Brauer algebras of type \mathrm{I}_2^n(n\ge 5)

We will present an algebra related to the Coxeter group of type \(\mathrm{I}_2^n\) which can be taken as the twisted subalgebra in Brauer algebra of type \(\mathrm{A}_{n-1}\) . Also, we will describe some properties of this algebra.     

Centers for the restricted category \mathcal {O} at the critical level over affine Kac–Moody algebras

The restricted category $\mathcal {O}$ at the critical level over an affine Kac–Moody algebra is a certain subcategory of the ordinary BGG-category $\mathcal {O}$ . We study a deformed version introduced by Arakawa and Fiebig and calculate the center of the deformed restricted category $\mathcal {O}$ . This complements a result of Fiebig which describes the center of the non-restricted category $\mathcal {O}$ outside the critical hyperplanes over a symmetrizable Kac–Moody algebra.     

Embedding median algebras in products of trees

We show that a metric median algebra satisfying certain conditions admits a bilipschitz embedding into a finite product of $\mathbb{R }$ -trees. This gives rise to a characterisation of closed connected subalgebras of finite products of complete $\mathbb{R }$ -trees up to bilipschitz equivalence. Spaces of this sort arise as asymptotic cones of coarse median spaces. This applies to a large class of finitely generated groups, via their Cayley graphs. We show that such groups satisfy the rapid decay property. We also recover the result of Behrstock, Dru?u and Sapir, that the asymptotic cone of the mapping class group embeds in a finite product of $\mathbb{R }$ -trees.     

Domination by ergodic elements in ordered Banach algebras

We recall the definition and properties of an algebra cone in an ordered Banach algebra (OBA) and continue to develop spectral theory for the positive elements. An element $a$ of a Banach algebra is called ergodic if the sequence of sums $\sum _{k=0}^{n-1} \frac{a^k}{n}$ converges. If $a$ and $b$ are positive elements in an OBA such that $0\le a\le b$ and if $b$ is ergodic, an interesting problem is that of finding conditions under which $a$ is also ergodic. We will show that in a semisimple OBA that has certain natural properties, the condition we need is that the spectral radius of $b$ is a Riesz point (relative to some inessential ideal). We will also show that the results obtained for OBAs can be extended to the more general setting of commutatively ordered Banach algebras (COBAs) when adjustments corresponding to the COBA structure are made.     

Applications of Matrix Algebra to Clifford Groups

In this paper, we will show that all of nonzero vectors and nonzero bivectors in the Clifford algebra ${\mathcal{C} \ell_{0,3}}$ are invertible and we will find some conditions for those objects to be element of the Clifford group ??_0,3 using the corresponding properties in the subalgebra L ₈ of the matrix algebra ${M_8 \mathbb{(R)}}$ .     

Commutative Subalgebras of Quantum Algebras

In the present paper, a general assertion is proved, claiming that, for every associative algebra $\mathcal{A}$ without zero divisors which admits a valuation and a seminorm concordant with the valuation, the transcendence degree of an arbitrary commutative subalgebra does not exceed the maximal number of independent pairwise pseudocommuting elements of some basis of the algebra $\mathcal{A}$ . The author shows that for such a algebra $\mathcal{A}$ one can take an arbitrary algebra of quantum Laurent polynomials, quantum analogs of the Weyl algebra, and also some universal coacting algebras. In the case of the algebra $\mathcal{L}$ of quantum Laurent polynomials, it is proved that the transcendence degree of a maximal commutative subalgebra of $\mathcal{L}$ coincides with the maximal number of independent pairwise commuting elements of the monomial basis of the algebra $\mathcal{L}$ .     

Bethe subalgebras of the group algebra of the symmetric group

We introduce families $ \mathcal{B}_n^S\left( {{z_1},\ldots,{z_n}} \right) $ and $ \mathcal{B}_{{n,\hbar}}^S\left( {{z_1},\ldots,{z_n}} \right) $ of maximal commutative subalgebras, called Bethe subalgebras, of the group algebra $ \mathbb{C}\left[ {\mathfrak{S}n} \right] $ of the symmetric group. Bethe subalgebras are deformations of the Gelfand?Zetlin subalgebra of $ \mathbb{C}\left[ {\mathfrak{S}n} \right] $ . We describe various properties of Bethe subalgebras.     

Cohomology and Support Varieties for Restricted Lie Superalgebras

Let $(\mathfrak{g}, [p]) $ be a restricted Lie superalgebra over an algebraically closed field k of characteristic p?>?2. Let $\mathfrak{u}(\mathfrak{g})$ denote the restricted enveloping algebra of $\mathfrak{g}$ . In this paper we prove that the cohomology ring $\operatorname{H}^\bullet(\mathfrak{u}(\mathfrak{g}), k)$ is finitely generated. This allows one to define support varieties for finite dimensional $\mathfrak{u}(\mathfrak{g})$ -supermodules. We also show that support varieties for finite dimensional $\mathfrak{u}(\mathfrak{g})$ - supermodules satisfy the desirable properties of a support variety theory.     

Two-dimensional Toda field equations related to the exceptional algebra \mathfrak{g}_2 : Spectral properties of the lax operators

We analyze spectral properties of the Lax operator corresponding to the two-dimensional Toda field equations related to the algebra $\mathfrak{g}_2 $ . We construct two minimal sets of scattering data $\mathcal{T}_s $ , s = 1, 2, understanding the map between the potential and each of the sets $\mathcal{T}_s $ as a generalized Fourier transformation. We construct explicit recursion operators with special factorization properties.     

The \mathfrak {sl}_{3}-web algebra

In this paper we use Kuperberg’s $\mathfrak {sl}_3$ -webs and Khovanov’s $\mathfrak {sl}_3$ -foams to define a new algebra $K^S$ , which we call the $\mathfrak {sl}_3$ -web algebra. It is the $\mathfrak {sl}_3$ analogue of Khovanov’s arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of $q$ -skew Howe duality, which allows us to prove that $K^S$ is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group $K^{\oplus }_0(\mathcal {W}^S)_{\mathbb {Q}(q)}$ , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that $K^S$ is a graded cellular algebra.     

On Homotopes of Novikov Algebras

Given a unital associative commutative ring Φ containing $\frac{1}{2}$ , we consider a homotope of a Novikov algebra, i.e., an algebra $A_\varphi $ that is obtained from a Novikov algebra A by means of the derived operation $x \cdot y = xy\varphi $ on the Φ-module A, where the mapping ? satisfies the equality $xy\varphi = x(y\varphi )$ . We find conditions for a homotope of a Novikov algebra to be again a Novikov algebra.     

On the homological classification of pomonoids by properties of cyclic S-posets

Between different and relatively well investigated so-called flatness properties of S-posets there is a property called property ( ${\rm P}_{w}$ ) which, so far, has not received much attention. In this paper, we characterize pomonoids from a subclass of completely simple semigroups with adjoined identity all of whose cyclic (Rees factor) S-posets satisfy property ( ${\rm P}_{w}$ ). Moreover, for the same class of pomonoids, we find necessary and sufficient conditions under which all Rees factor S-posets satisfying property ( ${\rm P}_{w}$ ) satisfy property (P).     

Symmetry and quasi-centrality of the operator space projective tensor product

For C*-algebras A and B, the operator space projective tensor product ${A\widehat{\otimes}B}$ and the Banach space projective tensor product ${A\otimes_{\gamma}B}$ are shown to be symmetric. We also show that ${A\widehat{\otimes}B}$ is a weakly Wiener algebra. Finally, quasi-centrality and the unitary group of ${A\widehat{\otimes}B}$ are discussed.     

Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra

Let M be aσ-finite von Neumann algebra and let AM be a maximal subdiagonal algebra with respect to a faithful normal conditional expectationΦ.Based on the Haagerup’s noncommutative Lpspace Lp(M)associated with M,we consider Toeplitz operators and the Hilbert transform associated with A.We prove that the commutant of left analytic Toeplitz algebra on noncommutative Hardy space H2(M)is just the right analytic Toeplitz algebra.Furthermore,the Hilbert transform on noncommutative Lp(M)is shown to be bounded for 1p∞.As an application,we consider a noncommutative analog of the space BMO and identify the dual space of noncommutative H1(M)as a concrete space of operators.     

On The Structure of A Commutative Banach Algebra Generated By Toeplitz Operators With Quasi-Radial Quasi-Homogeneous Symbols

Let ${\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}$ denote the standard weighted Bergman space over the unit ball ${\mathbb{B}^n}$ in ${\mathbb{C}^n}$ . New classes of commutative Banach algebras ${\mathcal{T}(\lambda)}$ which are generated by Toeplitz operators on ${\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}$ have been recently discovered in Vasilevski (Integr Equ Oper Theory 66(1):141?C152, 2010). These algebras are induced by the action of the quasi-elliptic group of biholomorphisms of ${\mathbb{B}^n}$ . In the present paper we analyze in detail the internal structure of such an algebra in the lowest dimensional case n?=?2. We explicitly describe the maximal ideal space and the Gelfand map of ${\mathcal{T}(\lambda)}$ . Since ${\mathcal{T}(\lambda)}$ is not invariant under the *-operation of ${\mathcal{L}(\mathcal{A}_{\lambda}^2(\mathbb{B}^n))}$ its inverse closedness is not obvious and is proved. We remark that the algebra ${\mathcal{T}(\lambda)}$ is not semi-simple and we derive its radical. Several applications of our results are given and, in particular, we conclude that the essential spectrum of elements in ${\mathcal{T}(\lambda)}$ is always connected.     

A minimal even type of the 2-adic Weil representation

The Weil representation is used to construct a minimal type of the two-fold central extension of $\mathrm{Sp }_{2n}({\mathbb {Q}}_2)$ . The corresponding Hecke algebra is shown to be isomorphic to the classical affine Hecke algebra of the split adjoint orthogonal group $\mathrm{SO }_{2n+1}({\mathbb {Q}}_2)$ .     

Zhelobenko invariants, Bernstein-Gelfand-Gelfand operators and the analogue Kostant Clifford algebra conjecture

Let $ \mathfrak{g} $ be a complex simple Lie algebra and $ \mathfrak{h} $ a Cartan subalgebra. The Clifford algebra C( $ \mathfrak{g} $ ) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2?m?+?1 is just the zero weight vector of the simple (2?m?+?1)-dimensional module of the principal s-triple obtained from the Langlands dual $ {\mathfrak{g}^\vee } $ . Bazlov [1] settled this conjecture positively in type A. The hard part of the Kostant Clifford algebra conjecture is a question concerning the Harish-Chandra map for the enveloping algebra U( $ \mathfrak{g} $ ) composed with evaluation at the half sum ?? of the positive roots. The analogue Kostant conjecture is obtained by replacing the Harish-Chandra map by a ??generalized Harish-Chandra?? map. This map had been studied notably by Zhelobenko [15]. The proof given here involves a symmetric algebra version of the Kostant conjecture, the Zhelobenko invariants in the adjoint case, and, surprisingly, the Bernstein-Gelfand-Gelfand operators introduced in their study [3] of the cohomology of the flag variety.     

On the Topology of Kac–Moody groups

We study the topology of spaces related to Kac–Moody groups. Given a Kac–Moody group over $\mathbb C $ , let $\text {K}$ denote the unitary form with maximal torus ${{\mathrm{T}}}$ having normalizer ${{\mathrm{N}}}({{\mathrm{T}}})$ . In this article we study the cohomology of the flag manifold $\text {K}/{{{\mathrm{T}}}}$ as a module over the Nil-Hecke algebra, as well as the (co)homology of $\text {K}$ as a Hopf algebra. In particular, if $\mathbb F $ has positive characteristic, we show that $\text {H}_*(\text {K},\mathbb F )$ is a finitely generated algebra, and that $\text {H}^*(\text {K},\mathbb F )$ is finitely generated only if $\text {K}$ is a compact Lie group . We also study the stable homotopy type of the classifying space $\text {BK}$ and show that it is a retract of the classifying space $\text {BN(T)}$ of ${{\mathrm{N}}}({{\mathrm{T}}})$ . We illustrate our results with the example of rank two Kac–Moody groups.     

A Few Comments on the Clifford Algebra {C\ell_2} of the Standard Euclidean Plane

This paper is a continuation of the author’s plenary lecture given at ICCA 9 which was held in Weimar at the Bauhaus University, 15–20 July, 2011. We want to study on both the mathematical and the epistemological levels the thought of the brilliant geometer W. K. Clifford by presenting a few comments on the structure of the Clifford algebra ${C\ell_2}$ associated with the standard Euclidean plane ${\mathbb{R}^2}$ . Miquel’s theorem will be given in the algebraic context of the even Clifford algebra ${C\ell^+_2}$ isomorphic to the real algebra ${\mathbb{C}}$ . The proof of this theorem will be based on the cross ratio (the anharmonic ratio) of four complex numbers. It will lead to a group of homographies of the standard projective line ${\mathbb{C}P^1 = P(\mathbb{C}^2)}$ which appeared so attractive to W. K. Clifford in his overview of a general theory of anharmonics. In conclusion it will be shown how the classical Clifford-Hopf fibration S ¹ → S ³ → S ² leads to the space of spinors ${\mathbb{C}^2}$ of the Euclidean space ${\mathbb{R}^3}$ and to the isomorphism ${{\rm {PU}(1) = \rm {SU}(2)/\{I,-I\} \simeq SO(3)}}$ .     

Explicit Matrix Realization of Clifford Algebras

In this paper, we construct associative subalgebras ${{L_{2}}{n}(\mathbb{R})}$ of the real ${2^{n} \times 2^{n}}$ matrix algebra ${{M_{2}}{n}(\mathbb{R})}$ , which is isomorphic to the real Clifford algebra ${C \ell_{0},n}$ for every ${n \in N}$ .