Questions tagged [bimodules]

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11
votes
1answer
125 views

Are algebras with invertible linear duals always Frobenius?

Let $A$ be a finite dimensional algebra over a ground field $k$. The linear dual $A^* = Hom_k(A,k)$ is naturally an $A$-$A$ bimodule. I am interested in those algebras such that $A^*$ is an invertible ...
1
vote
0answers
59 views

MAGMA-question concerning dual modules of bimodules

Let $G$ be a finite group and let $H_1,H_2\leq G$. Let char$(k)=p>0$, $k$ a field, large enough. Let $T$ be a $(kH_1, kH_2)$-bimodule given in MAGMA. Moreover, let $T$ be finitely generated ...
1
vote
2answers
249 views

A description of the kernel of projection map from the tensor algebra to the symmetric algebra $T(V)\to S(V)$

Let $V$ be a vector space over some arbitrary field. Let $T(V)$ and $S(V)$ be the tensor and symmetric algebras over $V$. We have the projection map $T(V)\to S(V)$, given by $x_1\otimes\cdots\otimes ...
7
votes
3answers
589 views

Dual of a bimodule

For a noncommutative ring $R$, and an $R$-$R$-bimodule $B$, is there a "correct/natural" notion of a dual bimodule? I am interested, really, when $B$ is projective as a left $R$-module. Note: ...
6
votes
1answer
180 views

“Left Brace Module”

Let $A$ be an algebra over the brace tree operad and $M$ a module over some base ring. Is there a good notion of a "left brace module" over a brace algebra? I do not think the definition of a module ...
6
votes
0answers
119 views

Characterizing fullness of a von Neumann algebra by the topology of its bimodules

Let $\mathcal{M}$ be a $\mathrm{II}_1$ factor. Among other characterizations, it is said to be full iff the adjoint map: $$ \mathrm{Ad}: U(\mathcal{M})/\mathbb{T} \longrightarrow \mathrm{Aut}(\...
2
votes
0answers
47 views

Is there an established name for bi-module morphisms that swap the module structures?

Let $P,Q$ denote two $R$-bimodules where $R$ is a ring, for my purposes commutative. I'll write the left $R$-module structure by multiplication on the left and analogously for the right structure. ...
1
vote
0answers
105 views

Decomposition of Banach bimodules of Banach algebras

Let $A$ and $B$ be Banach algebras, $\theta:A\rightarrow \mathbb{C}$ be a character (i.e., a multiplicative linear functional) and $A\oplus _{\theta} B$ be the $l^1$-direct sum of $A$ and $B$ equipped ...
5
votes
1answer
226 views

Slick construction of Hochschild complex

Let $R$ be a $k$-algebra and $M$ be an $(R,R)$-bimodule. Let $[n] \mapsto M \otimes R^{\otimes n}$ be the simplicial $k$-module which defines the Hochschild homology $H_*(R,M)$. Is it possible to ...
2
votes
0answers
54 views

Comparing right and left quasi-representable bimodules

Let $\mathcal V$ be your favourite (closed, symmetric) monoidal model category. To fix ideas, set $\mathcal V = \mathrm{Ch}(k)$, the category of chain complexes over a fixed commutative ring. Given a $...
5
votes
1answer
216 views

When are countably generated Hilbert modules generated by c.p.c. order zero maps?

Throughout let $B$ be a stable C*-algebra, i.e. $B\cong B\otimes K$, where $K$ is the C*-algebra of compact operators on an infinite dimensional separable Hilbert space. It is well-known that any ...
8
votes
0answers
249 views

Does this kind of non-noetherian bimodule exist?

Question: Do there exist simple rings $R$ and $S$ (i.e., rings with no proper nonzero ideals) and an $(R,S)$-bimodule $M$ such that $M$ is finitely generated both as a left $R$-module and a right $...
10
votes
0answers
252 views

When were bordered Heegaard Floer homology's DA bimodules invented?

This is less of a strictly mathematical question and more of a reference request. In their paper "Bordered Heegaard Floer Homology (http://arxiv.org/pdf/0810.0687.pdf)," Lipshitz-Ozsvath-Thurston (...
4
votes
2answers
748 views

Strong Morita Equivalence and Morphisms Between $ C^{*} $-Algebras

If $ A $ and $ B $ are $ C^{*} $-algebras, then they are strongly Morita equivalent if there exist a $ (B,A) $-bimodule $ E $ and an $ (A,B) $-bimodule $ F $ such that $$ E \otimes_{A} F \cong B \quad ...
3
votes
2answers
343 views

Morphisms between $K_0$

I suppose this is a question with a well known answer. Suppose $A$ and $B$ are two algebras over some field and there is a map $$ f: \operatorname{K_0}(A) \to \operatorname{K_0}(B) $$ is it ...
3
votes
1answer
229 views

Bimodule version of IBN

Hello all, Does anyone have an example in mind of a ring $R$ for which $R^n\cong R^m$ as $R,R$ bimodules for some positive integers $n\neq m$? I would be a little surprised if someone showed no such ...
13
votes
2answers
1k views

Intrinsic characterization of Soergel bimodules?

A Soergel bimodule (for $S_n$) is a bimodule over $R = \mathbb{Q}[x_1,\dots,x_n]$ which appears as a summand/grading shift of tensor products of the basic bimodules $$B_{i,i+1} = R \otimes_{i,i+1} R$$ ...
18
votes
2answers
776 views

Is the endomorphism algebra of a dualizable bimodule necessarily finite dimensional?

Let $k$ be field. Let $A$, $B$ be $k$-algebras, and let ${}_AM_B$ be a dualizable bimodule. Pre-Question (too naive): Is the algebra of $A$-$B$-bilinear endomorphisms of $M$ necessarily finite ...
2
votes
1answer
172 views

The growthrate of the homology of $H_*(M^{\otimes_A n})$ for a DG-bimodule $M$

Suppose you have an DG-algebra $A$, and a DG-bimodule $M$ over $A$. Under which conditions will the rank of the bimodules $H_*(M^{\otimes_A n})$ will grow exponentially in terms of $n$? Here $M^{\...
5
votes
1answer
314 views

One-parameter semigroups of bimodules

Suppose M is a von Neumann algebra. Consider a monoidal category of bimodules over M. Here a bimodule is a Hilbert space with two normal representations of M. The monoidal structure is given by Connes'...
24
votes
8answers
2k views

Bimodules in geometry

Grothendieck's approach to algebraic geometry in particular tells us to treat all rings as rings of functions on some sort of space. This can also be applied outside of scheme theory (e.g., Gelfand-...