# Questions tagged [super-algebra]

The super-algebra tag has no usage guidance.

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### How to formulate supercommutativity in a characteristic free way?

I would never dare posting this here, but the question https://math.stackexchange.com/q/3019853/214353 on math.SE did not receive any feedback (except for 13 views and one upvote) since November 30, ...

**0**

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**1**answer

128 views

### Classification of finite-dimensional real super C*-algebras

The title says it all. I feel like one should be able to find this somewhere, but every time I try to google, I just get results for "super Lie algebras". Does anybody know a reference? I am not so ...

**16**

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**1**answer

388 views

### Odd primary dual Steenrod algebra

My question is related to
this, this, and this older questions.
Let $\mathcal A_*$ be the dual Steenrod algebra.
This is a super-commutative Hopf algebra, and so its $Spec$ is an algebraic super-group....

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40 views

### symmetric polynomials for Super Hecke Clifford algebra

Fix a natural number $n$. In https://arxiv.org/abs/1107.1039, §3.5, Kang/Kashiwara/Tsuchioka define a (version of a) Hecke Clifford superalgebra. It is the superalgebra with the following generators:
...

**2**

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**1**answer

197 views

### Noncommutative cohomology of flag varieties

Consider the Grassmannian $Gr(n, N)$ of $n$-dimensional subspaces of $\mathbf C^N$. Its cohomology ring is isomorphic to $\mathbf C[x_1, \ldots, x_n, \bar x_1, \ldots \bar x_{N-n}]/I_{n,N}$, wherethe ...

**3**

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**1**answer

64 views

### Generators of odd polynomial superalgbras

I am getting myself acquainted to superalgebras. One often comes across odd polynomial rings of the form
$$k\langle x_i \rangle_{i\in I} / (x_ix_j -(-1)^{|x_i||x_j|} x_jx_i)$$
for some index set $I$,...

**3**

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**1**answer

139 views

### Definition of orthosymplectic supergroups

I found two versions of definitions of orthosymplectic supergroups. It seems that they are not equivalent. I don't know which version of the definition is standard.
The first version of the ...

**1**

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**1**answer

114 views

### Super version of Poisson brackets of tensor products

Let $A$ be a Poisson super algebra ($A$ is a super algebra and $A$ satisfies super Jacobi identity, super commutativity, super Leibniz rule).
Super version of the product of two tensor products is
\...

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**0**answers

52 views

### $\mathbb{Z}_2$ graded analog of row operations for supermatrices

I'm working on some research involving supermatrices, and I was wondering if there was a $\mathbb{Z}_2$ graded analog of row operations for supermatrices.
It seems to me that it makes sense to have ...

**9**

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**1**answer

272 views

### Super-plethysm?

Let $U$ be a representation of $S_m$ and $V$ a representation of $S_n$. Then the representation $\operatorname{Ind}_{S_m\wr S_n}^{S_{mn}}(U^{\otimes{n}}\otimes V)$ has a nice interpretation in terms ...

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201 views

### Is the (super-)symmetric power of the exterior algebra free?

Let $V$ be a vector space over $k$ of dimension $m$. (I'm only interested in the case $k=\mathbb{Q}$.) Let $R:=\Lambda^*V$ be the exterior algebra. It carries the structure of a supercommutative ring: ...

**3**

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**1**answer

190 views

### Supercommutator of exterior multiplication operators and their adjoints

Let $\mathfrak{h}$ be a complex Hilbert space and consider Grassmann algebra $\mathcal{F}=\bigwedge\mathfrak{h}$ with its induced inner product. For $\omega\in\mathcal{F}$ we also consider the ...

**1**

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**1**answer

119 views

### Non-graded representations over Lie superalgebra $\mathfrak{gl}(m,n)$

I have the following questions:
Let $m,n$ be positive integers. Consider representations over the general linear Lie super-algebra $\mathfrak{gl}(m,n)$. Namely, modules over the associative algebra $U(...

**4**

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**1**answer

289 views

### Quasicoherent sheaves on superschemes

I am interested in learning about super algebraic geometry (some objects I am studying seem to be naturally superstacks, at least in some sense). What would be the best reference for the subject? I am ...

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478 views

### What is the role of fiber functor in Deligne's theorem on Tannakian categories?

The theorem states that, for a field $k$ of characteristic 0, any $k$-linear tensor category with $End(1)=k$ satisfying a condition that each object is annihilated by a Schur functor, is equivalent to ...

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107 views

### How well is the classification of low-dimensional semisimple Hopf superalgebras (or braided Hopf algebras) understood?

As far as I know, low-dimensional semisimple Hopf algebras are classified (along with non-semisimple ones) up to dimension 60, with the first example of a semisimple Hopf algebra not coming from a ...

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126 views

### Supertrace on Weyl algebra

Consider Weyl algebra, i.e. the algebra of $x^i$ and $p_i=\frac{\partial}{\partial x^i}$,
its elements are differential operators $F(x,p)$. Weyl algebra is $\mathbb{Z}_2$ graded, hence one ask if ...

**4**

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100 views

### Tensor categories with integer rank

I wonder the state of the following conjecture in "Deformation theory, Kontsevich, Soibelman":
Conjecture 3.3.5. Rigid [abelian symmetric] tensor categories [over an algebraically closed field $k$] ...

**4**

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**1**answer

116 views

### Does every equivalence class in a Brauer-Wall group have a (graded) division algebra?

It is known that each equivalence class in a Brauer group has a division algebra (or, in other words, every central simple algebra is isomorphic to $\mathrm{Mat}(D)$ for some division algebra $D$). Is ...

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**1**answer

401 views

### 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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168 views

### FRT construction in the case of super algebras

I'm looking on papers which are talking about the super quantum algebra osp(2|1). I want to understand how one applies the FRT construction in the case of osp(2|1).
Of course there is a super ...

**3**

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**0**answers

189 views

### Super group GL(m,m) and Koszul (deRham) complex. (Is there brigde from super-math to usual-math ?)

Consider vector space with coordinates x1, ... xn. Consider polynomial deRham complex (also known as Koszul complex) which is generated by xi and dx_i. As an algebra it is just $C[x_i]\otimes \Lambda [...

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**2**answers

770 views

### How can I write down a point in the Berezinian of a super vector space?

A vector space $V$ of dimension $n$ has an associated determinant line $Det(V)$.
An element of $Det(V)$ is represented as a (formal limear combination) of expresstions of the form
$v_1 \wedge v_2 \...

**3**

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**1**answer

1k views

### Witten Index, letter partition function and superconformal representations.

Except in a few papers I have seen so little written about this that I am not sure I can even frame this question properly.
I would like to know of expository references and explanations on the ...

**6**

votes

**3**answers

737 views

### Derivations of C(X)? or Why Must Supermanifolds be Smooth?

What are the derivations of the algebra of continuous functions on a topological manifold?
A supermanifold is a locally ringed space (X,O) whose underlying space is a smooth manifold X, and whose ...

**7**

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**1**answer

476 views

### Does some version of U_q(gl(1|1)) have a basis like Lusztig's basis for \dot{U(sl_2)}?

There's a non-unital algebra $\dot{U}$ formed from $U_q (sl_2)$ by including a system of mutually orthogonal idempotents $1_n$, indexed by the weight lattice. You can think of this as a category with ...