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 ![]() ![]() |
[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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie-admissible_algebra&oldid=14803