algebra

superalgebra and (synthetic ) supergeometry The notion of Jordan superalgebra is the analog in superalgebra/supergeometry of that of Jordan algebra: A Jordan superalgebra is a supercommutative superalgebra with underlying -graded algebra , where: is an ordinary Jordan algebra, is a -bimodule with a “Lie bracket-like” product into , satisfying a super Jordan identity. Simple Jordan superalgebras o…

representation, 2-representation, ∞-representation Grothendieck group, lambda-ring, symmetric function, formal group principal bundle, torsor, vector bundle, Atiyah Lie algebroid Eilenberg-Moore category, algebra over an operad, actegory, crossed module The possible actions of well-behaved topological groups (such as compact Lie groups) on topological or smooth n-spheres display various interesti…

A Malcev operation on a set is a ternary operation, a function which satisfies the identities and . An important motivating example is the operation of a heap, for example the operation on a group defined by . An algebraic theory is a Malcev theory when contains a Malcev operation. An algebraic theory is Malcev iff one of the following equivalent statements is true: in the category of -algebras, …

Galois theory is, famously, typically what is used to prove certain results that are stateable without knowledge of it. For example, the following problems are usually proven in a first course in ...

Let $E/F$ be a finite separable extension of degree $n$. I am trying to prove that the number of intermediate fields $K$ (where $F \subseteq K \subseteq E$) is at most $2^n$. I have already seen a ...

symmetric monoidal (∞,1)-category of spectra The space of functions on the space of morphisms of a small category (with coefficients in some ring ) naturally inherits a convolution algebra structure from the composition operation on morphisms. This is called its category convolution algebra or just category algebra for short. Often this is considered specifically for groupoids, and hence accordin…

Special and general types group cohomology, nonabelian group cohomology, Lie group cohomology cohomology with constant coefficients / with a local system of coefficients Special notions Variants differential cohomology Extra structure Operations Theorems higher geometry / derived geometry Ingredients Concepts geometric little (∞,1)-toposes geometric big (∞,1)-toposes Constructions Examples derive…

Given a partially ordered set (poset) $(P,\leq)$, is there a lattice $L_P$ and an order-preserving map $h_P:P\to L_P$ with the following property? Whenever $L$ is a lattice and $f: P\to L$ is an ...

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical) quantum mechanical system, quantum probability interacting field quantization An -algebra is a (not necessarily unital) Banach algebra that is simultaneously a Hilbert space with a compatible star-algebra structure, namely with an anti-linear involution such that An -algebra (Def. ) is called proper if or equivalentl…

Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions Related concepts The projective general linear group (in some dimension and over some coefficients) is the quotient of the general linear group by its center, in other words by the subgroup consisting of scalar multiplies of the identity. Elements of are identified w…

Shinichi Mochizuki is a Japanese algebraic/arithmetic geometer, developing Grothendieck’s program of so-called anabelian geometry. His “inter-universal Teichmüller theory” claims to prove the abc conjecture.

I want to understand the difference between a "higher stack" and a "derived stack". I can state their definitions individually, but I don't understand what exactly are the ...

higher geometry / derived geometry Ingredients Concepts geometric little (∞,1)-toposes geometric big (∞,1)-toposes Constructions Examples derived smooth geometry Theorems symmetric monoidal (∞,1)-category of spectra Algebraic geometry is, in origin, a geometric study of solutions of systems of polynomial equations and generalizations. The set of zeros of a set of polynomial equations in finitely …

Let $R$ be a ring, and let $M_R$ and $N_R$ be right $R$-modules. I was taught to always write module homomorphisms on the opposite side from scalars. Thus, a map $\varphi\in {\rm Hom}_R(M,N)$ would ...

Fun with polynomials and linear algebra; or, slight abstract nonsense Posted This is mostly a bunch of notes to myself (with some slight expansion) and is a combination/extension/simplification of theorems/ideas/constructions from a bunch of texts, including Wistbauer’s “Foundations of Module and Ring Theory” and Fuhrmann’s “A Polynomial Approach to Linear Algebra”, along with others that at this…

Let be a differential graded category. A twisted complex in is a graded set of objects of , such that only finitely many are not the zero object; a set of morphisms such that ; . The differential graded category of twisted complexes in has as objects twisted complexes and with differential given on given by The construction of categories of twisted complexes is functorial in that for a dg-functor…

With braiding With duals for objects category with duals (list of them) dualizable object (what they have) ribbon category, a.k.a. tortile category With duals for morphisms With traces Closed structure Special sorts of products Semisimplicity Morphisms Internal monoids Examples Theorems In higher category theory A cartesian closed category (sometimes: ccc) is a category with finite products which…

On Lie integration of L-infinity algebras to smooth infinity-groups (Lie's third theorem in higher Lie theory): On simplicial principal bundles in descent categories as models for (smooth) principal -bundles: Jesse Wolfson: On Simplicial Principal Bundles in Descent Categories [arXiv:2305.01630] Jesse Wolfson: Descent for -bundles, Advances in Mathematics 288 (2016) 527–575 [doi:10.1016/j.aim.201…

John Greenlees is professor for mathematics in Sheffield. On equivariant complex oriented cohomology theory and equivariant formal group laws: Michael Cole, John Greenlees, Igor Kriz, Equivariant Formal Group Laws, Proceedings of the LMS, Volume81, Issue2 (2000)(doi:10.1112/S0024611500012466) John Greenlees, The coefficient ring of equivariant homotopical bordism classifies equivariant formal gro…

Odrzywołek's Exp-Minus-Log operator, eml(x, y) = exp(x) ln(y), is provably a complete basis for the entire class of elementary functions when paired with the constant 1 such that any finite expression built from variables, constants, arithmetic operations, and the exp and ln functions will eventually materialize given enough runtime. This paper strips away even this dependence to study a modified…