Cayley-Hamilton Identity Research Articles

Invariant Theory and wheeled PROPs

We study the category of wheeled PROPs using tools from Invariant Theory. A typical example of a wheeled PROP is the mixed tensor algebra V=T(V)⊗T(V⋆), where T(V) is the tensor algebra on an n-dimensional vector space over a field K of characteristic 0. First we classify all the ideals of the initial object Z in the category of wheeled PROPs. We show that non-degenerate sub-wheeled PROPs of V are exactly subalgebras of the form VG where G is a closed, reductive subgroup of the general linear group GL(V). When V is a finite dimensional Hilbert space, a similar description of invariant tensors for an action of a compact group was given by Schrijver. We also generalize the theorem of Procesi that says that trace rings satisfying the n-th Cayley-Hamilton identity can be embedded in an n×n matrix ring over a commutative algebra R. Namely, we prove that a wheeled PROP can be embedded in R⊗V for a commutative K-algebra R if and only if it satisfies certain relations.

Positive maps and trace polynomials from the symmetric group

With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials Xα1,…,Xαr and their traces tr(Xα1,…,Xαr). Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley–Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct permutation polynomials and tensor polynomial identities on tensor product spaces. We give connections to concepts in quantum information theory and invariant theory.

A power Cayley-Hamilton identity for n × n matrices over a Lie nilpotent ring of index k

For an n×n matrix A over a Lie nilpotent ring R of index k, with k≥2, we prove that an invariant “power” Cayley-Hamilton identity(Inλ0(2)+Aλ1(2)+⋯+An2−1λn2−1(2)+An2λn2(2))2k−2=0of degree n22k−2 holds. The right coefficients λi(2)∈R, 0≤i≤n2 are not uniquely determined by A, and the cosets λi(2)+D, with D the double commutator ideal R[[R,R],R]R of R, appear in the so-called second right characteristic polynomial pA‾,2(x) of the natural image A‾ of A in the n×n matrix ring Mn(R/D) over the factor ring R/D:pA‾,2(x)=(λ0(2)+D)+(λ1(2)+D)x+⋯+(λn2−1(2)+D)xn2−1+(λn2(2)+D)xn2.

Drinfeld–Sokolov reduction in quantum algebras: canonical form of generating matrices

We define the second canonical forms for the generating matrices of the Reflection Equation algebras and the braided Yangians, associated with all even skew-invertible involutive and Hecke symmetries. By using the Cayley–Hamilton identities for these matrices, we show that they are similar to their canonical forms in the sense of Chervov and Talalaev (J Math Sci (NY) 158:904–911, 2008).

Functional Identities on Matrix Algebras

Complete solutions of functional identities \({\sum }_{k\in K}F_{k}\left (\overline {x}_{m}^{k}\right )x_{k} = {\sum }_{l\in L}x_{l}G_{l}\left (\overline {x}_{m}^{l}\right )\) on the matrix algebra Mn (\(\mathbb {F}\)) are given. The nonstandard parts of these solutions turn out to follow from the Cayley-Hamilton identity.

Quasi-identities on matrices and the Cayley–Hamilton polynomial

We consider certain functional identities on the matrix algebra Mn that are defined similarly as the trace identities, except that the “coefficients” are arbitrary polynomials, not necessarily those expressible by the traces. The main issue is the question of whether such an identity is a consequence of the Cayley–Hamilton identity. We show that the answer is affirmative in several special cases, and, moreover, for every such an identity P and every central polynomial c with zero constant term there exists m∈N such that the affirmative answer holds for cmP. In general, however, the answer is negative. We prove that there exist antisymmetric identities that do not follow from the Cayley–Hamilton identity, and give a complete description of a certain family of such identities.

Generic super-orbits in [formula omitted] and their braided counterparts

We introduce some braided varieties–braided orbits–by considering quotients of the so-called Reflection Equation Algebras associated with Hecke symmetries (i.e. special type solutions of the quantum Yang–Baxter equation). Such a braided variety is called regular if there exists a projective module on it, which is a counterpart of the cotangent bundle on a generic orbit O ∈ g l ( m ) ∗ in the framework of the Serre approach. We give a criterium of regularity of a braided orbit in terms of roots of the Cayley–Hamilton identity valid for the generating matrix of the Reflection Equation Algebra in question. By specializing our general construction we get super-orbits in g l ( m | n ) ∗ and a criterium of their regularity.

KZ equation, G-opers, quantum Drinfeld–Sokolov reduction, and quantum Cayley–Hamilton identity

The Lax operator of Gaudin-type models is a 1-form at the classical level. In virtue of the quantization scheme proposed by D. Talalaev, it is natural to treat the quantum Lax operator as a connection; this connection is a partcular case of the Knizhnik–Zamolodchikov connection. In this paper, we find a gauge trasformation that produces the “second normal form,” or the “Drinfeld–Sokolov” form. Moreover, the differential operator nurally corresponding to this form is given precisely by the quantum characteristic polynomial of the Lax operator (this operator is called the G-oper or Baxter operator). This observation allows us to relate solutions of the KZ and Baxter equations in an obvious way, and to prove that the immanent KZ equation has only meromorphic solutions. As a corollary, we obtain the quantum Cayley–Hamilton identity for Gaudin-type Lax operators (including the general $ \mathfrak{gl}_{n}[t] $ case). The presented construction sheds a new light on the geometric Langlands correspondence. We also discuss the relation with the Harish-Chandra homomorphism. Bibliography: 19 titles.

Wave Operators on Quantum Algebras via Noncanonical Quantization

We suggest a method to quantize basic wave operators of Relativistic Quantum Mechanics (Laplace, Maxwell, Dirac ones) without using canonical coordinates. We define two-parameter deformations of the Minkowski space algebra and its 3-dimensional reduction via the so-called Reflection Equation Algebra and its modified version. Wave operators on these algebras are introduced by means of quantized partial derivatives described in two ways. In particular, they are given in so-called pseudospherical form which makes use of a q-deformation of the Lie algebra sl(2) and quantum versions of the Cayley-Hamilton identity.

Reflection equation algebra in braided geometry 1

Braided geometry is a sort of the noncommutative geometry related to a braiding. The central role in this geometry is played by the reflection equation algebra associated with a braiding of the Hecke type. Using this algebra, we introduce braided versions of the Lie algebras gl(n) and sl(n). We further define braided analogs of the coadjoint orbits and the vector fields on a q-hyperboloid which is the simplest example of a ”braided orbit”. Besides, we present a braided version of the Cayley-Hamilton identity generalizing the result of Kantor and Trishin on the super-matrix characteristic identities.

Quantum matrix algebras of the GL(m|n) type: The structure and spectral parameterization of the characteristic subalgebra

We continue the study of quantum matrix algebras of the GL(m|n) type. We find three alternative forms of the Cayley-Hamilton identity; most importantly, this identity can be represented in a factored form. The factorization allows naturally dividing the spectrum of a quantum supermatrix into subsets of “even” and “odd” eigenvalues. This division leads to a parameterization of the characteristic subalgebra (the subalgebra of spectral invariants) in terms of supersymmetric polynomials in the eigenvalues of the quantum matrix. Our construction is based on two auxiliary results, which are independently interesting. First, we derive the multiplication rule for Schur functions s λ (M) that form a linear basis of the characteristic subalgebra of a Hecke-type quantum matrix algebra; the structure constants in this basis coincide with the Littlewood-Richardson coefficients. Second, we prove a number of bilinear relations in the graded ring Λ of symmetric functions of countably many variables.

Berezinians, Exterior Powers and Recurrent Sequences

We study power expansions of the characteristic function of a linear operator A in a p|q-dimensional superspace V. We show that traces of exterior powers of A satisfy universal recurrence relations of period q. ‘Underlying’ recurrence relations hold in the Grothendieck ring of representations of GL(V). They are expressed by vanishing of certain Hankel determinants of order q+1 in this ring, which generalizes the vanishing of sufficiently high exterior powers of an ordinary vector space. In particular, this allows to express the Berezinian of an operator as a ratio of two polynomial invariants. We analyze the Cayley–Hamilton identity in a superspace. Using the geometric meaning of the Berezinian we also give a simple formulation of the analog of Cramer’s rule

Quantum line bundles on a noncommutative sphere

A noncommutative (NC) sphere is introduced as a quotient of the enveloping algebra of the Lie algebra su(2). Following Gurevich and Saponov (2001 J. Phys. A: Math. Gen.34 4533–69) and using the Cayley–Hamilton identities we introduce projective modules which are analogues of line bundles on the usual sphere (we call them quantum line bundles) and define a multiplicative structure in their family. Also, we compute a pairing between quantum line bundles and finite-dimensional representations of the NC sphere in the spirit of the NC index theorem. Moreover, we introduce projective modules being NC counterparts of tangent vector bundles and those of differential forms and define an analogue of the de Rham complex making use of these modules.

Quantum line bundles via Cayley-Hamilton identity

As shown by Pyatov and Saponov (1995 J. Phys. A: Math. Gen. 28 4415-21) and Gurevich et al (1997 Lett. Math. Phys. 41 255-64), the matrix L = || lij||, whose entries lij are generators of the so-called reflection equation algebra (REA), is subject to some polynomial identity resembling the Cayley-Hamilton identity for a numerical matrix. Here a similar statement is presented for a matrix whose entries are generators of a filtered algebra that is a `non-commutative analogue' of the REA. In an appropriate limit we obtain a similar statement for the matrix formed by the generators of the algebra U(gl(n)). This property is used to introduce the notion of line bundles over quantum orbits in the spirit of the Serre-Swan approach. The quantum orbits in question are presented explicitly as some quotients of one of the algebras mentioned above both in the quasiclassical case (i.e. that related to the quantum group Uq(sl(n))) and a non-quasiclassical one (i.e. that arising from a Hecke symmetry with non-standard Poincare series of the corresponding symmetric and skew-symmetric algebras).

A direct approach to second-order matrix non-classical vibrating equations

A direct framework is developed for second-order matrix equations without transforming them into a first-order companion equation. It is done in terms of its matrix impulse response that is directly related to the transfer matrix. We formulate an extension of the Cayley–Hamilton identity, derive the controllability and observability matrices, and discuss Krylov's method in terms of such matrix response. This formulation will allow to further discuss Arnoldi and Lanczos methods as well as time-integration by FFT.

The equivalence between the Mandelstam and the Cayley–Hamilton identities

Starting from the Cayley–Hamilton identities for n×n ordinary matrices which result from the application of the corresponding Cayley–Hamilton theorem, the generic Mandelstam identities are derived. The converse of the above statement is also proven. The method is extended to supermatrices and a definition of the generic Mandelstam identities in that case is proposed.

Computable identities in the algebra of formal matrices

In this paper, we want to give an explicit description of identities satisfied by matrices n × n over a field k of characteristic 0, in order to be able to compute with formal matrices (“forgetting” their representations with coefficients). We introduce a universal free algebra, where all formal manipulations are made. Using classical properties of identities in an algebra with trace, we reduce our problem to the study of identities among multilinear traces. These are closely linked with the action of the algebra of the symmetric group k[ S m ] on the mth tensor product of E = K n . By proving a theorem about the kernel of this action and its effective version, we can decompose all identities of matrices in an explicit way as linear combinations, substitutions, products or traces of the well-known Cayley-Hamilton identity. This leads to an algorithm of reduction to a canonical form, modulo the ideal of identities of matrices, in the free algebra. Moreover, this set of identities is in fact a prime ideal and a natural extension of the quotient by this ideal gives us a skew field, which is also the fraction ring of the algebra of generic matrices.

Semisimple Representations of Quivers

We discuss the invariant theory of the variety of representations of a quiver and present generators and relations. We connect this theory of algebras with a trace satisfying a formal CayleyHamilton identity

Semisimple representations of quivers

We discuss the invariant theory of the variety of representations of a quiver and present generators and relations. We connect this theory of algebras with a trace satisfying a formal Cayley-Hamilton identity

A formal inverse to the Cayley-Hamilton theorem

We prove that a Q-algebra R with formal trace can be realized as n × n matrices if and only if it satisfies the Cayley—Hamilton identity of degree n.