Algebra Homomorphism Research Articles

Presenting integral q-Schur algebras

Let [Formula: see text] be the Lusztig integral form of quantum [Formula: see text]. There is a natural surjective algebra homomorphism [Formula: see text] from [Formula: see text] to the integral [Formula: see text]-Schur algebra [Formula: see text]. We give a generating set for the kernel of [Formula: see text]. In particular, we obtain a presentation of the [Formula: see text]-Schur algebra by generators and relations over any field.

A strong splitting of the Frobenius morphism on the algebra of distributions of $SL_2$

Let $p$ be a prime number, and let $Dist(SL_2)$ be the algebra of distributions, supported at $1$, on the algebraic group $SL_2$ over $\mathbb{F}_p$. The Frobenius map $Fr:SL_2\to SL_2$ induces a map $Fr:Dist(SL_2)\to Dist(SL_2)$ which is in particular a surjective algebra homomorphism. In this note, we construct a section of this map, whenever $p\geq 3$. The main ingredient of this construction is a certain congruence modulo $p^3$, reminiscent of the congruence $\binom{np}{p}\equiv n\mod p^3$.

An Algebra Associated with a Flag in a Subspace Lattice over a Finite Field and the Quantum Affine Algebra

In this paper, we introduce an algebra $\mathcal{H}$ from a subspace lattice with respect to a fixed flag which contains its incidence algebra as a proper subalgebra. We then establish a relation between the algebra $\mathcal{H}$ and the quantum affine algebra $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$, where $q$ denotes the cardinality of the base field. It is an extension of the well-known relation between the incidence algebra of a subspace lattice and the quantum algebra $U_{q^{1/2}}(\mathfrak{sl}_2)$. We show that there exists an algebra homomorphism from $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$ to $\mathcal{H}$ and that any irreducible module for $\mathcal{H}$ is irreducible as an $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$-module.

Homotopy G-algebra structure on the cochain complex of hom-type algebras

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. We show that the Hochschild type cochain complex of a hom-associative algebra carries a homotopy G-algebra structure. As a consequence, we get a Gerstenhaber algebra structure on the cohomology of a hom-associative algebra. We also find similar results for hom-dialgebras.

On cobrackets on the Wilson loops associated with flat $\mathrm{GL}(1, \mathbf{R})$-bundles over surfaces

Let $S$ be a closed connected oriented surface of genus $g \gt 0$. We study a Poisson subalgebra $W_1(g)$ of $C^{\infty}(\mathrm{Hom}(\pi_1(S), \mathrm{GL}(1, \mathbf{R}))/\mathrm{GL}(1, \mathbf{R}))$, the smooth functions on the moduli space of flat $\mathrm{GL}(1, \mathbf{R})$-bundles over $S$. There is a surjective Lie algebra homomorphism from the Goldman Lie algebra onto $W_1(g)$. We classify all cobrackets on $W_1(g)$ up to coboundary, that is, we compute $H^1(W_1(g), W_1(g) \wedge W_1(g)) \cong \mathrm{Hom}(\mathbf{Z}^{2g}, \mathbf{R})$. As a result, there is no cohomology class corresponding to the Turaev cobracket on $W_1(g)$.

Coproduct for Yangians of affine Kac–Moody algebras

Given an affine Kac–Moody algebra, we explain how to construct a coproduct on its associated Yangian. In order to prove that this coproduct is an algebra homomorphism, we obtain, in the first half of this paper, a minimalistic presentation of the Yangian when the Kac–Moody algebra is, more generally, symmetrizable.

Microformal Geometry and Homotopy Algebras

We extend the category of (super)manifolds and their smooth mappings by introducing a notion of microformal or "thick" morphisms. They are formal canonical relations of a special form, constructed with the help of formal power expansions in cotangent directions. The result is a formal category so that its composition law is also specified by a formal power series. A microformal morphism acts on functions by an operation of pullback, which is in general a nonlinear transformation. More precisely, it is a formal mapping of formal manifolds of even functions (bosonic fields), which has the property that its derivative for every function is a ring homomorphism. This suggests an abstract notion of a "nonlinear algebra homomorphism" and the corresponding extension of the classical "algebraic-functional" duality. There is a parallel fermionic version. The obtained formalism provides a general construction of $L_{\infty}$-morphisms for functions on homotopy Poisson ($P_{\infty}$-) or homotopy Schouten ($S_{\infty}$-) manifolds as pullbacks by Poisson microformal morphisms. We also show that the notion of the adjoint can be generalized to nonlinear operators as a microformal morphism. By applying this to $L_{\infty}$-algebroids, we show that an $L_{\infty}$-morphism of $L_{\infty}$-algebroids induces an $L_{\infty}$-morphism of the "homotopy Lie--Poisson" brackets for functions on the dual vector bundles. We apply this construction to higher Koszul brackets on differential forms and to triangular $L_{\infty}$-bialgebroids. We also develop a quantum version (for the bosonic case), whose relation with the classical version is like that of the Schr\"odinger equation with the Hamilton--Jacobi equation. We show that the nonlinear pullbacks by microformal morphisms are the limits at $\hbar\to 0$ of certain "quantum pullbacks", which are defined as special form Fourier integral operators.

T-Unique Reductions for Mészáros's Subdivision Algebra

Fix a commutative ring $\mathbf{k}$, two elements $\beta,\alpha\in\mathbf{k}$ and a positive integer $n$. Let $\mathcal{X}$ be the polynomial ring over $\mathbf{k}$ in the $n(n-1)/2$ indeterminates $x_{i,j}$ for all $1\leq i<j\leq n$. Consider the ideal $\mathcal{J}$ of $\mathcal{X}$ generated by all polynomials of the form $x_{i,j}x_{j,k}-x_{i,k}(x_{i,j}+x_{j,k}+\beta)-\alpha$ for $1\leq i<j<k\leq n$. The quotient algebra $\mathcal{X}/\mathcal{J}$ (at least for a certain choice of $\mathbf{k}$, $\beta$ and $\alpha$) has been introduced by Karola M\'esz\'aros as a commutative analogue of Anatol Kirillov's quasi-classical Yang-Baxter algebra. A monomial in $\mathcal{X}$ is said to be pathless if it has no divisors of the form $x_{i,j}x_{j,k}$ with $1\leq i<j<k\leq n$. The residue classes of these pathless monomials span the $\mathbf{k}$-module $\mathcal{X}/\mathcal{J}$, but (in general) are $\mathbf{k}$-linearly dependent. Recently, the study of Grothendieck polynomials has led Laura Escobar and Karola M\'esz\'aros to defining a $\mathbf{k}$-algebra homomorphism $D$ from $\mathcal{X}$ into the polynomial ring $\mathbf{k}[t_{1},t_{2},\ldots,t_{n-1}]$ that sends each $x_{i,j}$ to $t_{i}$. We show the following fact (generalizing a conjecture of M\'esz\'aros): If $p\in\mathcal{X}$, and if $q\in\mathcal{X}$ is a $\mathbf{k}$-linear combination of pathless monomials satisfying $p\equiv q\operatorname{mod}\mathcal{J}$, then $D(q)$ does not depend on $q$ (as long as $\beta$, $\alpha$ and $p$ are fixed). Thus, reducing a $p\in\mathcal{X}$ modulo $\mathcal{J}$ may lead to different results depending on the choices made in the reduction process, but all of them become identical once $D$ is applied. We also find an actual basis of the $\mathbf{k}$-module $\mathcal{X}/\mathcal{J}$, using what we call forkless monomials.

Quantum character varieties and braided module categories

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants int _S{mathcal {A}} of a surface S, determined by the choice of a braided tensor category {mathcal {A}}, and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a braided module category for {mathcal {A}}, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called quantum moment maps. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided {mathcal {A}}-modules are objects of the torus category int _{T^2}{mathcal {A}}. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of {mathcal {A}}={text {Rep}}_{q}G with the category {mathcal {D}}_q(G/G)-mod of equivariant quantum {mathcal {D}}-modules. When G=GL_n, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra {mathbb {SH}}_{q,t}.

Jacobi-Trudi Determinants over Finite Fields

In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over the integers to a finite field $\mathbb{F}_q$, what is the probability that the Schur function $s_\lambda$ maps to zero? We show that this probability is always at least $1/q$ and is asymptotically $1/q$. Moreover, we give a complete classification of all shapes that can achieve probability $1/q$. In addition, we identify certain families of shapes where the corresponding Schur functions being sent to zero are independent events, and we look into the probability that a Schur functions is mapped to nonzero values in $\mathbb{F}_q$.

Involutions in split semi‐quaternions

A map is an involution (resp, anti‐involution) if it is a self‐inverse homomorphism (resp, antihomomorphism) of a field algebra. The main purpose of this paper is to show how split semi‐quaternions can be used to express half‐turn planar rotations in 3‐dimensional Euclidean space and how they can be used to express hyperbolic‐isoclinic rotations in 4‐dimensional semi‐Euclidean space . We present an involution and an anti‐involution map using split semi‐quaternions and give their geometric interpretations as half‐turn planar rotations in . Also, we give the geometric interpretation of nonpure unit split semi‐quaternions, which are in the form p = coshθ + sinhθi + 0j + 0k = coshθ + sinhθi, as hyperbolic‐isoclinic rotations in .

The q-Onsager Algebra and the Universal Askey-Wilson Algebra

Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the $q$-Onsager algebra $\mathcal O_q$. They defined the PBW basis elements recursively, and it is obscure how to express them in closed form. To mitigate the difficulty, we bring in the universal Askey-Wilson algebra $\Delta_q$. There is a natural algebra homomorphism $\natural \colon \mathcal O_q \to \Delta_q$. We apply $\natural $ to the above PBW basis, and express the images in closed form. Our results make heavy use of the Chebyshev polynomials of the second kind.

Effective descent morphisms for Banach modules

It is proved that a norm-decreasing homomorphism of commutative Banach algebras is an effective descent morphism for Banach modules if and only if it is a weak retract.

A decomposition of the group algebraof a hyperoctahedral group

The descent algebra of a Coxeter group is a subalgebra of the group algebra with interesting representation theoretic properties. For instance, the natural map from the descent algebra of the symmetric group to the character ring is a surjective algebra homomorphism, so the descent algebra implicitly encodes information about the representations of the symmetric group. However, this property does not hold for other Coxeter groups. Moreover, a complete set of primitive idempotents in the descent algebra of the symmetric group leads to a decomposition of the group algebra as a direct sum of induced linear characters of centralizers of conjugacy class representatives. In this dissertation, I consider the hyperoctahedral group. When the descent algebra of a hyperoctahedral group is replaced with a generalization called the Mantaci-Reutenauer algebra, the natural map to the character ring is surjective. In 2008, Bonnafe asked whether a complete set of idempotents in the Mantaci-Reutenauer algebra could lead to a decomposition of the group algebra of the hyperoctahedral group as a direct sum of induced linear characters of centralizers. In this dissertation, I will answer this question positively and go through the construction of the idempotents, conjugacy class representatives, and linear characters required to do so.

Higher cohomology operations and R–completion

Let $R=\mathbb{F}_p$ or a field of characteristic $0$. For each $R$-good topological space $Y$, we define a collection of higher cohomology operations which, together with the cohomology algebra $H^*(Y;R)$ suffice to determine $Y$ up to $R$-completion. We also provide a similar collection of higher cohomology operations which determine when two maps $f_0,f_1: Z\to Y$ between $R$-good spaces(inducing the same algebraic homomorphism $H^*(Y;R)\to H^*(Z;R)$) are $R$-equivalent.

On the second fundamental theorem of invariant theory for the orthosymplectic supergroup

Let OSp(V) be the orthosymplectic supergroup on an orthosymplectic vector superspace V of superdimension (m|2n). Lehrer and Zhang showed that there is a surjective algebra homomorphism Frr:Br(m−2n)→EndOSp(V)(V⊗r), where Br(m−2n) is the Brauer algebra of degree r with parameter m−2n. The second fundamental theorem of invariant theory in this setting seeks to describe the kernel KerFrr of Frr as a 2-sided ideal of Br(m−2n). In this paper, we show that KerFrr≠0 if and only if r≥rc:=(m+1)(n+1), and give a basis and a dimension formula for KerFrr. We show that KerFrr as a 2-sided ideal of Br(m−2n) is generated by KerFrcrc for any r≥rc, and we provide an explicit set of generators for KerFrcrc. These generators coincide in the classical case with those obtained in recent papers of Lehrer and Zhang on the second fundamental theorem of invariant theory for the orthogonal and symplectic groups. As an application we obtain the necessary and sufficient conditions for the endomorphism algebra Endosp(V)(V⊗r) over the orthosymplectic Lie superalgebra osp(V) to be isomorphic to Br(m−2n).

BLM realization for 𝒰ℤ(𝔤𝔩 ̂n)

In 1990, Beilinson–Lusztig–MacPherson (BLM) discovered a realization for quantum [Formula: see text] via a geometric setting of quantum Schur algebras. We will generalize their result to the classical affine case. More precisely, we first use Ringel–Hall algebras to construct an integral form [Formula: see text] of [Formula: see text], where [Formula: see text] is the universal enveloping algebra of the loop algebra [Formula: see text]. We then establish the stabilization property of multiplication for the classical affine Schur algebras. This stabilization property leads to the BLM realization of [Formula: see text] and [Formula: see text]. In particular, we conclude that [Formula: see text] is a [Formula: see text]-Hopf subalgebra of [Formula: see text]. As a bonus, this method leads to an explicit [Formula: see text]-basis for [Formula: see text], and it yields explicit multiplication formulas between generators and basis elements for [Formula: see text]. As an application, we will prove that the natural algebra homomorphism from [Formula: see text] to the affine Schur algebra over [Formula: see text] is surjective.

The classical dynamic symmetry for the U(1)-Kepler problems

For the Jordan algebra of hermitian matrices of order $n\ge 2$, we let $X$ be its submanifold consisting of rank-one semi-positive definite elements. The composition of the cotangent bundle map $\pi_X$: $T^*X\to X$ with the canonical map $X\to \mathbb{C}P^{n-1}$ (i.e., the map that sends a hermitian matrix to its column space), pulls back the K\"{a}hler form of the Fubini-Study metric on $\mathbb{C}P^{n-1}$ to a real closed differential two-form $\omega_K$ on $T^*X$. Let $\omega_X$ be the canonical symplectic form on $T^*X$ and $\mu$ be a real number. A standard fact says that $\omega_\mu:=\omega_X+2\mu\,\omega_K$ turns $T^*X$ into a symplectic manifold, hence a Poisson manifold with Poisson bracket $\{\, ,\,\}_\mu$. In this article we exhibit a Poisson realization of the simple real Lie algebra $\mathfrak {su}(n, n)$ on the Poisson manifold $(T^*X, \{\, ,\,\}_\mu)$, i.e., a Lie algebra homomorphism from $\mathfrak {su}(n, n)$ to $\left(C^\infty(T^*X, \mathbb R), \{\, ,\,\}_\mu\right)$. Consequently one obtains the Laplace-Runge-Lenz vector for the classical $\mathrm{U}(1)$-Kepler problem with level $n$ and magnetic charge $\mu$. Since the McIntosh-Cisneros-Zwanziger-Kepler problems (MICZ-Kepler Problems) are the $\mathrm{U}(1)$-Kepler problems with level $2$, the work presented here is a direct generalization of the work by A. Barut and G. Bornzin [ J. Math. Phys. $\bf 12$ (1971), 841-843] on the classical dynamic symmetry for the MICZ-Kepler problems.

Relative global dimensions of extensions

ABSTRACTAn extension of algebras is a homomorphism of algebras preserving identities. Given an extension f:B→A, the relative global dimension of f is defined to be the supremum of relative projective dimensions of all A-modules. In this paper, we compare relative homological dimensions of two extensions of ordinary algebras under certain conditions. As an application, for any natural number n, we present a general method for constructing non-trivial extensions of Artin algebras of relative global dimension at least n.

On a question of Külshammer for homomorphisms of algebraic groups

Let G be a linear algebraic group over an algebraically closed field of characteristic p≥0. We show that if H1 and H2 are connected subgroups of G such that H1 and H2 have a common maximal unipotent subgroup and H1/Ru(H1) and H2/Ru(H2) are semisimple, then H1 and H2 are G-conjugate. Moreover, we show that if H is a semisimple linear algebraic group with maximal unipotent subgroup U then for any algebraic group homomorphism σ:U→G, there are only finitely many G-conjugacy classes of algebraic group homomorphisms ρ:H→G such that ρ|U is G-conjugate to σ. This answers an analogue for connected algebraic groups of a question of B. Külshammer.In Külshammer's original question, H is replaced by a finite group and U by a Sylow p-subgroup of H; the answer is then known to be no in general. We obtain some results in the general case when H is non-connected and has positive dimension. Along the way, we prove existence and conjugacy results for maximal unipotent subgroups of non-connected linear algebraic groups. When G is reductive, we formulate Külshammer's question and related conjugacy problems in terms of the nonabelian 1-cohomology of unipotent radicals of parabolic subgroups of G, and we give some applications of this cohomological approach. In particular, we analyse the case when G is a semisimple group of rank 2.