Commutative Subalgebras Research Articles

Zeros of Zeon Polynomials and the Zeon Quadratic Formula

Zeon algebras arise as commutative subalgebras of fermions, and can be constructed as subalgebras of Clifford algebras of appropriate signature. Their combinatorial properties have been applied to graph enumeration problems, stochastic integrals, and even routing problems in communication networks. Analogous to real polynomial functions, zeon polynomial functions are defined as zeon-valued polynomial functions of a zeon variable. In this paper, properties of zeon polynomials and their zeros are considered. Nilpotent and invertible zeon zeros of polynomials with real coefficients are characterized, and necessary conditions are established for the existence of zeros of polynomials with zeon coefficients. Quadratic polynomials with zeon coefficients are considered in detail. A “zeon quadratic formula” is developed, and solutions of $$ax^2+bx+c=0$$ are characterized with respect to the “zeon discriminant” of the equation.

The maximality of certain commutative subalgebras in Yangians

В работе доказано, что всякая подалгебра Бете янгиана, отвечающая регулярному полупростому элементу, является максимальной коммутативной и, более того, централизатором своей квадратичной части. Как следствие, получено описание таких подалгебр как следов $R$-матрицы по всем конечномерным представлениям янгиана.

KZ equations and Bethe subalgebras in generalized Yangians related to compatible $R$-matrices

Abstract The notion of compatible braidings was introduced in Isaev et al. (1999, J. Phys. A, 32, L115–L121). On the base of this notion, the authors of Isaev et al. (1999, J. Phys. A, 32, L115–L121) defined certain quantum matrix algebras generalizing the RTT algebras and Reflection Equation ones. They also defined analogues of some symmetric polynomials in these algebras and showed that these polynomials generate commutative subalgebras, called Bethe. By using a similar approach, we introduce certain new algebras called generalized Yangians and define analogues of some symmetric polynomials in these algebras. We claim that they commute with each other and thus generate a commutative Bethe subalgebra in each generalized Yangian. Besides, we define some analogues (also arising from couples of compatible braidings) of the Knizhnik–Zamolodchikov equation—classical and quantum. Communicated by: Alexander Veselov

Representation Theorems for Operators on Free Banach Spaces of Countable Type

This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on free Banach spaces of countable type. The main goal of this work will be to formulate a representation theorem for these operators through integrals defined by spectral measures type. In order to get this objective, we will show that, under special conditions, each one of these algebras is isometrically isomorphic to some space of continuous functions defined over a compact set. Then, we will identify such compact sets developing the Gelfand space theory in the non-Archimedean setting. This fact will allow us to define a measure which is known as spectral measure. As a second goal, we will formulate a matrix representation theorem for this class of operators in which the entries of the matrices will be integrals coming from scalar measures.

A Family of Maximal Algebras of Block Toeplitz Matrices

Abstract The maximal commutative subalgebras containing only Toeplitz matrices have been identified as generalized circulants. A similar simple description cannot be obtained for block Toeplitz matrices. We introduce and investigate certain families of maximal commutative algebras of block Toeplitz matrices.

Higher rank classical analogs of the Askey-Wilson algebra from the slN Onsager algebra

The $sl_N$-Onsager algebra has been introduced by Uglov and Ivanov in 1995. In this letter, a FRT presentation of the $sl_N$-Onsager algebra is given, its current algebra and commutative subalgebra are constructed. Certain quotients of the $sl_N$-Onsager algebra are then considered, which produce `classical' analogs of higher rank extensions of the Askey-Wilson algebra. As examples, the cases $N=3$ and $N=4$ are described in details.

A Proof of the Gaudin Bethe Ansatz Conjecture

Gaudin algebra is the commutative subalgebra in $U(\mathfrak{g})^{\otimes N}$ generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra $\mathfrak{g}$. This algebra depends on a collection of pairwise distinct complex numbers $z_1,\ldots,z_N$. We prove that this subalgebra has a cyclic vector in the space of singular vectors of the tensor product of any finite-dimensional irreducible $\mathfrak{g}$-modules, for all values of the parameters $z_1,\ldots,z_N$. We deduce from this result the Bethe Ansatz conjecture in the Feigin-Frenkel form which states that the joint eigenvalues of the higher Gaudin Hamiltonians on the tensor product of irreducible finite-dimensional $\mathfrak{g}$-modules are in 1-1 correspondence with monodromy-free ${}^LG$-opers on the projective line with regular singularities at the points $z_1,\ldots,z_N,\infty$ and the prescribed residues at the singular points.

Cocommutative elements form a maximal commutative subalgebra in quantum matrices

In this paper, we prove that the subalgebras of cocommutative elements in the quantized coordinate rings of [Formula: see text], [Formula: see text] and [Formula: see text] are the centralizers of the trace [Formula: see text] in each algebra, for [Formula: see text] being not a root of unity. In particular, it is not only a commutative subalgebra as it was known before, but it is a maximal one.

Nonlocal symmetries of Frobenius sinh-Gordon systems

In this paper, we consider a weakly coupled sinh-Gordon equation which takes values in a commutative Frobenius subalgebra of gl(2,mathbb {C}). Then we construct some nonlocal symmetries of the Frobenius sinh-Gordon system using its Bäcklund transformation and infinitesimal transformations. Based on the nonlocal symmetries, we show some conserved densities of the Frobenius sinh-Gordon system. Using these symmetries, we also construct some new coupled integro-differential systems.

Surprising examples of nonrational smooth spectral surfaces

In this paper we study necessary and sufficient algebro-geometric conditions for the existence of a nontrivial commutative subalgebra of rank in , a completion of the algebra of partial differential operators in two variables, which can be thought of as a simple algebraic analogue of the algebra of analytic pseudodifferential operators on a manifold. These are conditions on a projective (spectral) surface; they are encoded in a new notion of pre-spectral data. For smooth surfaces the sufficient conditions look especially simple. On a smooth projective surface there should exist 1) an ample integral curve with and ; 2) a divisor with , , , and . Amazingly, there are examples of such surfaces for which the corresponding commutative subalgebras do not admit isospectral deformations. Bibliography: 45 titles.

On divisors of small canonical degree on Godeaux surfaces

Pre-spectral data coding the rank-1 commutative subalgebras of a certain completion of the algebra of differential operators , where is an algebraically closed field of characteristic 0, are shown to exist. Here is a Godeaux surface, is an effective ample divisor represented by a smooth curve, and is a divisor on satisfying the conditions , for and . Bibliography: 26 titles.

The commutative core of a Leavitt path algebra

For any unital commutative ring R and for any graph E, we identify the commutative core of the Leavitt path algebra of E with coefficients in R, which is a maximal commutative subalgebra of the Leavitt path algebra. Furthermore, we are able to characterize injectivity of representations which gives a generalization of the Cuntz–Krieger uniqueness theorem.

Representations of quantum groups at roots of unity, Whittaker vectors and q-W algebras

Let Uε(g) be the standard simply connected version of the Drinfeld–Jumbo quantum group at an odd m-th root of unity ε. The center of Uε(g) contains a huge commutative subalgebra isomorphic to the algebra ZG of regular functions on (a finite covering of a big cell in) a complex connected, simply connected algebraic group G with Lie algebra g. Let V be a finite-dimensional representation of Uε(g) on which ZG acts according to a non-trivial character ηg given by evaluation of regular functions at g∈G. Then V is a representation of the finite-dimensional algebra Uηg=Uε(g)/Uε(g)Kerηg. We show that in this case, under certain restrictions on m, Uηg contains a subalgebra Uηg(m−) of dimension m12dimO, where O is the conjugacy class of g, and Uηg(m−) has a one-dimensional representation Cχg. We also prove that if V is not trivial then the space of Whittaker vectors HomUηg(m−)(Cχg,V) is not trivial and the algebra Wηg=EndUηg(Uηg⊗Uηg(m−)Cχg) naturally acts on it which gives rise to a Schur-type duality between representations of the algebra Uηg and of the algebra Wηg called a q-W algebra.

Subalgebras of Solomon's descent algebra based on alternating runs

The number of alternating runs is a natural permutation statistic. We show it can be used to define some commutative subalgebras of the symmetric group algebra, and more precisely of the descent algebra. The Eulerian peak algebras naturally appear as subalgebras of our run algebras. We also calculate the orthogonal idempotents for run algebras in terms of noncommutative symmetric functions.

Completeness of Some Commutative Subalgebras Associated with Nijenhuis Operators on Lie Algebras

We prove the completeness of commutative subalgebras in S(gl(n)) constructed from the Sokolov–Odesskii Lie pencil with generic parameters.

Quantum Bounds for Option Prices

Option pricing is the most elemental challenge of mathematical finance. Knowledge of the prices of options at every strike is equivalent to knowing the entire pricing distribution for a security, as derivatives contingent on the security can be replicated using options. The available data may be insufficient to determine this distribution precisely, however, and the question arises: What are the bounds for the option price at a specified strike, given the market-implied constraints? Positivity of the price map imposed by the principle of no-arbitrage is here utilised, via the Gelfand-Naimark-Segal construction, to transform the problem into the domain of operator algebras. Optimisation in this larger context is essentially geometric, and the outcome is simultaneously super-optimal for all commutative subalgebras. This generates an upper bound for the price of a basket option. With innovative decomposition of the assets in the basket, the result is used to create converging families of price bounds for vanilla options, interpolate the volatility smile, price options on cross FX rates, and analyse the relationships between swaption and caplet prices.

The structure of the Kauffman bracket skein algebra at roots of unity

This paper examines the structure of the Kauffman bracket skein algebra of a punctured surface at roots of unity. A criterion that determines when a collection of skeins forms a basis of the skein algebra as a module over the $$SL(2,{{\mathbb {C}}})$$ characters of the fundamental group of the surface, with appropriate localization is given. This is used to prove that when the algebra is localized so that every nonzero element of the center has a multiplicative inverse, it is a division algebra. Finally, it is proved that the localized skein algebra can be split as a module over its center as a tensor product of two commutative subalgebras.

Bergman’s centralizer theorem and quantization

ABSTRACTWe prove Bergman’s theorem [10] on centralizers by using generic matrices and Kontsevich’s quantization method. For any field k of positive characteristics, set be a free associative algebra, then any centralizer 𝒞(f) of nontrivial element f∈A∖k is a ring of polynomials on a single variable. We also prove that there is no commutative subalgebra with transcendent degree ≥2 of A.

An integrable hierarchy including the AKNS hierarchy and its strict version

We present an integrable hierarchy that includes both the AKNS hierarchy and its strict version. We split the loop space g of gl2 into Lie subalgebras g≥0 and g<0 of all loops with respectively only positive and only strictly negative powers of the loop parameter. We choose a commutative Lie subalgebra C in the whole loop space s of sl2 and represent it as C = C≥0⊕C<0. We deform the Lie subalgebras C≥0 and C<0 by the respective groups corresponding to g<0 and g≥0. Further, we require that the evolution equations of the deformed generators of C≥0 and C<0 have a Lax form determined by the original splitting. We prove that this system of Lax equations is compatible and that the equations are equivalent to a set of zero-curvature relations for the projections of certain products of generators. We also define suitable loop modules and a set of equations in these modules, called the linearization of the system, from which the Lax equations of the hierarchy can be obtained. We give a useful characterization of special elements occurring in the linearization, the so-called wave matrices. We propose a way to construct a rather wide class of solutions of the combined AKNS hierarchy.

Commutative Nilpotent Subalgebras with Nilpotency Index n-1 in the Algebra of Matrices of Order n

The paper establishes the existence of an element with nilpotency index n − 1 in an arbitrary nilpotent commutative subalgebra with nilpotency index n−1 in the algebra of upper niltriangular matrices N n (𝔽) over a field 𝔽 with at least n elements for all n ≥ 5, and also, as a corollary, in the full matrix algebra M n (𝔽). The result implies an improvement with respect to the base field of known classification theorems due to D. A. Suprunenko, R. I. Tyshkevich, and I. A. Pavlov for algebras of the class considered.