The structure of integrable Lax flows on nonlocal manifolds: Dynamic systems with sources

We consider a dynamic system on the extended phase space to the initial Lie algebra and study its generalized Hamiltonian and integrability in the cases when the initial Lie algebra coincides with the Grassmann algebra of pseudodifferential operators on the real line and on the centrally extended affine Lie algebra.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 106–115.     

Construction of a new class of quantum integrable inhomogeneous models

Integrable inhomogenous or impurity models are usually constructed by either shifting the spectral parameter in the Lax operator or using another representation of the spin algebra. We propose a more involved general method for such construction in which the Lax operator contains generators of a novel quadratic algebra, a generalization of the known quantum algebra. In forming the monodromy matrix, we can replace any number of the local Lax operators with different realizations of the underlying algebra, which can result in spin chains with nonspin impurities causing changed coupling across the impurity sites, as well as with impurities in the form of bosonic operators. Following the same idea, we can also generate integrable inhomogeneous versions of the generalized lattice sine-Gordon model, nonlinear Schrödinger equation, Liouville model, relativistic and nonrelativistic Toda chains, etc.     

Über endliche galoissche schiefkörpererweiterungen ohne endliche galoisgruppen

We define a generalized Brauer-Severi variety to be the projective Variety whose closed points correspond to rank m right ideals in a central simple algebra. We show that these varieties are forms of the Grassmann variety and their function fields are generic partial splitting fields for the associated algebras. The subgroup of the Brauer group split by the function field is calculated. Following Schofield and van den Bergh, we calculate the index of a simple ailgebra extended by the function field of a generalized Brauer-Severi variety associated with any other central simple algebra.     

Bispinor Space

In this paper, we give identifications of bispinor space with Grassmann algebra, and with Clifford algebra. The multiplication in Clifford algebra provides an action on them. Lastly we have researched on the geometry of bispinor space, and define Dirac operators to get a Pythagoras equality.     

A note on the hodge star operator

The usual proof that the Hodge star operator on the Grassmann algebra is independent of the orthonormal basis used to define it requires a decomposition of the operator into a product of three maps, each of which is independent of the basis. The present note contains a very short proof of the result which depends on simple properties of the multiplication and induced inner product in the Grassmann algebra.     

Triangularization of a Jordan algebra of Schatten operators

We show that a Jordan algebra of compact quasinilpotent operators which contains a nonzero trace class operator has a common invariant subspace. As a consequence of this result, we obtain that a Jordan algebra of quasinilpotent Schatten operators is simultaneously triangularizable.

     

Banach algebra generated by a finite number of bergman polykernel operators,continuous coefficients,and a finite group of shifts

We study the Banach algebra generated by a finite number of Bergman polykernel operators with continuous coefficients that is extended by operators of weighted shift that form a finite group. By using an isometric transformation, we represent the operators of the algebra in the form of a matrix operator formed by a finite number of mutually complementary projectors whose coefficients are Toeplitz matrix functions of finite order. Using properties of Bergman polykernel operators, we obtain an efficient criterion for the operators of the algebra considered to be Fredholm operators.     

A generalized vertex operator algebra for Heisenberg intertwiners

We consider the extension of the Heisenberg vertex operator algebra by all its irreducible modules. We give an elementary construction for the intertwining vertex operators and show that they satisfy a complex parameterized generalized vertex operator algebra. We illustrate some of our results with the example of integral lattice vertex operator superalgebras.     

二阶算子矩阵代数中的全可导点V

研究二阶算子矩阵代数中的全可导点.利用线性映射与算子矩阵代数运算,以及套代数理论的相关结果,给出并证明了第二行第二列元素为可逆算子,其余元素为零算子的二阶矩阵是二阶算子矩阵代数的关于强算子拓扑的全可导点,推广了相关文献中的结果.     

Spherical functions and invariant differential operators on complex Grassmann manifolds

Proofs are given of two theorems of Berezin and Karpelevi?, which as far as we know never have been proved correctly. By using eigenfunctions of the Laplace-Beltrami operator it is shown that the spherical functions on a complex Grassmann manifold are given by a determinant of certain hypergeometric functions. By application of this result, it is proved that a certain system of operators, fow which explicit expressions are given, generates the algebra of radial parts of invariant differential operators.     

Point Particle with Extrinsic Curvature as a Boundary of a Nambu–Goto String: Classical and Quantum Model

It is shown how a string living in a higher dimensional space can be approximated as a point particle with squared extrinsic curvature. We consider a generalized Howe–Tucker action for such a “rigid particle” and consider its classical equations of motion and constraints. We find that the algebra of the Dirac brackets between the dynamical variables associated with velocity and acceleration contains the spin tensor. After quantization, the corresponding operators can be represented by the Dirac matrices, projected onto the hypersurface that is orthogonal to the direction of momentum. A condition for the consistency of such a representation is that the states must satisfy the Dirac equation with a suitable effective mass. The Pauli–Lubanski vector composed with such projected Dirac matrices is equal to the Pauli–Lubanski vector composed with the usual, non projected, Dirac matrices, and its eigenvalues thus correspond to spin one half states.     

Spherical functions and invariant differential operators on complex Grassmann manifolds

Proofs are given of two theorems of Berezin and Karpelevič, which as far as we know never have been proved correctly. By using eigenfunctions of the Laplace-Beltrami operator it is shown that the spherical functions on a complex Grassmann manifold are given by a determinant of certain hypergeometric functions. By application of this result, it is proved that a certain system of operators, fow which explicit expressions are given, generates the algebra of radial parts of invariant differential operators.     

Time-Dependent Correlators of Local Spins of the One-Dimensional XY Heisenberg Chain

Time-dependent temperature correlators of the anisotropic Heisenberg XY chain are calculated by the integration techniques with respect to Grassmann variables. For a chain of length M, the correlators are represented as the determinants of 2M × 2M matrices. In the thermodynamic limit, the correlation functions are expressed in terms of the Fredholm determinants of linear integral operators with matrix kernels. Bibliography: 32 titles.     

Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras

To each multiquiver Γ we attach a solution to the consistency equations associated to twisted generalized Weyl (TGW) algebras. This generalizes several previously obtained solutions in the literature. We show that the corresponding algebras \(\mathcal {A}({\Gamma })\) carry a canonical representation by differential operators and that \(\mathcal {A}({\Gamma })\) is universal among all TGW algebras with such a representation. We also find explicit conditions in terms of Γ for when this representation is faithful or locally surjective. By forgetting some of the structure of Γ one obtains a Dynkin diagram, D(Γ). We show that the generalized Cartan matrix of \(\mathcal {A}({\Gamma })\) coincides with the one corresponding to D(Γ) and that \(\mathcal {A}({\Gamma })\) contains graded homomorphic images of the enveloping algebra of the positive and negative part of the corresponding Kac-Moody algebra. Finally, we show that a primitive quotient U/J of the enveloping algebra of a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero is graded isomorphic to a TGW algebra if and only if J is the annihilator of a completely pointed (multiplicity-free) simple weight module. The infinite-dimensional primitive quotients in types A and C are closely related to \(\mathcal {A}({\Gamma })\) for specific Γ. We also prove one result in the affine case.     

Boundary value problems with global projection conditions

Parametrices of elliptic boundary value problems for differential operators belong to an algebra of pseudodifferential operators with the transmission property at the boundary. However, generically, smooth symbols on a manifold with boundary do not have this property, and several interesting applications require a corresponding more general calculus. We introduce here a new algebra of boundary value problems that contains Shapiro-Lopatinskij elliptic as well as global projection conditions; the latter ones are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. We show that every elliptic operator admits (up to a stabilisation) elliptic conditions of that kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. Moreover, we construct parametrices in the calculus.     

Rota–Baxter Operators on Quadratic Algebras

We prove that all Rota–Baxter operators on a quadratic division algebra are trivial. For nonzero weight, we state that all Rota–Baxter operators on the simple odd-dimensional Jordan algebra of bilinear form are projections on a subalgebra along another one. For weight zero, we find a connection between the Rota–Baxter operators and the solutions to the alternative Yang–Baxter equation on the Cayley–Dickson algebra. We also investigate the Rota–Baxter operators on the matrix algebras of order two, the Grassmann algebra of plane, and the Kaplansky superalgebra.     

Quantum group structure and local fields in the algebraic approach to 2D gravity

This review contains a summary of the work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables — the Liouville exponentials and the Liouville field itself — and the underlying algebra of chiral vertex operators. The double quantum group structure arising from the presence of two screening charges is discussed and the generalized algebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version of the classical group-theoretic representation of the Liouville fields by Leznov and Saveliev.Supported by DFG.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 158–191, July, 1995.     

Generalized multipliers for left-invertible analytic operators and their applications to commutant and reflexivity

We introduce generalized?multipliers for?left-invertible analytic operators. We show that they form a Banach algebra and characterize the commutant of such operators in its terms. In the special case, we describe the commutant of balanced weighted shift only in terms of its weights. In addition, we prove two independent criteria for reflexivity of weighted shifts on directed trees.     

Translation-invariant integrals,and Fourier analysis on Clifford and Grassmann algebras

The generalization of Berezin's Grassmann algebra integral to a Clifford algebra is shown to be translation-invariant in a certain sense. This enables the construction of analogs of twisted convolutions of Grassmann algebra elements and of the Fourier-Weyl transformation, which is an isomorphism from a Clifford algebra to the Grassmann algebra over the dual space, equipped with a twisted convolution product. As an application a noncommutative central limit theorem for states of a Clifford algebra is proved.