On a theorem on infinite dimensionality of an associative algebra

1969 ◽  
pp. 237-242 ◽  
Author(s):  
È. B. Vinberg
Keyword(s):  
Associative Algebra ◽  
Infinite Dimensionality

Post-quantum digital signature schemes: setting a hidden group with two-dimensional cyclicity

2020 ◽  
Vol 4 ◽  
pp. 75-82
Author(s):  
D.Yu. Guryanov ◽  
◽  
D.N. Moldovyan ◽  
A. A. Moldovyan ◽  
Keyword(s):  
Associative Algebra ◽  
Digital Signature ◽  
Quantum Signature ◽  
Commutative Group ◽  
Two Dimensional ◽  
Signature Scheme ◽  
Signature Schemes ◽  
Fixed Element ◽  
Commutative Associative Algebra ◽  
Quantum Digital Signature

For the construction of post-quantum digital signature schemes that satisfy the strengthened criterion of resistance to quantum attacks, an algebraic carrier is proposed that allows one to define a hidden commutative group with two-dimensional cyclicity. Formulas are obtained that describe the set of elements that are permutable with a given fixed element. A post-quantum signature scheme based on the considered finite non-commutative associative algebra is described.

scholarly journals CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

2021 ◽  
Author(s):  
MÁTYÁS DOMOKOS ◽  
VESSELIN DRENSKY
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Invariant Theory ◽  
Reductive Group ◽  
Symmetric Tensor ◽  
Free Associative Algebra ◽  
Tensor Algebra ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Degree Bounds

AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

scholarly journals On the lower central series quotients of a graded associative algebra

2011 ◽  
Vol 328 (1) ◽  
pp. 287-300 ◽  
Author(s):  
Martina Balagović ◽  
Anirudha Balasubramanian
Keyword(s):  
Associative Algebra ◽  
Lower Central Series ◽  
Central Series

scholarly journals Products of several commutators in a Lie nilpotent associative algebra

2017 ◽  
Vol 27 (08) ◽  
pp. 1027-1040 ◽  
Author(s):  
Galina Deryabina ◽  
Alexei Krasilnikov
Keyword(s):  
General Formula ◽  
Associative Algebra ◽  
Present Note ◽  
Commutator Formula ◽  
Positive Integers

Let [Formula: see text] be a field of characteristic [Formula: see text] and let [Formula: see text] be a unital associative [Formula: see text]-algebra. Define a left-normed commutator [Formula: see text] [Formula: see text] recursively by [Formula: see text], [Formula: see text] [Formula: see text]. For [Formula: see text], let [Formula: see text] be the two-sided ideal in [Formula: see text] generated by all commutators [Formula: see text] ([Formula: see text]. Define [Formula: see text]. Let [Formula: see text] be integers such that [Formula: see text], [Formula: see text]. Let [Formula: see text] be positive integers such that [Formula: see text] of them are odd and [Formula: see text] of them are even. Let [Formula: see text]. The aim of the present note is to show that, for any positive integers [Formula: see text], in general, [Formula: see text]. It is known that if [Formula: see text] (that is, if at least one of [Formula: see text] is even), then [Formula: see text] for each [Formula: see text] so our result cannot be improved if [Formula: see text]. Let [Formula: see text]. Recently, Dangovski has proved that if [Formula: see text] are any positive integers then, in general, [Formula: see text]. Since [Formula: see text], Dangovski’s result is stronger than ours if [Formula: see text] and is weaker than ours if [Formula: see text]; if [Formula: see text], then [Formula: see text] so both results coincide. It is known that if [Formula: see text] (that is, if all [Formula: see text] are odd) then, for each [Formula: see text], [Formula: see text] so in this case Dangovski’s result cannot be improved.

Quantum Homomorphisms of Associative Algebras

1997 ◽  
Vol 12 (38) ◽  
pp. 2963-2974
Author(s):  
A. E. F. Djemai
Keyword(s):  
Quantum Group ◽  
Associative Algebra ◽  
Associative Algebras ◽  
Type Formula ◽  
Algebra Of Functions ◽  
Inhomogeneous Quantum

Given an associative algebra A generated by {ek, k=1, 2,…} and with an internal law of type: [Formula: see text], we first show that it is possible to construct a quantum bi-algebra [Formula: see text] with unit and generated by (non-necessarily commutative) elements [Formula: see text] satisfying the relations: [Formula: see text]. This leads one to define a quantum homomorphism[Formula: see text]. We then treat the example of the algebra of functions on a set of N elements and we show, for the case N=2, that the resulting bihyphen;algebra is an inhomogeneous quantum group. We think that this method can be used to construct quantum inhomogeneous groups.

Frobenius algebra with relaxed associativity constraint

2018 ◽  
Vol 27 (07) ◽  
pp. 1841008
Author(s):  
Zbigniew Oziewicz ◽  
William Stewart Page
Keyword(s):  
Associative Algebra ◽  
Monoidal Category ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Frobenius Algebra ◽  
Frobenius Algebras ◽  
Frobenius Structures ◽  
Necessary And Sufficient

Frobenius algebra is formulated within the Abelian monoidal category of operad of graphs. A not necessarily associative algebra [Formula: see text] is said to be a Frobenius algebra if there exists a [Formula: see text]-module isomorphism. A new concept of a solvable Frobenius algebra is introduced: an algebra [Formula: see text] is said to be a solvable Frobenius algebra if it possesses a nonzero one-sided [Formula: see text]-module morphism with nontrivial radical. In the category of operad of graphs, we can express the necessary and sufficient conditions for an algebra to be a solvable Frobenius algebra. The notion of a solvable Frobenius algebra makes it possible to find all commutative nonassociative Frobenius algebras (Conjecture 10.1), and to find all Frobenius structures for commutative associative Frobenius algebras. Frobenius algebra allows [Formula: see text]-permuted opposite algebra to be extended to [Formula: see text]-permuted algebras.

FIXED ELEMENTS OF NONINJECTIVE ENDOMORPHISMS OF POLYNOMIAL ALGEBRAS IN TWO VARIABLES

2016 ◽  
Vol 95 (2) ◽  
pp. 209-213
Author(s):  
YUEYUE LI ◽  
JIE-TAI YU
Keyword(s):  
Associative Algebra ◽  
Characteristic Zero ◽  
Polynomial Algebra ◽  
Free Associative Algebra ◽  
Polynomial Algebras ◽  
Fixed Element ◽  
The Given

Let $A_{2}$ be a free associative algebra or polynomial algebra of rank two over a field of characteristic zero. The main results of this paper are the classification of noninjective endomorphisms of $A_{2}$ and an algorithm to determine whether a given noninjective endomorphism of $A_{2}$ has a nontrivial fixed element for a polynomial algebra. The algorithm for a free associative algebra of rank two is valid whenever an element is given and the subalgebra generated by this element contains the image of the given noninjective endomorphism.

scholarly journals A 5-Engel associative algebra whose group of units is not 5-Engel

2019 ◽  
Vol 519 ◽  
pp. 101-110
Author(s):  
Galina Deryabina ◽  
Alexei Krasilnikov
Keyword(s):  
Associative Algebra ◽  
Group Of Units
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