All Questions
Tagged with universal-algebra lie-algebras
10 questions
17
votes
1
answer
2k
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So, did Poincaré prove PBW or not?
This seems to be a question whose answer depends on whom you ask. Maybe we can come up with a final answer?
It is known that Poincaré, at least, invented something that can be called Poincaré-...
- 33.8k
14
votes
1
answer
411
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characterization of subalgebras of universal enveloping algebra coming from Lie subalgebras
Let $\mathfrak{g}$ be a Lie algebra and $\mathfrak{g}'$ its subalgebra. Then the universal enveloping algebra $U(\mathfrak{g}')$ can be canonically embedded into $U(\mathfrak{g})$, that of $\mathfrak{...
- 2,709
9
votes
2
answers
377
views
Reference for an old result of P. M. Cohn
As it was shown by Malcev, unlike the commutative case, in which every domain can be embedded in a field, there are noncommutative domains that can't be embedded in a division ring.
For noncommutative ...
- 3,318
9
votes
1
answer
437
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$U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ over a ring containing $\mathbb{Q}$
While the Poincaré-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is known (see also here) that it holds for arbitrary modules if the ground ring is ...
- 33.8k
4
votes
1
answer
214
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The question about elementary equivalence of free products
Let $A,B,C,D$ be algebraic systems and $A$ and $B$ be elementary equivalent as well as $C$ and $D$. Are free products of $A,C$ and $B,D$ elementary equivalent if
$A,B,C,D$ are groups, or
$A,B,C,D$ ...
- 41
4
votes
0
answers
172
views
Poincaré-Birkhoff-Witt theorem for Leibniz algebras
Leibniz algebras can be seen as a non-skew-symmetric generalization of Lie algebras. I have already taken a look at some papers related to Leibniz algebras and extending main results of Lie algebras ...
- 367
3
votes
0
answers
95
views
Lie structure over $R$-module
In Higgins' paper Baer invariants and the Birkhoff-Witt theorem (J. Algebra 11 (1969) 469–482, doi:10.1016/0021-8693(69)90086-6) the following definition is given:
A Lie structure over the $R$-module ...
- 427
1
vote
0
answers
106
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How does $R \equiv 1\ (\text {mod}\ h)\ $?
Definition $:$ Let $H$ be a Hopf algebra. An invertible element $R \in H \otimes H$ is called a coboundary structure on $H$ if
$(1)$ $\Delta^{\text {op}} = R \Delta R^{-1},$
$(2)$ $R_{21} = R^{-1},$
$(...
- 41
1
vote
0
answers
55
views
Questions of the paper "PBW-pairs of varieties of linear algebras"
I am reading this paper "PBW-pairs of varieties of linear algebras", the link is here:https://www.tandfonline.com/doi/abs/10.1080/00927872.2012.720867.
At page 672, there is a definition of PBW-pair.
...
- 775
0
votes
1
answer
654
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Book on algebraic structures
What is the most complete book on algebraic structures that deals with the complete taxonomy from magmas to Lie algebras and inner product spaces?
- 19