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Lie superalgebra
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(Definition)
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Definition 1 A Lie superalgebra is a vector superspace equipped with a bilinear map \begin{equation} \begin{split} [\cdot,\cdot]: V \otimes V &\rightarrow V, \\ v \otimes w &\mapsto [v, w], \end{split} \end{equation}satisfying the following properties:
- If $v$ and $w$ are homogeneous vectors, then $[v,w]$ is a homogeneous vector of degree $|v| + |w| \pmod 2$ ,
- For any homogeneous vectors $v, w$ , $[v,w] = (-1)^{|v||w| + 1} [w,v]$ ,
- For any homogeneous vectors $u,v,w$ , $(-1)^{|u||w|}[u, [v,w]] + (-1)^{|v||u|} [v, [w, u]] + (-1)^{|w||v|} [w,[u,v]]$ = 0.
The map $[\cdot,\cdot]$ is called a Lie superbracket.
Example 1 A Lie algebra $V$ can be considered as a Lie superalgebra by setting $V = V_0$ and, therefore, $V_1 = \{0\}$ .
Example 2 Any associative superalgebra $A$ has a Lie superalgebra structure where, for any homogeneous elements $a,b \in A$ , the Lie superbracket is defined by the equation \begin{equation}\label{supercomm} [a, b] = ab - (-1)^{|a||b|}ba. \end{equation} The Lie superbracket ( ![[*]](http://images.planetmath.org:8080/cache/objects/7509/js//usr/share/latex2html/icons/crossref.png "[*]") ) is called the supercommutator bracket on $A$ .
Let $V = V_0 \oplus V_1$ be a vector superspace. Then any element of $V_0$ is said to be even, and any element of $V_1$ is said to be odd. By the definition of the direct sum, any element $v$ of $V$ can be uniquely written as $v = v_0 + v_1$ , where $v_0 \in V_0$ and $v_1 \in V_1$ .
Definition 3 A vector $v \in V$ is homogeneous of degree $i$ if $v \in V_i$ for $i = 0$ or $1$ .
If $v \in V$ is homogeneous, then the degree of $v$ is denoted by $|v|$ . In other words, if $v \in V_i$ , then $|v| = i$ by definition.
Remark 1 The vector $0$ is homogeneous of both degree $0$ and $1$ , and thus $|0|$ is not well-defined.
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"Lie superalgebra" is owned by bci1. [ full author list (2) | owner history (1) ]
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See Also: Cartan calculus, superalgebra, graded algebra, Lie algebroids, superspace and supergravity superfields, supersymmetry, Lie algebroids, Jordan-Banach and Jordan-Lie algebras, Lie algebra, Lie algebra cohomology, superalgebra, Cartan calculus, quantum gravity theories, symmetry and groupoid representations in functional biology
| Other names: |
Lie super algebra, graded Lie algebra |
| Also defines: |
vector superspace, Lie superbracket, supercommutator bracket |
| Keywords: |
supergeometry, supersymmetry |
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Cross-references: well-defined, direct sum, odd, even, decomposition, vector space, supercommutative, graded derivations, equation, homogeneous elements, structure, superalgebra, associative, Lie algebra, map, degree, vectors, homogeneous, properties, bilinear map
There are 13 references to this entry.
This is version 13 of Lie superalgebra, born on 2005-11-30, modified 2009-02-01.
Object id is 7509, canonical name is LieSuperalgebra3.
Accessed 4962 times total.
Classification:
| AMS MSC: | 17B01 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Identities, free Lie algebras) | | | 81Q60 (Quantum theory :: General mathematical topics and methods in quantum theory :: Supersymmetric quantum mechanics) | | | 17B60 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Lie ) | | | 81R50 (Quantum theory :: Groups and algebras in quantum theory :: Quantum groups and related algebraic methods) |
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Pending Errata and Addenda
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