-dimensional modules. The main tools are Capelli-type identities and filtered algebra. [Copyright &y& Elsevier]

We prove that each (possibly not finitely generated) associative PI-algebra does not coincide with its commutant. We thus solve I. V. L'vov's problem in the Dniester Notebook. The result follows from the fact (also established in this article) that, in every T-prime variety, some weak identity holds and there exists a central polynomial (the existence of a central polynomial was earlier proved by A. R. Kemer). Moreover, we prove stability of T-prime varieties (in the case of characteristic zero, this was done by S. V. Okhitin who used A. R. Kemer's classification of T-prime varieties). [ABSTRACT FROM AUTHOR]