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1.
Commutator identities on associative algebras and the integrability of nonlinear evolution equations
A. K. Pogrebkov 《Theoretical and Mathematical Physics》2008,154(3):405-417
We show that commutator identities on associative algebras generate solutions of the linearized versions of integrable equations.
In addition, we introduce a special dressing procedure in a class of integral operators that allows deriving both the nonlinear
integrable equation itself and its Lax pair from such a commutator identity. The problem of constructing new integrable nonlinear
evolution equations thus reduces to the problem of constructing commutator identities on associative algebras.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 477–491, March, 2008. 相似文献
2.
L. M. Samoilov 《Mathematical Notes》1999,65(2):208-213
In the paper, trinomial identities of associative algebras over an infinite field are considered (in general, the algebras
may have no unit), i.e., identities of the formα
m
1+β
m
2+γ
m
3=0, where α, β, and γ are scalars andm
1,m
2, andm
3 are different monomials. It is shown that any nontrivial identity if this kind implies a semigroup identity. The algebras
with trinomial identities are characterized in the language of varieties.
Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 254–260, February, 1999. 相似文献
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4.
S. Saito 《Theoretical and Mathematical Physics》1999,118(3):369-377
An extension of the Hirota bilinear difference equation to a multilinear, multidimensional lattice space is discussed. This extension admits linear Bäcklund transformations. A preliminary result on solutions is presented in the case of trilinear equations. 相似文献
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A. V. Zabrodin 《Theoretical and Mathematical Physics》2008,155(1):567-584
We consider GL(K|M)-invariant integrable supersymmetric spin chains with twisted boundary conditions and demonstrate the role of Bäcklund transformations in solving the difference Hirota equation for eigenvalues of their transfer matrices. We show that the nested Bethe ansatz technique is equivalent to a chain of successive Bäcklund transformations “undressing” the original problem to a trivial one. 相似文献
8.
A. S. Gordienko 《Mathematical Notes》2008,83(5-6):744-752
A result confirming the Regev conjecture for the codimension of associative algebras with unit which are of PI-exponent 2 is obtained. It is proved that the sequence of multiplicities of irreducible summands in proper cocharacters of algebras of PI-exponent 2 is of period 2, beginning with some index, whereas this sequence is constant for the ordinary cocharacters of the algebras of PI-exponent 1. 相似文献
9.
This paper studies the concepts of a totally compatible dialgebra and a totally compatible Lie dialgebra,defined to be a vector space with two binary operations that satisfy individual and mixed associativity conditions and Lie algebra conditions respectively.We show that totally compatible dialgebras are closely related to bimodule algebras and semi-homomorphisms.More significantly,Rota-Baxter operators on totally compatible dialgebras provide a uniform framework to generalize known results that Rota-Baxter related operators give tridendriform algebras.Free totally compatible dialgebras are constructed.We also show that a Rota-Baxter operator on a totally compatible Lie dialgebra gives rise to a PostLie algebra,generalizing the fact that a Rota-Baxter operator on a Lie algebra gives rise to a PostLie algebra. 相似文献
10.
11.
Chia-Hsin Liu 《代数通讯》2013,41(10):4783-4801
We study locally finite algebras and twisted group algebras with units satisfying a group identity. As a preliminary result, we obtain a necessary condition for twisted group algebras to satisfy a generalized polynomial identity. 相似文献
12.
We study the exponential rate of growth of the sequence of proper, Lie and Jordan codimensions of an associative algebra.
We show that for any finite dimensional associative algebra, the exponential rates of growth can be explicitly computed and
are strictly related to the PI-exponent of the algebra.
The first author was partially supported by MIUR of Italy. The second author was partially supported by RFBR grant No 06-01-00485
and SSC-5666.2006.1 相似文献
13.
S. M. Ratseev 《Mathematical Notes》2014,96(3-4):538-547
In the paper, the varieties of Poisson algebras whose ideals of identities contain the identity {x, y}· {z, t} = 0 are studied, and the correlation of these varieties with varieties of Lie algebras is investigated. A variety of Poisson algebras of almost exponential growth is presented. An example of a variety of Poisson algebras with fractional exponent is also given. 相似文献
14.
ABSTRACTAn algebra with identities a(bc)?=?b(ac), (ab)c?=?(ac)b is called bicommutative. We construct list of identities satisfied by commutator and anti-commutator products in a free bicommutative algebra. We give criterions for elements of a free bicommutative algebra to be Lie or Jordan. 相似文献
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17.
Given a finite, simple, vertex-weighted graph, we construct a graded associative (noncommutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Noncommutative Gr?bner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment?Cangle complex. We apply our result to the loop-space homology of this space. 相似文献
18.
The object of this work is to investigate the initial-boundary value problem for coupled Hirota equation on the half-line.We show that the solution of the coupled Hirota equation can be expressed in terms of the solution of a 3×3 matrix Riemann-Hilbert problem formulated in the complex k-plane.The relevant jump matrices are explicitly given in terms of the matrix-valued spectral functions s(k)and S(k)that depend on the initial data and boundary values,respectively.Then,applying nonlinear steepest descent techniques to the associated 3×3 matrix-valued Riemann-Hilbert problem,we can give the precise leading-order asymptotic formulas and uniform error estimates for the solution of the coupled Hirota equation. 相似文献
19.
V. S. Luchko 《Ukrainian Mathematical Journal》2009,61(7):1187-1191
Let I be a nilpotent ideal of an associative algebra A over a field F and let D be a derivation of A. We prove that the ideal I + D(I) is nilpotent if char F = 0 or the nilpotency index I is less than char F = p in the case of the positive characteristic of the field F. In particular, the sum N(A) of all nilpotent ideals of the algebra A is a characteristic ideal if char F = 0 or N(A) is a nilpotent ideal of index < p = char F. 相似文献
20.
A. N. Krasil'nikov 《Mathematical Notes》1997,62(4):426-433
To each associative ringR we can assign the adjoint Lie ringR
(−) (with the operation(a,b)=ab−ba) and two semigroups, the multiplicative semigroupM(R) and the associated semigroupA(R) (with the operationaob=ab+a+b). It is clear that a Lie ringR
(−) is commutative if and only if the semigroupM(R) (orA(R)) is commutative. In the present paper we try to generalize this observation to the case in whichR
(−) is a nilpotent Lie ring. It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR
(−) is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in whichR is an algebra without identity element overF, this assertion remains valid forA(R), but fails forM(R). Another similar results are obtained.
Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–519, October, 1997.
Translated by A. I. Shtern 相似文献