We prove that, if A is an algebra (over a field of characteristic zero) with no nonzero joint divisor of zero and containing a nonzero central idempotent, and if A satisfies the identities
(
x
p
,
x
...q
,
x
r
)
=
0
and
(
x
p
′
,
x
q
′
,
x
r
′
)
=
0
with exponents
p
,
q
,
r
,
p
′
,
q
′
,
r
′
belonging to {1, 2} such that
p
≠
r
,
(
p
′
,
q
′
,
r
′
)
≠
(
p
,
q
,
r
)
and
(
p
′
,
q
′
,
r
′
)
≠
(
3
−
r
,
3
−
q
,
3
−
p
)
,
then A is a unital power-associative algebra.
We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an ...associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E_8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.
The area of coalgebra has emerged within theoretical computer science with a unifying claim: to be the mathematics of computational dynamics. It combines ideas from the theory of dynamical systems ...and from the theory of state-based computation. Although still in its infancy, it is an active area of research that generates wide interest. Written by one of the founders of the field, this book acts as the first mature and accessible introduction to coalgebra. It provides clear mathematical explanations, with many examples and exercises involving deterministic and non-deterministic automata, transition systems, streams, Markov chains and weighted automata. The theory is expressed in the language of category theory, which provides the right abstraction to make the similarity and duality between algebra and coalgebra explicit, and which the reader is introduced to in a hands-on manner. The book will be useful to mathematicians and (theoretical) computer scientists and will also be of interest to mathematical physicists, biologists and economists.
We consider a generalization of (pro)algebraic loops defined on general categories of algebras and the dual notion of a coloop bialgebra suitable to represent them as functors. We prove that the ...natural loop of formal diffeomorphisms with associative coefficients is proalgebraic, and we give the closed formulas of the codivisions on its coloop bialgebra. This result provides a generalization of the Lagrange inversion formula to series with non-commutative coefficients, and a loop-theoretic explanation to the existence of the non-commutative Faà di Bruno Hopf algebra.
In this work, we classify the group gradings on finite-dimensional incidence algebras over a field, where the field has characteristic zero, or the characteristic is greater than the dimension of the ...algebra, or the grading group is abelian.
Moreover, we investigate the structure of G-graded (D1,D2)-bimodules, where G is an abelian group, and D1 and D2 are the group algebra of finite subgroups of G. As a consequence, we can provide a more profound structure result concerning the group gradings on the incidence algebras, and we can classify their isomorphism classes of group gradings.
We present an eight-dimensional even sub-algebra of the
-dimensional associative Clifford algebra
and show that its eight-dimensional elements denoted as
and
respect the norm relation
thus forming an ...octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere.
In the paper, we consider
dimensional naturally graded nilpotent associative algebras with the characteristic sequence
over an algebraically closed field of characteristic zero. There are two types ...of such algebras. We give the algebraic classification up to isomorphism of both types of algebras. Moreover, we prove that there are no such algebras of the second type for
.
We introduce a new approach to the classification of operator identities, based on basic concepts from the theory of algebraic operads together with computational commutative algebra applied to ...determinantal ideals of matrices over polynomial rings. We consider operator identities of degree 2 (the number of variables in each term) and multiplicity 1 or 2 (the number of operators in each term), but our methods apply more generally. Given an operator identity with indeterminate coefficients, we use partial compositions to construct a matrix of consequences, and then use computer algebra to determine the values of the indeterminates for which this matrix has submaximal rank. For multiplicity 1 we obtain six identities, including the derivation identity. For multiplicity 2 we obtain eighteen identities and two parametrized families, including the left and right averaging identities, the Rota-Baxter identity, the Nijenhuis identity, and some new identities which deserve further study.
The paper is devoted to give a complete classification of all n-dimensional non-associative Jordan algebras with (n−3)-dimensional annihilator over an algebraically closed field of characteristic ≠2. ...We also give a complete classification of all n-dimensional Jordan algebras with (n−1)- and (n−2)-dimensional annihilator.
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