Lie-admissible algebra
A (non-associative) algebra (cf. Non-associative rings and algebras) whose commutator algebra becomes a Lie algebra. It was first introduced by A.A. Albert in 1948 and originated from one of the defining identities for standard algebras [a1]. For an algebra 
over a field 
, the commutator algebra 
of 
is the anti-commutative algebra with multiplication 
defined on the vector space 
. If 
is a Lie algebra, i.e. 
satisfies the Jacobi identity 
, then 
is called Lie-admissible (LA). Much of the structure theory of Lie-admissible algebras has been carried out initially under additional conditions such as the flexible identity 
or power-associativity (i.e. every element generates an associative subalgebra), or both. An algebra 
is flexible Lie-admissible (FLA) if and only if it satisfies the identity
 | (a1) |
if and only if the mapping 
is a Lie module homomorphism of 
to 
for 
under the adjoint action. For this reason, representations of Lie algebras play a main role in the structure theory of FLA algebras [a2]. Lie and associative algebras are examples of FLA algebras.
Beginning with Albert's problem of classifying all power-associative FLA algebras 
with 
semi-simple [a1], a common theme of the structure theory in various mathematical, physical and geometrical settings has been to focus on the case of a prescribed Lie algebra structure on 
. Albert's problem was first solved in 1962 for finite-dimensional algebras 
over an algebraically closed field 
of characteristic 0, and such algebras turned out to be Lie algebras [a3]. This result was extended to the case of 
when 
is a classical Lie algebra or a generalized Witt algebra [a2], [a4] (cf. Witt algebra). In 1981, these algebras were classified without the assumption of power-associativity [a5], [a6]: When 
is simple over the base field 
as above, the multiplication 
in 
is given by
 | (a2) |
for a fixed scalar 
, where 
for 
not of type 
(
), and for 
of type 
(
), 
and 
is defined on 
by
 |
where 
denotes the matrix product of 
and 
and 
is the identity matrix. Thus, the algebra 
with 
of type 
(
) can not be power-associative. If 
is semi-simple, 
is a direct sum of simple algebras given by (a2). The classification was extended to the case where the solvable radical (cf. Radical of rings and algebras) of 
is a direct summand of 
or Abelian [a2]. In 1984, the algebras 
in Albert's problem were determined in the absence of flexibility [a7]: If 
is semi-simple with decomposition 
(
), where the 
are simple ideals of 
, then the multiplication 
in 
has the form 
for 
, 
, where the 
are linear functionals on the 
and satisfy certain conditions prescribed in terms of graphs having 2, 3 or 4 vertices.
R.M. Santilli in 1978 obtained LA algebras (brackets) from a modified form of Hamilton's equations with external terms which represent a general non-self-adjoint Newtonian system in classical mechanics [a8]. Such a form leads to a time evolution
 | (a3) |
where 
is a local chart in a manifold of dimension 
, 
is a Hamiltonian, and 
a non-singular 
-tensor in a region with decomposition 
for 
and 
(
). The symmetric tensor 
represents the presence of non-self-adjoint forces in the system. The commutator 
is given by 
in terms of the classical Poisson brackets 
and thus, by (a3), 
defines a Lie-admissible product 
on the 
-space of 
-functions in 
, where 
denotes the field of real numbers. The bracket 
, or 
, is called a fundamental Lie-admissible bracket, or tensor. More generally, if 
for a skew-symmetric non-singular 
-tensor 
in a region, Lie-admissibility of 
or the bracket (a3) is described by partial differential equations of first order in 
. The general solution 
, called the general cosymplectic tensor, to these equations exists under certain conditions and plays a central role in Birkhoffian mechanics (a generalization of Hamiltonian mechanics) [a9]. In this case 
, or 
, is called a general Lie-admissible bracket, or tensor. A quantum mechanical version of this leads to a time-development equation 
in an associative algebra 
of operators in a physical system, where 
and 
are in general non-Hermitian non-singular operators in 
which represent non-self-adjoint forces [a8]. From this equation, regarded as a generalization of the Heisenberg equation, one obtains an LA algebra 
, called the 
-mutation of 
, with product 
defined on an associative algebra with identity for fixed invertible 
. 
is not in general flexible or power-associative. In fact, any one of these conditions is equivalent to the relation 
for some invertible 
in the centre of 
[a10]. A special case of the above approach has been investigated by Santilli in 1967 [a11], where he first introduced LA algebras into physics: For real numbers 
, the bracket (a3) with 
and the algebra 
were considered for a generalization of Hamiltonian and quantum mechanics. According to Santilli [a8], the aim of this Lie-admissible approach is to make a transition from contemporary physical models based on Lie algebras or their graded-supersymmetric extensions to the general Lie-admissible models, which transition essentially permits the treatment of particles as being extended and therefore admits additional contact, non-potential and non-Hamiltonian interactions.
From a different point of view, S. Okubo [a12] in 1978 used FLA algebras 
to generalize the framework of the consistent canonical quantization procedure based on the associative law. A quantization is called consistent if the Hamiltonian equation of motion 
can reproduce the original Lagrange equation. Such a quantization can be done in 
based only on the canonical commutation relation and the identity (a1). If 
consists of operators in a physical system, then using (a1) it can be shown that the Heisenberg equation 
for some 
is essentially the most general time-development equation in 
, where 
is the commutator in 
. If the Hamiltonian 
is power-associative in 
, then the time-development operator 
is well defined for the Schrödinger formulation in 
with state vector 
satisfying 
. If, in addition, 
is weakly associative in 
, i.e. 
for all positive integers 
and 
, then the solution to the Heisenberg equation in 
has the form 
, as in the usual quantum mechanics. An example of such an algebra is the real pseudo-octonion algebra 
, which has the multiplication 
defined on the 
-space of 
Hermitian matrices of trace 0, where 
[a2]. 
is an FLA division algebra, with 
isomorphic to 
, and has some relevance to 
particle physics. It also plays an important role in the structure theory of real division algebras [a2].
An algebra 
over a field 
is called Mal'tsev-admissible (MA) if its commutator algebra 
becomes a Mal'tsev algebra, i.e., 
satisfies the Mal'tsev identity
 |
 |
It arises as a natural generalization of LA algebras as well as Mal'tsev algebras, and its structure theory is parallel to that of LA algebras [a2]. Alternative algebras (cf. Alternative rings and algebras) are examples of flexible Mal'tsev-admissible (FMA) algebras, and octonion algebras (also called Cayley–Dickson algebras, cf. Cayley–Dickson algebra) are FMA but not LA. For an octonion algebra 
over a field 
of characteristic 
with standard involution 
, one obtains an algebra 
with multiplication 
defined on 
. The algebra 
, called a para-octonion algebra, is a simple FMA algebra without identity and so not alternative [a2]. An algebra 
(not necessarily with identity) is called a composition algebra if there exists a non-degenerate quadratic form 
on 
such that 
for all 
. Any finite-dimensional flexible composition algebra (
) is an MA algebra of dimension 1, 2, 4, or 8, and for dimension 8 octonion, pseudo-octonion and para-octonion algebras are the only such algebras [a13]. For an MA algebra 
, let 
, where 
is the adjoint mapping 
given by 
. Then 
is a Lie subalgebra of the derivation algebra 
of 
(cf. also Derivation in a ring), and if 
is FMA, then 
is also a subalgebra of 
and the mapping 
is a Lie module homomorphism of 
to 
for 
[a2]. Let 
be finite dimensional over a field 
of characteristic 0. If 
is semi-simple, then so is 
. Because of this, virtually all results about FLA algebras can be extended to FMA algebras [a2]. If 
is FMA with 
central simple, non-Lie over 
, then 
is a Mal'tsev algebra isomorphic to a 
-dimensional simple Mal'tsev algebra obtained from an octonion algebra (cf. Mal'tsev algebra). If 
is semi-simple and 
is algebraically closed, then 
is the direct sum of simple algebras given by (a2) and simple Mal'tsev algebras. Some of the work on MA algebras was motivated by algebraic formalisms in physics aimed at generalizing both the Lie-admissible and the octonionic approach in quantum mechanics.
LA and MA algebras also arise from differential geometry on Lie groups and reductive homogeneous spaces. For a (connected) Lie group 
with Lie algebra 
, the determination of all (left) invariant affine connections (cf. Affine connection) 
on 
reduces to the problem of classifying all algebras 
with a multiplication 
defined on 
; the relation is given by 
for 
. In this case, 
is called the connection algebra of 
. Those connections which are torsion free correspond to the LA algebras 
with 
(i.e., 
) [a14]. If, in addition, the curvature tensor of 
is zero (i.e., 
is flat), then 
satisfies the left-symmetric identity
 |
The classification of left-invariant affine structures on 
reduces to that of left-symmetric algebras 
with 
[a15]. Other geometrical properties of 
on 
, such as geodesic, holonomy, pseudo-Riemannian structure, and infinitesimal generator, can be described in terms of 
. For example, if every vector field in 
is an infinitesimal generator for a one-parameter group of affine diffeomorphisms on 
for 
, then the connection algebra 
of 
is FLA [a14].
Let 
be a reductive homogeneous space with a fixed decomposition 
(direct sum), where 
is the Lie algebra of a closed Lie subgroup 
of 
and 
is a subspace of 
such that 
(or, equivalently, 
). There is a one-one correspondence between the set of 
-invariant affine connections 
on 
and the set of algebras 
with 
, i.e., 
, the automorphism group of 
. The projection 
of 
onto 
for 
converts 
into an anti-commutative algebra 
, called a reductive algebra. More generally, an algebra 
is called reductive-admissible if 
is isomorphic to 
for some reductive decomposition 
of a Lie algebra 
. Those connections on 
which are torsion free correspond to the reductive-admissible algebras 
such that 
and 
or 
. Any MA algebra 
is reductive-admissible with 
, where 
is a Lie algebra with multiplication 
for 
and 
. Geometrical properties of 
on 
such as those above are described in terms of the connection algebra 
. For a detailed account of these, see [a15]–[a17].
References
| [a1] | A.A. Albert, "Power associative rings" Trans. Amer. Math. Soc. , 64 (1948) pp. 552–593 |
| [a2] | H.C. Myung, "Malcev-admissible algebras" , Birkhäuser (1986) |
| [a3] | P.J. Laufer, M.L. Tomber, "Some Lie admissible algebras" Canad. J. Math. , 14 (1962) pp. 287–292 |
| [a4] | H.C. Myung, "Some classes of flexible Lie-admissible algebras" Trans. Amer. Math. Soc. , 167 (1972) pp. 79–88 |
| [a5] | S. Okubo, H.C. Myung, "Adjoint operators in Lie algebras and the classification of simple flexible Lie-admissible algebras" Trans. Amer. Math. Soc. , 264 (1981) pp. 459–472 |
| [a6] | G.M. Benkart, J.M. Osborn, "Flexible Lie-admissible algebras" J. of Algebra , 71 (1981) pp. 11–31 |
| [a7] | G.M. Benkart, "Power-associative Lie-admissible algebras" J. of Algebra , 90 (1984) pp. 37–58 |
| [a8] | R.M. Santilli, "Lie-admissible approach to the hadronic structure" , II , Hadronic Press (1982) |
| [a9] | R.M. Santilli, "Foundations of theoretical mechanics" , II , Springer (1982) |
| [a10] | J.M. Osborn, "The Lie-admissible mutation  of an associative algebra  " Hadronic J. , 5 (1982) pp. 904–930 |
| [a11] | R.M. Santilli, "Imbedding of Lie algebras in nonassociative structures" Nuovo Cimento A (10) , 51 (1967) pp. 570–576 |
| [a12] | S. Okubo, "Non-associative quantum mechanics via flexible Lie-admissible algebras" R. Casabuoni (ed.) G. Domokos (ed.) S. Koveski-Domokos (ed.) , Proc. 3-rd Workshop Current Problems in High Energy Physics , Johns Hopkins Univ. Press (1979) pp. 103–120 |
| [a13] | S. Okubo, "Classification of flexible composition algebras, I, II" Hadronic J. , 5 (1982) pp. 1564–1612 |
| [a14] | H.C. Myung, A.A. Sagle, "Lie-admissible algebras and affine connections on Lie groups" S.A. Park (ed.) , Proc. Workshops in Pure Math. , 7. Algebraic Structures , Pure Math. Res. Assoc. (1988) pp. 115–148 |
| [a15] | H. Kim, "Complete left-invariant affine structures on nilpotent Lie groups" J. Differential Geom. , 24 (1986) pp. 373–394 |
| [a16] | A.A. Sagle, "Invariant Lagrangian mechanics, connections, and non-associative algebras" Algebras, Groups Geom. , 3 (1986) pp. 199–263 |
| [a17] | H.C. Myung, A.A. Sagle, "On the construction of reductive Lie-admissible algebras" J. Pure Appl. Algebra , 53 (1988) pp. 75–91 |
Lie-admissible algebra. H.C. Myung (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Lie-admissible_algebra&oldid=14803