Conjunctive normal form to simple elementary algebra

I’m curious to know the computational complexity class of each step in this method of converting a CNF formula into simple elementary algebra.

An example: $ $ \phi=\left(x_1 \vee x_2 \right) \wedge \left(\neg x_1 \vee x_3 \right) \wedge \left(\neg x_2 \vee \neg x_3 \right) \tag{CNF} $ $ Let $ \neg a = 1-a$

Let $ a \vee b= a+b-ab$

Let $ a \wedge b = ab$

Then: $ $ \phi=\left(x_1 + x_2 -x_1x_2\right) \left(1-x_1+x_1x_3\right) \left(1-x_2x_3 \right) \tag{AFF} $ $ I refer to this step as algebraic factor form (AFF) (I am unfamiliar with any canonical terminology) Then expanding these brackets out gives $ $ \phi = x_1-{x_1}^2+x_2 – 2x_1 x_2 +{x_1}^2x_2 + {x_1}^2x_3-{x_2}^2x_3+2x_1{x_2}^2x_3-x_1^2x_2^2x_3-x_1^2x_2x_3^2-x_1x_2^2x_3^2 + x_1^2x_2^2x_3^2 \tag{EAF}$ $ Which is in elementary algebra form.

Finally, using $ {x_1}^2=x_1, \; \; {x_2}^2=x_2, \; \; {x_3}^2=x_3$ we get $ $ \phi = x_1-{x_1}+x_2 – 2x_1 x_2 +{x_1}x_2 + {x_1}x_3-{x_2}x_3+2x_1{x_2}x_3-x_1x_2x_3-x_1x_2x_3-x_1x_2x_3 + x_1x_2x_3$ $ Which simplifies to: $ $ \phi = x_2 – x_1x_2 + x_1x_3 – x_2x_3 \tag{SEAF}$ $ Which I call simple elementary algebra form.

If there are already established names for these formulas please let me know and I will amend asap.

So my question is: What are the computational complexity classes of each transformation in (CNF) $ \rightarrow$ (AFF) $ \rightarrow$ (EAF) $ \rightarrow$ (SEAF)

I’m interested to know which parts are P and which parts are NP

Thanks in advance for any answers, Ben

Minimal injective coresolution in the stable Auslander algebra

Let $ A$ be a finite dimensional (connected) quiver algebra. Let $ T(A)$ denote the full subcategory of coherent functors from $ mod-A$ to $ Ab$ that vanish on projective objects. $ T(A)$ is equivalent to the module category of the stable Auslander algebra of $ A$ when $ A$ is representation-finite.

The indecomposable projective objects of $ T(A)$ are $ \underline{Hom_A}(-,X)$ for any indecomposable non-projective object $ X$ and this is non-injective if and only if $ X$ has codominant dimension equal to zero. So assume in the following that $ X$ is non-projective with codominant dimension zero (meaning that the projective cover of $ X$ is not injective).

Auslander and Reiten showed (see for example their article “stable equivalence of artin algebras”) that a injective coresolution of $ \underline{Hom_A}(-,X)$ is given as follows when $ 0 \rightarrow \Omega^1(X) \rightarrow P \rightarrow X \rightarrow 0$ determines the projective cover of $ X$ :

$ 0 \rightarrow \underline{Hom_A}(-,X) \rightarrow Ext_A^1(-,\Omega^1(X)) \rightarrow Ext_A^1(-,P) \rightarrow Ext_A^1(-,X) \rightarrow Ext_A^2(-,\Omega^1(X)) \rightarrow Ext_A^2(-,P) \rightarrow Ext_A^2(-,X) \rightarrow Ext_A^3(-,\Omega^1(X)) ….$

Questions:

  1. In general this injective coresolution will not be minimal. Is it known what the minimal injective coresolution looks like?

  2. When is the above injective coreslution minimal (or at least minimal until the first term is zero)? I suppose that this is for example the case when $ P$ and $ \Omega^1(X)$ are indecomposable.

  3. Is there are formula for calculation the injective dimension of $ \underline{Hom_A}(-,X)$ or at least when it is finite?

  4. In case every indecomposable summand of $ X, P$ and $ \Omega^1(X)$ has infinite injective dimension, does also $ \underline{Hom_A}(-,X)$ has infinite injective dimension?

infinite fold tensor product of universal enveloping algebra

Let $ \mathfrak a$ be a Lie algebra graded by the abelian semigroup $ S$ , then the universal enveloping algebra $ U(\mathfrak a)$ of $ \mathfrak a$ is $ S \sqcup \{0\}$ graded. I have the following questions.

  1. What is the definition of infinite fold tensor product ($ U(\mathfrak a)^{\otimes \infty}$ ) of $ U(\mathfrak a)$ and is this also $ S \sqcup \{0\}$ graded?
  2. If so, how to express the grade spaces of this infinite tensor product in terms of grade spaces of $ U(\mathfrak a)$ ?
  3. Is it a good notation $ U(\mathfrak a)^{\otimes \infty}$ ?

Thank you.

Absorption Law Proof by Algebra

I’m struggling to understand the absorption law proof and I hope maybe you could help me out.

The absorption law states that: $ X + XY = X$
Which is equivalent to $ (X \cdot 1) + (XY) = X$

No problem yet, it’s this next step that stumps me. How can I apply the distributive law when there are two “brackets”?

How can I manipulate $ (X \cdot 1) + (XY) = X$ to give me $ X \cdot (1+Y)$ ?

An image to clarify

I understand that the absorption law works. I would just like to see how the algebra proof works.

Thank you!

Finding the value of x in shapes, and finding the size of each unknown angle with algebra and variables [on hold]

CHAPTER 7 – SHAPES 1. Find the value of x. a) (3x −10)° (2x−5)° (x−5)° b) c) (x + 45)° (2x+40)° (2x+10)° (3 x )° 2. Find the size of each unknown angle. 3. Determine the value of x and find the size of each of the angles represented by an algebraic expression. 4. Determine the measure of each unknown angle.

Cute/striking application(s) of Snake Lemma outside homological algebra

I already asked this question on MSE here https://math.stackexchange.com/questions/3254184/cute-striking-applications-of-snake-lemma-outside-homological-algebra, but still received no answer. I hope I will be more lucky here.

When you teach algebra to students, it’s often easy to find cute/direct applications of “big” theorems to motivate how useful these results can be.

For example, group actions, Sylow theorems, or the first isomorphism theorems have nice applications and can provide non trivial results in few lines.

However, I’m struggling to find such applications concerning the Snake Lemma outside homological algebra, so an answer like “we can use it to prove the $ n$ -lemma” (pick for $ n$ your favorite integer), is not the kind of answer I’m looking for.

Precisely, I would like to know applications of the Snake Lemma, even direct and/or not very sophisticated, but which would have been difficult or lengthy to prove without it.

I found such an example here, which is rather sophisticated: https://math.stackexchange.com/questions/682777/is-an-abelian-group-characterized-by-its-localizations/712351#712351

Maybe you will have other examples , even shorter or simpler?

Thanks for your help !

Definition of integrable representation of Kac-Moody algebra

I have seen several definitions of integrable representation $ V$ of Kac-Moody algebra $ \mathfrak{g}$ online. Which one is the standard one? Are they actually equivalent?

First one is $ e_i, f_i$ acts nilpotently on $ V$ , where $ e_i, f_i$ are the Chevalley basis of $ \mathfrak{g}$ .

Second one is for any root $ \alpha$ , restriction of $ V$ to the $ sl_2$ corresponding to $ \alpha$ can be integrated to a $ SL_2$ representation.

Third one is $ V$ can be lifted to a representation of the (maximal) simply connected Kac-Moody group whose lie algebra is $ \mathfrak{g}$ .

The case I am interested most is untwisted affine Lie algebra. So feel free to restrict to this case if it helps.