Generalization of a Lie algebra
Algebraic structure
In mathematics, a Lie algebra has been generalized in several ways.
Graded Lie algebra and Lie superalgebra
A graded Lie algebra is a Lie algebra with grading. When the grading is
Z
/
2
{\displaystyle \mathbb {Z} /2}
, it is also known as a Lie superalgebra.
Lie-isotopic algebra
A Lie-isotopic algebra is a generalization of Lie algebras proposed by physicist R. M. Santilli in 1978.
Definition
Recall that a finite-dimensional Lie algebra
L
{\displaystyle L}
with generators
X
1
,
X
2
,
.
.
.
,
X
n
{\displaystyle X_{1},X_{2},...,X_{n}}
and commutation rules
[
X
i
X
j
]
=
X
i
X
j
−
X
j
X
i
=
C
i
j
k
X
k
,
{\displaystyle [X_{i}X_{j}]=X_{i}X_{j}-X_{j}X_{i}=C_{ij}^{k}X_{k},}
can be defined (particularly in physics) as the totally anti-symmetric algebra
A
(
L
)
−
{\displaystyle A(L)^{-}}
attached to the universal enveloping associative algebra
A
(
L
)
=
{
X
1
,
X
2
,
.
.
.
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