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Results tagged with ra.rings-and-algebras
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user 496277
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
9
votes
1
answer
232
views
Formal smoothness of path algebras and connections
Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if
$$
\Omega^1_kA = \operatorname{Ker}(\mat …
- 985
7
votes
0
answers
280
views
Lie algebra cohomology of the space of vector fields
For a (closed and oriented) manifold $M$, the first Lie algebra cohomology $H^1(\mathrm{Vect}(M),C^\infty(M))$ of the space of vector fields with coefficients in smooth functions is isomorphic to $H^1 …
- 985
5
votes
0
answers
299
views
Connections in non-commutative geometry
Let $K$ be a field, $A$ a unital associative $K$-algebra and $M$ a left $A$-module. A connection on $M$ is a $K$-linear map $\nabla:M\to \Omega^1A\otimes_AM$ which satisfies the Leibniz rule. Accordin …
- 985
5
votes
1
answer
242
views
Infinite-dimensional, non-unital Frobenius algebras
A Frobenius algebra is a tuple $(A,\mu,\delta,\eta,\varepsilon)$, where $A$ is a vector space over some field, $(A,\mu,\eta)$ a unital associative algebra, and $(A,\delta,\varepsilon)$ a counital coas …
- 985
5
votes
0
answers
123
views
Lie algebra cohomology of formal non-commutative vector fields
Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is isomorph …
- 985
5
votes
0
answers
205
views
Cohomology of representation varieties and the Hochschild cohomology
Let $k$ be a field, $A$ a $k$-algebra, and $V$ a $k$-vector space. Then we can consider the representation varieties of $A$ on $V$: $\mathrm{Hom}_{k\textrm{-alg}}(A, \mathfrak{gl}(V))$ and $\mathrm{Ho …
- 985
4
votes
1
answer
87
views
Lie algebra with finitely generated envelope
If a Lie algebra $\mathfrak g$ is finitely generated, its enveloing algebra $U\mathfrak g$ is finitely generated as an associative algebra. In fact, taking the enveloping algebra of the surjection $\m …
- 985
4
votes
2
answers
391
views
Units of the group algebra of a free group
Let $K$ be a field of characteristic zero and $F_n$ be a free group of rank $n$. What is known about the group of units $K[F_n]^\times$?
In the case of $n=1$, there are only trivial units: $K[F_1]^\ti …
- 985
3
votes
1
answer
417
views
Is Malcev completion an embedding?
The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by
$$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$
the group-like part of the completed (by the augmentation …
- 985
3
votes
0
answers
100
views
Pairing on a Koszul dual pair
Let $A$ be a graded quadratic algebra over a field $k$, and suppose that it admits the Koszul dual $A^!$. I want to know if there is a natural pairing $A\otimes A^!\to k$ or something similar to this. …
- 985
3
votes
0
answers
125
views
Trace map on Ext group
Let $R$ be a (possibly non-commutative) unital ring and $M$ be a perfect left $R$-module. Then, we have the trace map
$$
\operatorname{Tr}\colon \mathrm{Hom}_R(M,M)\to R/[R,R]\,.
$$
According to the a …
- 985
2
votes
Accepted
Infinite-dimensional, non-unital Frobenius algebras
It is studied under the name "nearly Frobenius algebras" in this paper. In Example 3.3, the algebra $\mathbb{C}[[x,x^{-1}]]$ of Laurent series can be endowed with (countably) many unital but not co-un …
- 985