Capelli Identities Research Articles

Applications of hook Young diagrams to P.I. Algebras

(a) The multiplicities m λ in the cocharacter χ n ( A) of (any P.I. algebra A) are polynomially bounded, (b) A hook containing χ n ( A ⊗ B) is obtained from the hooks containing χ n ( A) and χ n ( B). These results are obtained by applying a theory of hook Young diagrams to P.I. algebras, and they generalize results known for algebras satisfying Capelli identities.

PI-algebras and their cocharacters

All PI-algebras R satisfy identities of the form: f ∗[x, y] = Σ α σx σ(1)y 1x σ(2)y 2 … y n − 1x σ(n) (Theorem A.) The existence of these identities imply also that cocharacters x n ( R) lie in a hook-shaped strip of width depending on the degree of the minimal identity of R (Theorem C). This extends a characterization of rings satisfying a Capelli identity (Theorem B).

ALGEBRAS SATISFYING CAPELLI IDENTITIES

In this paper the author considers the representation of an algebra L of a certain signature in an algebra A (generally of a different signature) satisfying identity relations of Capelli type. A criterion for the Capelli identities to hold in the pair (A,L) is indicated, and a structural description of such pairs is given. The results are applied for the case where L is a Lie algebra and A is its associative enveloping algebra. In addition, from these results it is deduced that an algebraicity identity over a field of characteristic zero implies nilpotence of the Jacobson radical of a finitely generated associative algebra. Bibliography: 10 titles.

On an identity from classical invariant theroy

A simple derivation is given of a well-known relation involving the so-called "Cayley Operator" of classical invariant theory. The proof is induction-free and independent of Capelli's identity; it makes use only of a known-theorem in the theory of determinants and some elementary combinatorics.

Symmetry operators, polarizations, and a generalized Capelli identity

The classical Capelli identity [1,5, p. 88, 7, p. 39], a fundamental tool in the theory of invariants, shows that, in the case of the symmetric group, the alternating symmetry operator acting on multilinear forms is the determinant of polarizations in the variables of the forms [5, p. 88]. Rota [5, p. 88] has posed the question as to whether or not other symmetry operators of the symmetric group are expressible explicitly in a similar manner in terms of polarizations of variables of the forms. In this paper we answer this question in the affirmative by showing that the symmetry operator Tm associated with any representation M of the symmetric group is a Schur matrix function [6] of polarizations in the variables of the form. In the particular case where M is of degree one we have either M = sgn, the sign or alternating character, or M = id, the trivial character. In the former case we obtain the classical result expressing the alternating symmetry operator as a determinant of polarizations. In the latte...

The Kronecker product of S n-characters and an A ⊗ B theorem for Capelli identities

If A and B satisfy a Capelli identity then so does A ⊗ B. To prove this we show that the height of [λ] ⊗ [μ] is bounded by the product of the heights of [λ] and [μ].

Algebras satisfying a Capelli identity

The sequence of cocharacters (c.c.s.) of a P.I. algebra is studied. We prove that an algebra satisfies a Capelli identity if, and only if, all the Young diagrams associated with its cocharacters are of a bounded height. This result is then applied to study the identities of certain P.I. algebras, includingF k .

Remark on the standard identity

It is proved that the standard identity of degree n implies all Capelli identities of some order m depending on n.