Search Results

I think the answer is that we can say very little about a generalized Cartan matrix of indefinite type in general. Most of them will be invertible in the reals (because invertible matrices form a den …

Harish-Chandra pairs and their use in localization is discussed in Beilinson-Bernstein A proof of Jantzen Conjectures, available on Joseph Bernstein's web page. See in particular sections 1.8 and 3.3 …

The answer is "yes". This is (the compact version of) Proposition 2.20 b on page 139 of the Deligne-Milne article on Tannakian categories in Hodge cycles, Shimura Varieties, and Motives (Springer LNM …

These are the finite linearly reductive groups. In characteristic zero, they are just the finite algebraic groups. In positive characteristic, Nagata's theorem gives the following characterization o …

I'm not an expert in this area, but I'm told that the key phrase in the superalgebra world is "Weyl groupoid" rather than Weyl group. I did not look at the construction long enough to understand it. …

Suppose $A \in PGL_n(\mathbb{R})$ lies in the kernel of the adjoint representation. Then for any lift $\tilde{A}$ of $A$ in $GL_n(\mathbb{R})$, and any traceless matrix $B$, we have $\tilde{A}B = B\t …

The formulas you are writing seem to arise from Cartier duality, which gives a collection of antiequivalences of categories of group schemes and sheaves of various types. For the case of tori, one ge …

If you have any free abelian group with an integral bilinear form embedded in the Lorentz space $\mathbb{R}^{n,1}$, you may consider the group of automorphisms generated by roots, i.e., reflections in …

I don't think the statement that you linked in your revised question needs a reference, or even an explicit proof. The fact that comodule structures can be pushed forward along coalgebra homomorphism …

If we just consider central representations, i.e., those for which $z$ acts by a nonzero scalar, then up to a certain kind of equivalence (given by conjugation with algebra isomorphisms) there is a un …

I'll give a description on the level of the polynomial Lie algebra, and then wave my hands about integrating and completing. As Victor Protsak mentioned in the comments, you can find a more precise t …

Symmetric algebras (aka free commutative associative unital algebras) are given by a functor, and they satisfy a universal property: If M is a module over a commutative ring k and R is a commutative k …

Given a finite group, the sums of squares of dimensions of irreducible representations add up to the order of the group, so the dimension of an irreducible representation is at most the square root of …

There are some nice Morita equivalences arising from Hecke algebras in representation theory - they arise as algebras of bi-invariant functions on a locally compact group under convolution. Good, wor …

For any finite dimensional representation, you can restrict to $T^p$ to get a decomposition into a direct sum of one-dimensional characters, which are given by $p$-tuples of integers. If a given char …