Algebra notes

• λ-rings: Definitions and basic properties.
  At the moment, this just contains the basics, with long-winded but complete and (more or less) detailed proofs. I wrote this up mainly because I was very annoyed by the sloppiness with which the subject is treated in standard literature (though I have seen a lot better references by now). I hope the text will grow with my own understanding of lambda-rings. As for now, I refer to Michiel Hazewinkel's Witt vectors. Part 1 for a more informative (but less detailed) treatise of λ-rings; however, it should be kept in mind that he uses the word "λ-ring" in a different way (his "λ-ring" is my "special λ-ring", and his "pre-λ-ring" is my "λ-ring"), and his Λ (A) is slightly different from mine (for example, where I set Π (K^~, [u₁, u₂, ..., u_n]) = product (1 + u_iT) from 1 till n, he sets Π (K^~, [u₁, u₂, ..., u_n]) = ∏ (1 - u_iT)^-1 from 1 till n).

• Hans-Jürgen Schneider: Hopfalgebren (German).
  Mitschrift von Prof. Schneiders Vorlesungen über "Hopfalgebren und Quantengruppen" an der LMU München im WS 2008/2009 und im SS 2009. Das Skript ist gegen Ende des zweiten Semesters unvollständig, aber stellenweise von mir ergänzt.

• Pavel Etingof: 18.747: Infinite-dimensional Lie algebras (Spring term 2012 at MIT).
  Notes I am taking in Etingof's MIT lectures. Under construction!

• Darij Grinberg, A few facts on integrality (version 30 November 2010).
  PDF file.
  Detailed version with all the proofs much more formalized. Old version (version 20 August 2009) with slightly weaker Theorem 1. Sourcecode of all the files.
  
  See main page for an abstract.

• Darij Grinberg, Poincaré-Birkhoff-Witt type results for inclusions of Lie algebras (version 1.4, 29 November 2011).
  PDF file.
  Detailed version with all the proofs (warning: extremely long and hard to read; warning 2: only chapters 1-4 are detailed). This was written in order to make sure the calculations really work out as they should. No reader should seriously need to read it.
  Sourcecode of both files. It uses the LaTeX packages framed and comment. Toggle the "in" and "ex" parts of "\excludecomment{verlong}" and "\includecomment{vershort}" to switch between detailed (long) and normal (short) version.
  
  This is (a slightly updated version of) my diploma thesis.
  
  This paper provides detailed proofs of the main results of PBW for an inclusion of Lie algebras (arXiv:1010.0985) by Damien Calaque, Andrei Caldararu, and Junwu Tu. In particular, the fundamental lemma (Lemma 3.4) is proven in a new and significantly more elementary way, and the results are shown to hold over commutative rings (under appropriate splitting or flatness conditions) rather than over fields only.
  
  The leading question of the paper is in how far the classical PBW (Poincaré-Birkhoff-Witt) theorem, stating that the associated graded algebra gr(U(g)) of the universal enveloping algebra U(g) of a Lie algebra g (over a field k) is isomorphic to the symmetric algebra Sym(g) of g, can be extended to the situation of a Lie algebra g with a Lie subalgebra h. The most logical guess for such an extension would be that the associated graded space of the vector space U(g) / (U(g)h) is isomorphic to Sym(g/h). In the Calaque-Caldararu-Tu paper, this was proven, along with some strengthenings (for example, we even get an isomorphism of h-modules, and with an additional condition on g and h, it turns out that the filtered h-module U(g) / (U(g)h) itself is isomorphic to Sym(g/h)) and generalizations. Here we reprove the main results of that paper by elementary means (nothing more advanced than the classical PBW theorem is used) and extend them to the case of k-modules (for k a commutative rings) rather than k-vector spaces (for k a field). We need additional assumptions for the results to hold in this generality, but flatness of g and g/h and splitting of the injection h -> g are enough for almost everything (and for some results, the splitting of h -> g alone suffices).
  
  
• Darij Grinberg, The Clifford algebra and the Chevalley map - a computational approach (version 0.4, 15 June 2011).
  PDF file.
  Detailed version with all the proofs (warning: extremely long and hard to read). Sourcecode of all the files.
  
  Let k be a commutative ring (with 1), L some k-module, and f : L × L -> k be a bilinear form (not necessarily symmetric). We define the Clifford algebra Cl(L, f) as the tensor algebra ⊗ L of L, divided by the two-sided ideal generated by all terms of the form u ⊗ u - f(u, u) with u being a vector in L.
  
  This note shows that, as a k-module, Cl(L, f) is isomorphic to the exterior algebra /\ L of L. This is a standard result in case of k being a field of characteristic 0 and f being symmetric, but we establish it independently of these assumptions. Moreover, the isomorphism /\ L -> Cl(L, f) that we construct (inductively) is a projection of a k-module automorphism α^f : ⊗ L -> ⊗ L. Considering this α^f for different f, we notice the surprising fact that the composition α^f α^g equals α^f+g for any two bilinear forms f and g on L. We show that our isomorphism /\ L -> Cl(L, f) is indeed the antisymmetrization map that is usually constructed in textbooks on Clifford algebras, if k is a field of characteristic 0 (or, at least, (dim L)! is invertible in k). We also show that the k-module Cl(L, f) has a basis similar to the standard basis of /\ L if L itself is a free k-module.
  
  A further section describes some elements of Fix α^symm, which is the space of all tensors in ⊗ L that are fixed under α^f for all symmetric bilinear forms f. Finally it is shown that if L is a direct sum of two k-submodules M and N and h is a bilinear form on L such that h (M × M) = 0, then there exists an isomorphism of k-modules /\ L -> Cl(L, h) which sends (/\ L) M to (Cl(L, h)) M.
  
  
• Darij Grinberg, A representation-theoretical solution to MathOverflow question #88399 (version 1, 18 February 2012).
  PDF file.
  Sourcecode.
  
  An answer to MathOverflow question #88399. (A combinatorial determinant evaluated using the representation theory of symmetric groups.)
  
  
• Darij Grinberg, A note on lifting isomorphisms of modules over PIDs (version 0.1, 18 April 2013).
  PDF file.
  Sourcecode.
  
  This proves a few properties of finite free modules over principal ideal domains. The main result is: Let R be a PID. Let M be a finite free R-module, and A₁ and A₂ two R-linear maps from M to M. Then, A₁(M) = A₂(M) if and only if there exists an R-module automorphism U of M such that A₁ = A₂U. This result is then used to extend the claims from Keith Conrad's note Simultaneously aligned bases to a less restrictive case.
  
  
• Darij Grinberg, A few classical results on tensor, symmetric and exterior powers (version 0.2, 22 December 2011).
  PDF file.
  Sourcecode.
  
  Some basic facts about tensor products over commutative rings that I have written down with detailed proofs. "Detailed", as usual, means "a pain to read".
  
  Let k be a commutative ring.
  In §0.9, it is shown that if f : V -> V' and g : W -> W' are two surjective maps of k-modules, then Ker (f ⊗ g) = (the image of (Ker f) ⊗ W -> V ⊗ W) + (the image of V ⊗ (Ker g) -> V ⊗ W). This is extended to non-surjective maps under appropriate flatness conditions, and counterexamples are given for the general case.
  In §0.11, the surjective case is extended to n modules.
  In §0.12, the "pseudoexterior algebra" of a k-module is defined; this is the tensor algebra modulo identifying each tensor with each permutation of its tensorand times the sign of the permutation. This is almost the exterior algebra, but not the same if 2 is not invertible in k. In §0.13, the kernel of the map between pseudoexterior algebras induced by a surjective k-module map is computed.
  In §0.14, the same is done for the symmetric algebra.
  In §0.15, the same is done for the exterior algebra.
  
  Back when I wrote this note, I hadn't realized that the claim about Ker (f ⊗ g) for surjective f and g (as well as its n-modules generalization) are in Keith Conrad's Tensor Products, II. They are a lot better explained there, so go there if you want to see them proven in a human-readable form.
  
  
• Darij Grinberg, A problem on bilinear maps (problem U228 in Mathematical Reflections) (version 0.7, 30 October 2012).
  PDF file.
  Sourcecode.
  
  This note provides three solutions to the following problem:
  
  Let L/K be a separable algebraic extension of fields and let V, W and U be L-vector spaces.
  Let h : V × W -> U be a K-bilinear map satisfying
  h(xa, xb) = x² h(a, b) for every x ∈ L, a ∈ V and b ∈ W.
  Prove that h is L-bilinear.
  
  One of the solution generalizes the problem from separable algebraic field extensions to commutative separable algebras over commutative rings. Along the way, some basic properties of separable field extensions are shown.

• Witt#0: Teichmüller representatives: This gives detailed proofs of the results in Section 4 of Hazewinkel's paper.


• Witt#1: The Burnside Theorem: Proof of the Burnside theorem, stating that a G-set (for a finite group G) is uniquely characterized (up to isomorphism) by specifying its number of H-invariant elements for each subgroup H of G. (This is Theorem 19.10 in Hazewinkel's paper.) This is easy and old, but I haven't seen a proof of this in freely accessible online sources, so I wrote it up myself.


• Witt#2: Polynomials that can be written as w_n: This proves a simple assertion: A polynomial τ (in one or several variables) over the integers has the form w_n(τ₀,τ₁,...,τ_n) (for some polynomials τ_i over the integers) if and only if its derivative with respect to every indeterminate is divisible by pⁿ. (Here, w_n is the n-th p-adic Witt polynomial.) Generalized in Witt#5a below.


• Witt#3: Ghost component computations: This formalizes the principle of "working with ghost components" in Section 5 of Hazewinkel's article. In particular, Theorem 5.2 and some assertions on Frobenius and Verschiebung for p-adic Witt vectors are proven en detail.


• Witt#4: Some computations with symmetric functions: Caution, very boring / close to unreadable. I wanted to give proofs for some of the formulas in Section 9, but ended up proving the easy stuff ((9.37), (9.44), (9.57), (9.62), (9.70)) and leaving the hard stuff (particularly, the Schur polynomial identities) untouched. The only things of interest in this sidenote are Theorem 5 (b) (which generalizes (9.62) and can be used in combinatorics) and Theorem 9 (which yields an analogue to (9.70)).


• Witt#4a: Equigraded power series: Lemma for Witt#4. Can be used as an exercise in commutative algebra.


• Witt#4b: A combinatorial identity proven using symmetric functions identities: Theorem 5 (b) is used to prove the following combinatorial identity: For any natural n and real k, the sum of sign σ · k^{cycle σ} over all permutations σ of {1, 2, ..., n} is equal to n! binom(k,n). Here, cycle σ means the number of cycles (including those of length 1) in the cycle decomposition of σ. This comes from a MathLinks thread. Warning: sloppy writing.


• Witt#5: Around the integrality criterion 9.93: The integrality criterion 9.93 is proven and generalized. Some more criteria for ghost-Witt vectors are added. The result is applied to different circumstances. For instance (part of Theorem 20), we get that if n and r are positive integers, and q is an integer, then the sum of binom (q gcd(i, n), r gcd(i, n)) over all i from 1 till n is divisible by qn/r (in other words, if we divide it by qn/r, we get an integer). This generalizes a wellknown combinatorics exercise and many others.


• Witt#5a: Polynomials that can be written as big w_n: We prove that a polynomial τ (in one or several variables) over the integers has the form w_m(τ₀,τ₁,...,τ_m) (for some polynomials τ_i over the integers) if and only if its derivative with respect to every indeterminate is divisible by m. Here, w_m is the m-th big (a. k. a. universal) Witt polynomial. This is more general than Witt#2, but harder to prove (in particular, I need a result from Witt#5). Still it's rather simple and I doubt that it is new.


• Witt#5b: Some divisibilities for big Witt polynomials: We show some divisibility relations for the (big a. k. a. universal) Witt polynomials w_n. For example (Theorem 7 (c)), for any positive integer n, the sum of w_nk/gcd(i,n)^gcd(i,n) over all i = 1, 2, ..., n is divisible by n as a polynomial (i. e., every coefficient is divisible by n).


• Witt#5c: The Chinese Remainder Theorem for Modules: Proof of the Chinese Remainder Theorem for modules (only a part of it; the rest was proven in Witt#5). Was written for use in Witt#5b. Now that I know that the Chinese Remainder Theorem for modules follows from the ring version by tensoring, and the ring version is known, this note is not of much use.


• Witt#5d: Analoga of integrality criteria for radical Witt polynomials and Witt#5e: Generalizing integrality theorems for ghost-Witt vectors: A rather esoteric generalization of Witt#5. Almost all proofs are exactly the same as in Witt#5 (and often are copypasted from Witt#5).


• Witt#5f: Ghost-Witt integrality for binomial rings: This generalizes some of the statements made in Witt#5 about the ring of integers to general binomial rings, and adds a couple more. Here is the only novel result (the equivalence of G_bin and I_bin, formulated for the nest {1,2,3,...} for the sake of simplicity): A sequence (b₁, b₂, b₃, ...) of elements of a binomial ring A (for instance, of the ring ℤ) satisfies
  n | ∑_d|n φ(d) b_n/d for every positive integer n
  (where φ stands for Euler's totient function) if and only if there exists a sequence (q₁, q₂, q₃, ...) of elements of A such that every positive integer n satisfies
  b_n = ∑_d|n d binom(q_d n/d, n/d),
  with binom(x, y) denoting the binomial coefficient "x choose y". This is part of a long list of equivalent assertions, most of which are classical (and characterize the so-called "ghost-Witt vectors".

• Rep#1: Deformations of a bimodule algebra: An exercise in Introduction to representation theory on deformations of algebras is generalized and solved.


• Rep#2: An algebraic proof of an analytic lemma: The famous representation-theoretical proof of Burnside's p^aq^b theorem requires a lemma about roots of unity (stating that the arithmetic mean of finitely many roots of unity is never an algebraic integer unless it is = 0 or all the roots are equal). This is trivial using the triangle inequality in the complex numbers, but in contrast to many other uses of complex numbers, here we actually need to work in complex numbers and not just in any field extension of the rationals (in particular, we need to know that complex numbers have absolute values, and these absolute values are reals). This note was written as a remedy, giving a purely algebraical proof of the lemma. However, the proof grew longer and longer in writing, and is now probably ugly enough to serve as an example why algebraic proofs are not always better than analytical ones...


• Rep#2a: Finite subgroups of multiplicative groups of fields: A proof of a lemma for Rep#2. Rather ugly.

• Quadratic reciprocity for function fields, through determinant identities. This will take some time, I fear, if I'll write it up at all (there are much simpler proofs known).
  
• More on integrality.
• Tensor products and exterior powers (proofs of some folkore results).
  
• Any nilpotent matrix over a commutative ring has a nilpotent trace. But which is the lowest power of the trace to be identically 0? See here for an answer with a beautiful proof by Gert Almkvist. I have a different proof and if I succeed to write it up in a readable way, it will land here.
• If two polynomials over a commutative a ring have product 1, then both of them have the form (constant coefficient) + (nilpotent polynomial). This is known (one of the very first exercises in Atiyah/Macdonald) and easy, but a good example for how a proof using the axiom of choice (prime ideals etc.) can be turned into a constructive one using the method of trees. Another proof illustrates the usefulness of root adjunction to commutative rings.
• The set { (a₁x₁ + a₂x₂ + ... + a_nx_n)^m | a₁, a₂, ..., a_n are nonnegative integers with sum m } is a basis for the vector space of all m-th degree homogeneous polynomials in n variables x₁, x₂, ..., x_n over a field of characteristic 0. This was conjectured by t0rajir0u on MathLinks. I believe I have a proof now.

Algebra notes

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Darij Grinberg