Editing Superalgebra

{{Short description|Algebraic structure used in theoretical physics}}
In [[mathematics]] and [[theoretical physics]], a '''superalgebra''' is a '''Z'''<sub>2</sub>-[[graded algebra]].<ref>{{harvnb|Kac|Martinez|Zelmanov|2001|p=3}}</ref> That is, it is an [[algebra (ring theory)|algebra]] over a [[commutative ring]] or [[field (mathematics)|field]] with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.

The prefix ''super-'' comes from the theory of [[supersymmetry]] in theoretical physics. Superalgebras and their representations, [[supermodule]]s, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called [[super linear algebra]]. Superalgebras also play an important role in related field of [[supergeometry]] where they enter into the definitions of [[graded manifold]]s, [[supermanifold]]s and [[superscheme]]s.

==Formal definition==

Let ''K'' be a [[commutative ring]]. In most applications, ''K'' is a [[field (mathematics)|field]] of [[Characteristic (algebra)|characteristic]] 0, such as '''R''' or '''C'''.

A '''superalgebra''' over ''K'' is a [[module (mathematics)|''K''-module]] ''A'' with a [[direct sum of modules|direct sum]] decomposition
:<math>A = A_0\oplus A_1</math>
together with a [[bilinear map|bilinear]] multiplication ''A'' &times; ''A'' &rarr; ''A'' such that
:<math>A_iA_j \sube A_{i+j}</math>
where the subscripts are read [[Modular arithmetic|modulo]] 2, i.e. they are thought of as elements of '''Z'''<sub>2</sub>.

A '''superring''', or '''Z'''<sub>2</sub>-[[graded ring]], is a superalgebra over the ring of [[integer]]s '''Z'''.

The elements of each of the ''A''<sub>''i''</sub> are said to be '''homogeneous'''. The '''parity''' of a homogeneous element ''x'', denoted by {{abs|''x''}}, is 0 or 1 according to whether it is in ''A''<sub>0</sub> or ''A''<sub>1</sub>. Elements of parity 0 are said to be '''even''' and those of parity 1 to be '''odd'''. If ''x'' and ''y'' are both homogeneous then so is the product ''xy'' and <math>|xy| = |x| + |y|</math>.

An '''associative superalgebra''' is one whose multiplication is [[associative]] and a '''unital superalgebra''' is one with a multiplicative [[identity element]]. The identity element in a unital superalgebra is necessarily even. Unless otherwise specified, all superalgebras in this article are assumed to be associative and unital.

A '''[[commutative superalgebra]]''' (or supercommutative algebra) is one which satisfies a graded version of [[commutativity]]. Specifically, ''A'' is commutative if

:<math>yx = (-1)^{|x||y|}xy\,</math>

for all homogeneous elements ''x'' and ''y'' of ''A''. There are superalgebras that are commutative in the ordinary sense, but not in the superalgebra sense. For this reason, commutative superalgebras are often called ''supercommutative'' in order to avoid confusion.<ref>{{harvnb|Varadarajan|2004|p=87}}</ref>

==Sign conventions==
When the '''Z'''<sub>2</sub> grading arises as a "rollup" of a '''Z'''- or '''N'''-[[graded algebra]] into even and odd components, then two distinct (but essentially equivalent) sign conventions can be found in the literature.<ref name="deligne">See [http://www.math.ias.edu/QFT/fall/bern-appen1.ps Deligne's discussion] of these two cases.</ref> These can be called the "cohomological sign convention" and the "super sign convention". They differ in how the antipode (exchange of two elements) behaves. In the first case, one has an exchange map
:<math>xy\mapsto (-1)^{mn+pq} yx</math>
where <math>m=\deg x</math> is the degree ('''Z'''- or '''N'''-grading) of <math>x</math> and <math>p</math> the parity. Likewise, <math>n=\deg y</math> is the degree of <math>y</math> and with parity <math>q.</math> This convention is commonly seen in conventional mathematical settings, such as differential geometry and differential topology.  The other convention is to take 
:<math>xy\mapsto (-1)^{pq} yx</math>
with the parities given as <math>p=m\bmod 2</math> and <math>q=n\bmod 2</math> the parity. This is more often seen in physics texts, and requires a parity functor to be judiciously employed to track isomorphisms. Detailed arguments are provided by [[Pierre Deligne]]<ref name="deligne"/>

==Examples==
*Any algebra over a commutative ring ''K'' may be regarded as a purely even superalgebra over ''K''; that is, by taking ''A''<sub>1</sub> to be trivial.
*Any '''Z'''- or '''N'''-[[graded algebra]] may be regarded as superalgebra by reading the grading modulo 2. This includes examples such as [[tensor algebra]]s and [[polynomial ring]]s over ''K''.
*In particular, any [[exterior algebra]] over ''K'' is a superalgebra. The exterior algebra is the standard example of a [[supercommutative algebra]].
*The [[symmetric polynomials]] and [[alternating polynomials]] together form a superalgebra, being the even and odd parts, respectively. Note that this is a different grading from the grading by degree.
*[[Clifford algebra]]s are superalgebras. They are generally noncommutative.
*The set of all [[endomorphism]]s (denoted <math>\mathbf{End} (V) \equiv \mathbf{Hom}(V,V)</math>, where the boldface <math>\mathrm {Hom}</math> is referred to as ''internal'' <math>\mathrm {Hom}</math>, composed of ''all'' linear maps) of a [[super vector space]] forms a superalgebra under composition.
*The set of all square [[supermatrices]] with entries in ''K'' forms a superalgebra denoted by ''M''<sub>''p''|''q''</sub>(''K''). This algebra may be identified with the algebra of endomorphisms of a free supermodule over ''K'' of rank ''p''|''q'' and is the internal Hom of above for this space.
*[[Lie superalgebra]]s are a graded analog of [[Lie algebra]]s. Lie superalgebras are nonunital and nonassociative; however, one may construct the analog of a [[universal enveloping algebra]] of a Lie superalgebra which is a unital, associative superalgebra.

==Further definitions and constructions==

===Even subalgebra===

Let ''A'' be a superalgebra over a commutative ring ''K''. The [[submodule]] ''A''<sub>0</sub>, consisting of all even elements, is closed under multiplication and contains the identity of ''A'' and therefore forms a [[subalgebra]] of ''A'', naturally called the '''even subalgebra'''. It forms an ordinary [[algebra (ring theory)|algebra]] over ''K''.

The set of all odd elements ''A''<sub>1</sub> is an ''A''<sub>0</sub>-[[bimodule]] whose scalar multiplication is just multiplication in ''A''. The product in ''A'' equips ''A''<sub>1</sub> with a [[bilinear form]]
:<math>\mu:A_1\otimes_{A_0}A_1 \to A_0</math>
such that
:<math>\mu(x\otimes y)\cdot z = x\cdot\mu(y\otimes z)</math>
for all ''x'', ''y'', and ''z'' in ''A''<sub>1</sub>. This follows from the associativity of the product in ''A''.

===Grade involution===

There is a canonical [[Involution (mathematics)|involutive]] [[automorphism]] on any superalgebra called the '''grade involution'''. It is given on homogeneous elements by
:<math>\hat x = (-1)^{|x|}x</math>
and on arbitrary elements by
:<math>\hat x = x_0 - x_1</math>
where ''x''<sub>''i''</sub> are the homogeneous parts of ''x''. If ''A'' has no [[torsion (algebra)|2-torsion]] (in particular, if 2 is invertible) then the grade involution can be used to distinguish the even and odd parts of ''A'':
:<math>A_i = \{x \in A : \hat x = (-1)^i x\}.</math>

===Supercommutativity===

The '''[[supercommutator]]''' on ''A'' is the binary operator given by
:<math>[x,y] = xy - (-1)^{|x||y|}yx</math>
on homogeneous elements, extended to all of ''A'' by linearity. Elements ''x'' and ''y'' of ''A'' are said to '''supercommute''' if {{nowrap|1=[''x'', ''y''] = 0}}.

The '''supercenter''' of ''A'' is the set of all elements of ''A'' which supercommute with all elements of ''A'':
:<math>\mathrm{Z}(A) = \{a\in A : [a,x]=0 \text{ for all } x\in A\}.</math>
The supercenter of ''A'' is, in general, different than the [[center of an algebra|center]] of ''A'' as an ungraded algebra. A commutative superalgebra is one whose supercenter is all of ''A''.

===Super tensor product===

The graded [[tensor product of algebras|tensor product]] of two superalgebras ''A'' and ''B'' may be regarded as a superalgebra ''A'' &otimes; ''B'' with a multiplication rule determined by:
:<math>(a_1\otimes b_1)(a_2\otimes b_2) = (-1)^{|b_1||a_2|}(a_1a_2\otimes b_1b_2).</math>
If either ''A'' or ''B'' is purely even, this is equivalent to the ordinary ungraded tensor product (except that the result is graded). However, in general, the super tensor product is distinct from the tensor product of ''A'' and ''B'' regarded as ordinary, ungraded algebras.

==Generalizations and categorical definition==

One can easily generalize the definition of superalgebras to include superalgebras over a commutative superring. The definition given above is then a specialization to the case where the base ring is purely even.

Let ''R'' be a commutative superring. A '''superalgebra''' over ''R'' is a [[supermodule|''R''-supermodule]] ''A'' with a ''R''-bilinear multiplication ''A'' &times; ''A'' &rarr; ''A'' that respects the grading. Bilinearity here means that
:<math>r\cdot(xy) = (r\cdot x)y = (-1)^{|r||x|}x(r\cdot y)</math>
for all homogeneous elements ''r'' &isin; ''R'' and ''x'', ''y'' &isin; ''A''.

Equivalently, one may define a superalgebra over ''R'' as a superring ''A'' together with an superring homomorphism ''R'' &rarr; ''A'' whose image lies in the supercenter of ''A''.

One may also define superalgebras [[category theory|categorically]]. The [[category (mathematics)|category]] of all ''R''-supermodules forms a [[monoidal category]] under the super tensor product with ''R'' serving as the unit object. An associative, unital superalgebra over ''R'' can then be defined as a [[monoid (category theory)|monoid]] in the category of ''R''-supermodules. That is, a superalgebra is an ''R''-supermodule ''A'' with two (even) morphisms
:<math>\begin{align}\mu &: A\otimes A \to A\\ \eta &: R\to A\end{align}</math>
for which the usual diagrams commute.

== Notes ==
<references/>

==References==
*{{cite conference | author-link = Pierre Deligne | first1 = P. | last1 = Deligne |first2=J. W.|last2 = Morgan  | title = Notes on Supersymmetry (following Joseph Bernstein) | book-title = Quantum Fields and Strings: A Course for Mathematicians | volume = 1 | pages = 41–97 | publisher = American Mathematical Society | year = 1999 | isbn = 0-8218-2012-5}}

*{{cite book | first1 = V. G. | last1 = Kac | author-link1 = Victor Kac | first2 = C.| last2 = Martinez | first3 = E. | last3 = Zelmanov | author-link3 = Efim Zelmanov | year = 2001 | title = Graded simple Jordan superalgebras of growth one | series = Memoirs of the AMS Series | volume = 711 | publisher = AMS Bookstore | isbn = 978-0-8218-2645-4 | url = https://books.google.com/books?id=aJHUCQAAQBAJ&q=bibliogroup:%22Graded+simple+Jordan+superalgebras+of+growth+one%22}}

*{{cite book | last = Manin | first = Y. I. | author-link = Yuri Manin| title = Gauge Field Theory and Complex Geometry | publisher = Springer | location = Berlin | year = 1997 | edition = (2nd ed.) | isbn = 3-540-61378-1}}

*{{cite book|last=Varadarajan|first=V. S.|author-link=V. S. Varadarajan|title=Supersymmetry for Mathematicians: An Introduction|year=2004|publisher=American Mathematical Society|isbn=978-0-8218-3574-6|url=https://books.google.com/books?id=sZ1-G4hQgIIC&q=supersymmetry+for+mathematicians&pg=PA1|series=Courant Lecture Notes in Mathematics|volume=11}}

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[[Category:Algebras]]
[[Category:Super linear algebra]]