Questions tagged [superalgebra]

For questions about superalgebra, which is a kind of graded algebra.

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What (mathematical) field does a (physics) superfield belong?

A superfield, $\phi(x,\theta)$, is define by a Grassman-even (i.e. commuting) function of a set of commuting variables ($x^\mu$) and a set of Grassman variables ($\theta^\alpha$ where $\theta^\alpha\...
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103 views

super commutation of matrices

Let $M_{p|q}(\mathbb{C}) = M_{p|q}(\mathbb{C})_0 \oplus M_{p|q}(\mathbb{C})_1$ be the super algebra of all $(p+q) \times (p+q)$ matrices. Let $A, B \in M_{p|q}(\mathbb{C})$ (not necessarily ...
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How to formulate supercommutativity in a characteristic free way?

I believe I've seen an answer to this somewhere (some Deligne notes?) but cannot recover it. In the $\mathbb Z/2$-graded formulation, supercommutativity means that two odd degree elements anticommute ...
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39 views

Defining supermanifolds by equations

I would like to in what sense supermanifolds may be defined by systems of equations in the ordinary flat superspace. I am particularly interested in the approach to supergeometry via ringed spaces. ...
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64 views

Is Whitehead lemma true for super Lie algebras?

Classical Whitehead lemma states that if $\mathfrak g$ is a finite-dimensional complex Lie algebra and $M$ is a finite-dimensional $\mathfrak g$-module, then first cohomology group $H^1(\mathfrak g, M)...
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22 views

Is every algebra generated by some elements $\mathbb{Z}_2$-graded?

Let $A$ be an unital associative algebra generated by some elements $a_1,...,a_n$. Is it always possible to come up with a $\mathbb{Z}_2$-grading for $A$? For example does it work if I consider the $...
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59 views

Representations of superalgebras

If I have a superalgebra $A$ and consider the category of finite-dimensional $A$-modules. Is this category the same as the category of finite-dimensional $A$-modules which are super-vector spaces? ...
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82 views

The super group $GL(1|1)$

It is difficult to find information on super groups and I have built my knowledge from various sources. I have the following questions. $GL(1|1)$ is defined as the group of invertible linear ...
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38 views

What is contragredient about contragrediant Lie superalgebras

Among the Lie superalgebras there is the class of contragredient Lie superalgebras. Roughly speaking these are those Lie superalgebras that can be defined with a matrix $a_{ij}$ and commutation ...
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282 views

A Grassmann-Variable Identity from Wikipedia

I found this identity on Wikipedia: $$\int\exp\left[\theta^T A\eta+\theta^T J+K^T\eta\right]d\theta d\eta =\det A\exp\left[-K^TA^{-1}J\right]$$ where the integration variables are Grassmann variable....
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Supercommutative algebras except commutative algebras and exterior algebras

When I first saw a definition of a supercommutative algebra, first example that came to my mind was an exterior algebra on some vector space. Of course, purely even supercommutative algebra is just a ...
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77 views

Representations of $\mathbb{Z}/2$ as super vector spaces?

I heard someone say that representations of $\mathbb{Z}/2$ "are" super vector spaces. As far as I understand, super vector spaces are $\mathbb{Z}/2$-graded vector spaces, so my question is whether ...
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344 views

What is the $\frac{1}{2}$ representation of $U(1)$?

This may be a silly question, and so I apologize in advance. But it stems from a reading of section 2 (page 5) of the physics paper, "Counting chiral primaries in N=1 d=4 superconformal field ...
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81 views

What is the commutator in $gl(m|n)$?

For Lie algebra $gl(m)$, the commutator is \begin{align} [E_{ij}, E_{kl}] = \delta_{jk}E_{il} - \delta_{li}E_{kj}. \end{align} What is the commutator $[E_{ij}, E_{kl}]$ in Lie superalgebra $gl(m|n)$? ...
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Orthosymplectic Lie Superalgebra

I am trying to work out a presentation for the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2n)$. I am following Musson's book "Lie Superalgebras and Enveloping Algebras". From what I ...
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196 views

Best texts on supermathematics for a mathematician?

I'm an undergraduate who's doing some summer mathematics research, and it looks like I need some information on Berezenians and supermatrices as well as supermathematics in general. The only text I ...
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32 views

How is this algebra a superalgebra?

In this set of notes http://arxiv.org/pdf/0809.1380.pdf on page ix he seems to be claiming that the algebra $\mathrm{End}(V)[[z,z^{-1}]]$ is a superalgebra (where $V$ is any vector space over $\...
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Finite dimensional, irreducible representations of the Lie superalgebra gl(1|1)

I am wondering how the finite dimensional, irreducible representations of the Lie superalgebra gl(1|1) are parametrized. I understand that they are all highest weight, and that the only non-trivial ...
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66 views

When are minimal faithful modules over algebras unique?

Let $A$ be a unital associative algebra over $\mathbb{C}$. We say an $A$-module $V$ is "minimal faithful" if (i) $V$ is faithful and (ii) $V$ does not have a proper submodule that is faithful. First ...
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Classical notions on super Riemann surfaces

A super Riemann surface $M$ is a complex supermanifold of dimension 1|1 with a superconformal structure given locally by an odd vector field $D=\frac{\partial}{\partial\theta}+\theta\frac{\partial}{\...
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166 views

The exterior algebra is a superalgebra?

Can someone explain how the exterior algebra of a vector space or a module over a commutative ring is a superalgebra? The exterior algebra has an obvious $\mathbb{Z}$-grading, but I don't see where ...
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59 views

Sign convention for derivatives in a $\mathbb{Z}_2$ graded space

Suppose $V=V_0\oplus\theta V_1$ is a $\mathbb{Z}_2$ graded super vector space. Note: Since $\theta^2=0$, this implies $\theta\mathrm{d}\theta=-\mathrm{d}\theta\cdot\theta$. However, I wish to know ...
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256 views

Geometric interpretation of Supersymmetry

Is there a geometric interpretation of supersymmetry? I.e., if one has a manifold $\mathcal {M} $, and there are $\mathcal {N} $ SUSY generators, then is there a geometric interpretation of the SUSY ...
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A good reference for learning about super-differentiation & super-integration?

I've looked at a couple of books for basic information for super-differentiation & super-integration - Rogers Supermanifolds, and Khrennikovs Superanalysis. Unfortunately both books lack a clear ...
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98 views

Question about the parity of the ghost number operator in BRST quantization

Given a Lie algebra $[K_i,K_j]=f_{ij}^k K_k$, and ghost fields satisfying the anticommutation relations $\{c^i,b_j\}=\delta_j^i$, the ghost number operator is then $U=c^ib_i$ (duplicate indices are ...
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Grassmann numbers as eigenvalues of nilpotent operators?

The following question is motivated by the construction of the fermionic path/field integral, as done for example in Altland & Simons "Condensed Matter Field Theory". Consider the vector space $\...