View source for Superalgebra

A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s0912001.png" />-graded algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s0912002.png" /> (see [[Graded algebra|Graded algebra]]), i.e. a [[Super-space|super-space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s0912003.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s0912004.png" /> endowed with an even linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s0912005.png" />. A superalgebra is said to be commutative (graded-commutative or supercommutative) if

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here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s0912007.png" /> is a parity, i.e. a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s0912008.png" />-grading.

The definition of a superalgebra can be generalized to include the case where the domain of scalars is an arbitrary commutative associative superalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s0912009.png" />.

Examples of associative superalgebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120010.png" /> are: the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120011.png" /> of matrices of the form

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120014.png" />, endowed with the natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120015.png" />-grading (cf. [[Super-space|Super-space]]); the [[Tensor algebra|tensor algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120016.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120017.png" />-graded module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120018.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120019.png" />; the [[Symmetric algebra|symmetric algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120020.png" /> of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120022.png" /> is the ideal generated by the elements of the form

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and the [[Exterior algebra|exterior algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120024.png" /> of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120025.png" /> (the latter two superalgebras are commutative).

A superalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120026.png" /> with a multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120027.png" /> is called a Lie superalgebra if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120028.png" />,

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120030.png" /></td> </tr></table>

(and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120031.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120033.png" />). In particular, there are no Lie superalgebras in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120034.png" />, only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120035.png" />-graded Lie algebras.

Examples. Any associative superalgebra endowed with commutation (the supercommutator difference)

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as the bracket operation; the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120037.png" /> of derivations of an arbitrary superalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120038.png" /> (i.e. of linear transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120039.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120040.png" />) with the operation of commutation. For any Lie superalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120041.png" /> there is an associative universal enveloping superalgebra, and the straightforward generalization of the [[Birkhoff–Witt theorem|Birkhoff–Witt theorem]] holds.

The classification of finite-dimensional simple Lie superalgebras over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120042.png" /> is known (see [[#References|[2]]], [[#References|[3]]]). They are divided into Lie superalgebras of classical type (characterized by the fact that the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120043.png" /> is reductive) and Lie superalgebras of Cartan type. The Lie superalgebras of classical type are exhausted by the following series of matrix algebras:

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for an even symmetric non-degenerate bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120048.png" />;

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for an odd symmetric non-degenerate bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120050.png" />;

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120052.png" /></td> </tr></table>

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where

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and certain exceptional algebras (of dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120059.png" />). The superalgebras of Cartan type are the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120060.png" /> and its supersubalgebras, analogues to the simple Lie graded algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120063.png" /> (cf. [[Lie algebra, graded|Lie algebra, graded]]).

The classification of real structures of simple Lie superalgebras and a description of semi-simple Lie superalgebras in terms of simple ones are also known.

The theory of linear representations of Lie superalgebras is essentially more complex than for Lie algebras in that representations of simple Lie superalgebras, as a rule, are not completely reducible, while irreducible representations of solvable Lie superalgebras need not be one-dimensional. A classification exists of the irreducible representations of simple finite-dimensional Lie superalgebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120064.png" /> in terms of the highest weights (see , [[#References|[2]]]), and an explicit description is known of the finite-dimensional representations, as well as for the [[Character formula|character formula]] for certain series of these algebras .

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  D.A. Leites,   "Lie superalgebras"  ''Josmar'' , '''30'''  (1984)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  D.A. Leites (ed.) , ''Seminar on supermanifolds'' , Kluwer  (1990)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.G. Kac,   "Lie superalgebras"  ''Adv. Math.'' , '''26'''  (1977)  pp. 8–96</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Scheunert,   "The theory of Lie superalgebras. An introduction" , Springer  (1979)</TD></TR></table>



====Comments====
A classification of the simple finite-dimensional Lie superalgebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120065.png" /> was obtained by V.G. Kac in 1975 (see [[#References|[2]]]).

A classification of the irreducible finite-dimensional representations of solvable Lie superalgebras may be found in [[#References|[2]]].

The irreducible finite-dimensional representations of a simple Lie superalgebra are divided into two classes: typical and atypical (exceptional). Characters of typical representations were computed in [[#References|[a1]]]. Characters of atypical representations are not known, not even in the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091200/s09120066.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.G. Kac,   "Representations of classical Lie superalgebras"  K. Bleuler (ed.)  et al. (ed.) , ''Differential Geometrical Methods in Mathematical Physics II'' , ''Lect. notes in math.'' , '''676''' , Springer  (1978)  pp. 597–626</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F.A. Berezin,   M.A. Shubin,   "The Schrödinger equation" , Kluwer  (1991)  (Translated from Russian)  (Supplement 3: D.A. Leites, Quantization and supermanifolds)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F.A. Berezin,   "Introduction to superanalysis" , Reidel  (1987)  (Translated from Russian)</TD></TR></table>