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Register a new userAtomistic subsemirings of the lattice of subspaces of an algebra
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Authors
Daniel S. Sage
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Let A be an associative algebra with identity over a field k. An atomistic subsemiring R of the lattice of subspaces of A, endowed with the natural product, is a subsemiring which is a closed atomistic sublattice. When R has no zero divisors, the set of atoms of R is endowed with a multivalued product. We introduce an equivalence relation on the set of atoms such that the quotient set with the induced product is a monoid, called the condensation monoid. Under suitable hypotheses on R, we show that this monoid is a group and the class of k1_A is the set of atoms of a subalgebra of A called the focal subalgebra. This construction can be iterated to obtain higher condensation groups and focal subalgebras. We apply these results to G-algebras for G a group; in particular, we use them to define new invariants for finite-dimensional irreducible projective representations.
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A paper of U. First & Z. Reichstein proves that if $R$ is a commutative ring of dimension $d$, then any Azumaya algebra $A$ over $R$ can be generated as an algebra by $d+2$ elements, by constructing such a generating set, but they do not prove that this number of generators is required, or even that for an arbitrarily large $r$ that there exists an Azumaya algebra requiring $r$ generators. In this paper, for any given fixed $nge 2$, we produce examples of a base ring $R$ of dimension $d$ and an Azumaya algebra of degree $n$ over $R$ that requires $r(d,n) = lfloor frac{d}{2n-2} rfloor + 2$ generators. While $r(d,n) < d+2$ in general, we at least show that there is no uniform upper bound on the number of generators required for Azumaya algebras. The method of proof is to consider certain varieties $B^r_n$ that are universal varieties for degree-$n$ Azumaya algebras equipped with a set of $r$ generators, and specifically we show that a natural map on Chow group $CH^{(r-1)(n-1)}_{PGL_n} to CH^{(r-1)(n-1)}(B^r_n)$ fails to be injective, which is to say that the map fails to be injective in the first dimension in which it possibly could fail. This implies that for a sufficiently generic rank-$n$ Azumaya algebra, there is a characteristic class obstruction to generation by $r$ elements.
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Riordan matrices are infinite lower triangular matrices determined by a pair of formal power series over the real or complex field. These matrices have been mainly studied as combinatorial objects with an emphasis placed on the algebraic or combinatorial structure. The present paper contributes to the linear algebraic discussion with an analysis of Riordan matrices by means of the interaction of the properties of formal power series with the linear algebra. Specifically, it is shown that if a Riordan matrix $A$ is an $ntimes n$ pseudo-involution then the singular values of $A$ must come in reciprocal pairs in $Sigma$ of a singular value decomposition $A=USigma V^T$. Moreover, we give a complete analysis of the existence and nonexistence of eigenvectors of Riordan matrices. As a result, we obtain a surprising partition of the group of Riordan matrices into three different types of eigenvectors. Finally, given a nonzero vector $v$, we investigate the algebraic structure of Riordan matrices $A$ that stabilize the vector $v$, i.e. $Av=v$.
Let $mathfrak{sp}_{2n}(mathbb {K})$ be the symplectic Lie algebra over an algebraically closed field of characteristic zero. We prove that for any nonzero nilpotent element $X in mathfrak{sp}_{2n}(mathbb {K})$ there exists a nilpotent element $Y in mathfrak{sp}_{2n}(mathbb {K})$ such that $X$ and $Y$ generate $mathfrak{sp}_{2n}(mathbb {K})$.
In this paper, the tensor product of highest weight modules with intermediate series modules over the Neveu-Schwarz algebra is studied. The weight spaces of such tensor products are all infinitely dimensional if the highest weight module is nontrivial. We find that all such tensor products are indecomposable. We give the necessary and sufficient conditions for these tensor product modules to be irreducible by using shifting technique established for the Virasoro case in [13]. The necessary and sufficient conditions for any two such tensor products to be isomorphic are also determined.
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