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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 $\mathfrak A$ over a field $F$, the commutator algebra $\mathfrak A ^ {-}$ of $\mathfrak A$ is the anti-commutative algebra with multiplication $[ x , y ] = xy - yx$ defined on the vector space $\mathfrak A$. If $\mathfrak A ^ {-}$ is a Lie algebra, i.e. $\mathfrak A ^ {-}$ satisfies the Jacobi identity $[[ x , y ] , z ] + [[ y , z ] , x ] + [[ z , x ] , y ] = 0$, then $\mathfrak A$ 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 $( xy ) x = y ( yx )$ or power associativity (i.e. every element generates an associative subalgebra), or both. An algebra $\mathfrak A$ is flexible Lie-admissible (FLA) if and only if it satisfies the identity

$$\tag{a1 } [ x , yz ] = y [ x , z] + [ x , y ] z$$

if and only if the mapping $x \otimes y \rightarrow xy$ is a Lie module homomorphism of $\mathfrak A \otimes \mathfrak A$ to $\mathfrak A$ for $\mathfrak A ^ {-}$ 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 $\mathfrak A$ with $\mathfrak A ^ {-}$ 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 $\mathfrak A ^ {-}$. Albert's problem was first solved in 1962 for finite-dimensional algebras $\mathfrak A$ over an algebraically closed field $F$ of characteristic 0, and such algebras turned out to be Lie algebras [a3]. This result was extended to the case of $\mathop{\rm char} F \neq 0$ when $\mathfrak A ^ {-}$ 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 $\mathfrak A ^ {-}$ is simple over the base field $F$ as above, the multiplication $\star$ in $\mathfrak A$ is given by

$$\tag{a2 } x \star y = \frac{1}{2} [ x , y ] + \beta x \# y$$

for a fixed scalar $\beta \in F$, where $\beta = 0$ for $\mathfrak A ^ {-}$ not of type $A _ {n}$( $n \geq 2$), and for $\mathfrak A ^ {-}$ of type $A _ {n}$( $n \geq 2$), $\beta \neq 0$ and $\#$ is defined on $\mathfrak A ^ {-} = \mathfrak s \mathfrak l ( n+ 1, F )$ by

$$x \# y = xy + yx - \frac{2}{n+} 1 ( \mathop{\rm Tr} xy) I ,$$

where $xy$ denotes the matrix product of $x$ and $y$ and $I$ is the identity matrix. Thus, the algebra $\mathfrak A$ with $\mathfrak A ^ {-}$ of type $A _ {n}$( $n \geq 2$) can not be power-associative. If $\mathfrak A ^ {-}$ is semi-simple, $\mathfrak A$ 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 $\mathfrak A ^ {-}$ is a direct summand of $\mathfrak A ^ {-}$ or Abelian [a2]. In 1984, the algebras $\mathfrak A$ in Albert's problem were determined in the absence of flexibility [a7]: If $\mathfrak A ^ {-}$ is semi-simple with decomposition $\mathfrak A ^ {-} = \mathfrak S _ {1} + \dots + \mathfrak S _ {n}$( $n \geq 2$), where the $\mathfrak S _ {i}$ are simple ideals of $\mathfrak A ^ {-}$, then the multiplication $\star$ in $\mathfrak A$ has the form $x \star y = [ x , y ] / 2 + \tau _ {ij} ( y) x + \tau _ {ji} ( x) y$ for $x \in \mathfrak S _ {i}$, $y \in \mathfrak S _ {j}$, where the $\tau _ {ij}$ are linear functionals on the $\mathfrak S _ {j}$ 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

$$\tag{a3 } \frac{dA ( a) }{dt} = \sum _ {i , j = 1 } ^ { 2n } \frac{\partial A }{\partial a ^ {i} } S ^ {ij} ( t , a ) \frac{\partial H }{\partial a ^ {j} } \equiv ( A , H ),$$

where $a = ( a ^ {1} \dots a ^ {2n} )$ is a local chart in a manifold of dimension $2n$, $H$ is a Hamiltonian, and $( S ^ {ij} )$ a non-singular $C ^ \infty$- tensor in a region with decomposition $S ^ {ij} = \omega ^ {ij} + T ^ {ij}$ for $( \omega ^ {ij} ) = ( {} _ {-} 1 ^ {0} {} _ {0} ^ {1} )$ and $T ^ {ij} = T ^ {ji}$( $i , j = 1 , \dots , 2n$). The symmetric tensor $( T ^ {ij} )$ represents the presence of non-self-adjoint forces in the system. The commutator $( A , H ) - ( H , A )$ is given by $2 [ A , H ]$ in terms of the classical Poisson brackets $[ , ]$ and thus, by (a3), $( S ^ {ij} )$ defines a Lie-admissible product $( , )$ on the $\mathbf R$- space of $C ^ \infty$- functions in $a ^ {1} \dots a ^ {2n}$, where $\mathbf R$ denotes the field of real numbers. The bracket $( , )$, or $( S ^ {ij} )$, is called a fundamental Lie-admissible bracket, or tensor. More generally, if $S ^ {ij} = \Omega ^ {ij} + T ^ {ij}$ for a skew-symmetric non-singular $C ^ \infty$- tensor $( \Omega ^ {ij} )$ in a region, Lie-admissibility of $( S ^ {ij} )$ or the bracket (a3) is described by partial differential equations of first order in $\Omega ^ {ij}$. The general solution $( \Omega ^ {ij} )$, 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 $( S ^ {ij} )$, is called a general Lie-admissible bracket, or tensor. A quantum mechanical version of this leads to a time-development equation $idA / dt = ARH - HSA$ in an associative algebra $\mathfrak A$ of operators in a physical system, where $R$ and $S$ are in general non-Hermitian non-singular operators in $\mathfrak A$ which represent non-self-adjoint forces [a8]. From this equation, regarded as a generalization of the Heisenberg equation, one obtains an LA algebra $\mathfrak A ( r , s )$, called the $( r , s )$- mutation of $\mathfrak A$, with product $x \star y = xry - ysx$ defined on an associative algebra with identity for fixed invertible $r , s \in \mathfrak A$. $\mathfrak A ( r , s)$ is not in general flexible or power-associative. In fact, any one of these conditions is equivalent to the relation $r = as$ for some invertible $a$ in the centre of $\mathfrak A$[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 $\lambda , \mu$, the bracket (a3) with $( S ^ {ij} ) = ( {} _ {\mu 1 } ^ {0} {} _ {0} ^ {\lambda 1 } )$ and the algebra $\mathfrak A ( \lambda , \mu )$ 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 $\mathfrak A$ 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 $dQ / dt = i [ H , Q ]$ can reproduce the original Lagrange equation. Such a quantization can be done in $\mathfrak A$ based only on the canonical commutation relation and the identity (a1). If $\mathfrak A$ consists of operators in a physical system, then using (a1) it can be shown that the Heisenberg equation $dx / dt = i [ H , x ]$ for some $iH \in \mathfrak A$ is essentially the most general time-development equation in $\mathfrak A$, where $[ H, x ] = Hx - xH$ is the commutator in $\mathfrak A$. If the Hamiltonian $H$ is power-associative in $\mathfrak A$, then the time-development operator $\mathop{\rm exp} ( itH )$ is well defined for the Schrödinger formulation in $\mathfrak A$ with state vector $\psi$ satisfying $id \psi / dt = H \psi$. If, in addition, $H$ is weakly associative in $\mathfrak A$, i.e. $( H ^ {m} x ) H ^ {n} = H ^ {m} ( x H ^ {n} )$ for all positive integers $m , n$ and $x \in \mathfrak A$, then the solution to the Heisenberg equation in $\mathfrak A$ has the form $x = e ^ {itH} x ( 0) e ^ {-} itH$, as in the usual quantum mechanics. An example of such an algebra is the real pseudo-octonion algebra $P _ {8}$, which has the multiplication $x \star y = \mu xy + ( 1 - \mu ) yx - 1/3 ( \mathop{\rm Tr} xy ) I$ defined on the $\mathbf R$- space of $( 3 \times 3 )$ Hermitian matrices of trace 0, where $\mu = 1/2 \pm ( \sqrt 3 / 6 ) i$[a2]. $P _ {8}$ is an FLA division algebra, with $P _ {8} ^ {-}$ isomorphic to $\mathfrak s \mathfrak u ( 3)$, and has some relevance to $\mathop{\rm SU} ( 3)$ particle physics. It also plays an important role in the structure theory of real division algebras [a2].

An algebra $\mathfrak A$ over a field $F$ is called Mal'tsev-admissible (MA) if its commutator algebra $\mathfrak A ^ {-}$ becomes a Mal'tsev algebra, i.e., $\mathfrak A ^ {-}$ satisfies the Mal'tsev identity

$$[[ x , y ] , [ x , z ]] =$$

$$= \ [[[ x , y ] , z ] , x] + [[[ y , z ] , x ] , x] + [[[ z, x], x] , y ].$$

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 $\mathfrak A$ over a field $F$ of characteristic $\neq 2$ with standard involution $x \rightarrow \overline{x}\;$, one obtains an algebra $\mathfrak A _ \star$ with multiplication $x \star y = \overline{x}\; \overline{y}\;$ defined on $\mathfrak A$. The algebra $\mathfrak A _ \star$, called a para-octonion algebra, is a simple FMA algebra without identity and so not alternative [a2]. An algebra $\mathfrak A$( not necessarily with identity) is called a composition algebra if there exists a non-degenerate quadratic form $q$ on $\mathfrak A$ such that $q ( xy ) = q ( x) q ( y)$ for all $x , y \in \mathfrak A$. Any finite-dimensional flexible composition algebra ( $\mathop{\rm char} F \neq 2$) 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 $\mathfrak A$, let $d ( x , y ) = \mathop{\rm ad} [ x , y ] + [ \mathop{\rm ad} x , \mathop{\rm ad} y ]$, where $\mathop{\rm ad} x$ is the adjoint mapping $\mathfrak A \rightarrow \mathfrak A$ given by $y \rightarrow [ x , y]$. Then $d ( \mathfrak A , \mathfrak A )$ is a Lie subalgebra of the derivation algebra $\mathop{\rm Der} \mathfrak A ^ {-}$ of $\mathfrak A ^ {-}$( cf. also Derivation in a ring), and if $\mathfrak A$ is FMA, then $d ( \mathfrak A , \mathfrak A )$ is also a subalgebra of $\mathop{\rm Der} \mathfrak A$ and the mapping $x \otimes y \rightarrow xy$ is a Lie module homomorphism of $\mathfrak A \otimes \mathfrak A$ to $\mathfrak A$ for $d ( \mathfrak A , \mathfrak A )$[a2]. Let $\mathfrak A$ be finite dimensional over a field $F$ of characteristic 0. If $\mathfrak A ^ {-}$ is semi-simple, then so is $d ( \mathfrak A , \mathfrak A )$. Because of this, virtually all results about FLA algebras can be extended to FMA algebras [a2]. If $\mathfrak A$ is FMA with $\mathfrak A ^ {-}$ central simple, non-Lie over $F$, then $\mathfrak A$ is a Mal'tsev algebra isomorphic to a $7$- dimensional simple Mal'tsev algebra obtained from an octonion algebra (cf. Mal'tsev algebra). If $\mathfrak A ^ {-}$ is semi-simple and $F$ is algebraically closed, then $\mathfrak A$ 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 $G$ with Lie algebra $\mathfrak g$, the determination of all (left) invariant affine connections (cf. Affine connection) $\nabla$ on $G$ reduces to the problem of classifying all algebras $( \mathfrak g , \star )$ with a multiplication $\star$ defined on $\mathfrak g$; the relation is given by $\nabla _ {X} Y = X \star Y$ for $X , Y \in \mathfrak g$. In this case, $( \mathfrak g , \star )$ is called the connection algebra of $\nabla$. Those connections which are torsion free correspond to the LA algebras $( \mathfrak g , \star )$ with $( \mathfrak g , \star ) ^ {-} = \mathfrak g$( i.e., $X \star Y - Y \star X = [ X , Y ]$) [a14]. If, in addition, the curvature tensor of $\nabla$ is zero (i.e., $\nabla$ is flat), then $( \mathfrak g , \star )$ satisfies the left-symmetric identity

$$X \star ( Y \star Z ) - ( X \star Y ) \star Z = Y \star ( X \star Z ) - ( Y \star X ) \star Z .$$

The classification of left-invariant affine structures on $G$ reduces to that of left-symmetric algebras $( \mathfrak g , \star )$ with $( \mathfrak g , \star ) ^ {-} = \mathfrak g$[a15]. Other geometrical properties of $\nabla$ on $G$, such as geodesic, holonomy, pseudo-Riemannian structure, and infinitesimal generator, can be described in terms of $( \mathfrak g , \star )$. For example, if every vector field in $\mathfrak g$ is an infinitesimal generator for a one-parameter group of affine diffeomorphisms on $G$ for $\nabla$, then the connection algebra $( \mathfrak g , \star )$ of $\nabla$ is FLA [a14].

Let $G / H$ be a reductive homogeneous space with a fixed decomposition $\mathfrak g = \mathfrak m + \mathfrak h$( direct sum), where $\mathfrak h$ is the Lie algebra of a closed Lie subgroup $H$ of $G$ and $\mathfrak m$ is a subspace of $\mathfrak g$ such that $[ \mathfrak h , \mathfrak m ] \subset \mathfrak m$( or, equivalently, $( \mathop{\rm Ad} H ) \mathfrak m \subset \mathfrak m$). There is a one-one correspondence between the set of $G$- invariant affine connections $\nabla$ on $G / H$ and the set of algebras $( \mathfrak m , \star )$ with $\mathop{\rm ad} \mathfrak h \subset \mathop{\rm Der} ( \mathfrak m , \star )$, i.e., $\mathop{\rm Ad} H \subset \mathop{\rm Aut} ( \mathfrak m , \star )$, the automorphism group of $( \mathfrak m , \star )$. The projection $X Y$ of $[ X , Y ]$ onto $\mathfrak m$ for $X , Y \in \mathfrak m$ converts $\mathfrak m$ into an anti-commutative algebra $( \mathfrak m , X Y )$, called a reductive algebra. More generally, an algebra $\mathfrak A$ is called reductive-admissible if $\mathfrak A ^ {-}$ is isomorphic to $( \mathfrak m , X Y )$ for some reductive decomposition $\mathfrak g = \mathfrak m + \mathfrak h$ of a Lie algebra $\mathfrak g$. Those connections on $G / H$ which are torsion free correspond to the reductive-admissible algebras $( \mathfrak m , \star )$ such that $( \mathfrak m , \star ) = ( \mathfrak m , X Y )$ and $\mathop{\rm ad} \mathfrak h \subset \mathop{\rm Der} ( \mathfrak m , \star )$ or $\mathop{\rm Ad} H \subset \mathop{\rm Aut} ( \mathfrak m , \star )$. Any MA algebra $\mathfrak A$ is reductive-admissible with $\mathfrak g = \mathfrak A ^ {-} \oplus d ( \mathfrak A , \mathfrak A )$, where $\mathfrak g$ is a Lie algebra with multiplication $[ X + D , Y + D ^ \prime ] = [ X , Y ] + D ( Y) - D ^ \prime ( X) + d ( X , Y ) + [ D , D ^ \prime ]$ for $X , Y \in \mathfrak A$ and $D , D ^ \prime \in d ( \mathfrak A , \mathfrak A )$. Geometrical properties of $\nabla$ on $G / H$ such as those above are described in terms of the connection algebra $( \mathfrak m , \star )$. For a detailed account of these, see [a15][a17].

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