Stereotype group algebra

In mathematics, specifically in functional analysis, a stereotype group algebra is an analog in stereotype theory of a group algebra of a locally compact group.

Idea: group algebras for finite groups

If $G$ is a finite group then for any field $K$ the set $K^{G}$ of all functions $u:G\to K$ possesses a natural structure of (finite-dimensional) Hopf algebra over $K$ with (the usual pointwise summation of functions and multiplication them by scalars, and)

the multiplication $\nabla :K^{G}\otimes K^{G}\to K^{G}$ , generated by the pointwise multiplication of functions,^[1]

\nabla (u\otimes v)(t)=u(t)\cdot v(t),\quad u,v\in K^{G},\quad t\in G,

the comultiplication $\Delta :K^{G}\to K^{G\times G}\cong K^{G}\otimes K^{G}$ , generated by the operation of multiplication in $G$ ^[2]

\Delta (u)(s,t)=u(s\cdot t),\quad u\in K^{G},\quad s,t\in K,

and the antipode $S:K^{G}\to K^{G}$ generated by the operation of taking the inverse element in $G$

S(u)(t)=u(t^{-1}),\quad u\in K^{G},\quad t\in K.

The dual space $K_{G}=(K^{G})'$ (of all linear functionals $\alpha :K^{G}\to K$ ) is the dual Hopf algebra. Since the delta-functionals

\delta _{t}\in K_{G},\quad \delta _{t}(u)=u(t),\quad u\in K^{G},\quad t\in G

,

form a basis in the vector space $K_{G}=(K^{G})'$ , one can describe the Hopf operations in $K_{G}$ by their actions on the elements $\delta _{t}$ :^[3]

the multiplication $\nabla :K_{G}\otimes K_{G}\to K_{G}$ :^[4]

\nabla (\delta _{s}\otimes \delta _{t})=\delta _{s\cdot t},\quad s,t\in G,

the comultiplication $\Delta :K_{G}\to K_{G}\otimes K_{G}$ :^[5]

\Delta (\delta _{t})=\delta _{t}\otimes \delta _{t},\quad t\in K,

and the antipode $S:K_{G}\to K_{G}$ :

S(\delta _{t})=\delta _{t^{-1}},\quad u\in K^{G},\quad t\in K.

The algebra $K_{G}$ (denoted also as $K[G]$ ) is called the group algebra (or the group ring) of the group $G$ over the field $K$ . The following construction connects the representations of the group $G$ with the representations of the algebra $K_{G}$ :

A map $\pi :G\to A$ of a group $G$ into a unital associative algebra $A$ over $K$ is called a representation of the group $G$ in the algebra $A$ , if it preserves the unit and the multiplication:

\pi (1_{G})=1_{A},\quad \pi (s\cdot t)=\pi (s)\cdot \pi (t),\quad s,t\in G.

Example: the operation of passage to the delta-functional

$\delta :G\to K_{G},\quad t\in G\mapsto \delta _{t}\in K_{G}$

is a representation of $G$ in $K_{G}$ :

$\delta _{1_{G}}=1_{K_{G}},\quad \delta _{s\cdot t}=\delta _{s}*\delta _{t},\quad s,t\in G.$

There is a natural correspondence between the representations of the group $G$ in (unital associative) algebras $A$ and the homomorphisms of the unital algebras $\varphi :K_{G}\to A$ :

Theorem (universal property)^[6]^[7].

For any finite group $G$ and for any unital associative algebra $A$ over $K$ the formula

\pi =\varphi \circ \delta

establishes a one-to-one correspondence between the representations $\pi :G\to A$ of the group $G$ in $A$ and the homomorphisms of the unital associative algebras $\varphi :K_{G}\to A$ .

This observation has a series of important corollaries, which allow to reduce the theory of representations of finite groups to the theory of representations of finite-dimensional algebras.^[8]^[7]

The construction of the group algebra $K_{G}$ can be easily generalized to arbitrary (not necessarily finite) groups $G$ (in the purely algebraic sense, without topology) with the same purposes: the generalization is called group ring $K[G]$ , and many results are preserved in this way, including the fact that $K[G]$ possesses the universal property (and is a Hopf algebra when $K$ is a field). But up to the recent time the generalizations to the topological groups faced numerous difficulties because of the lack of the convenient categories of topological vector spaces with dualities. The generalizations were mostly constructed in the category of Banach spaces, but the absence of a suitable duality in this category led to various distortions of the properties of these constructions, in particular, they were not Hopf algebras, and even the correspondence between the representations of groups and the homomorphisms of their group algebras was usually violated (however, this correspondence sometimes could be understood in some special sense).

Examples

The full analogy with the purely algebraic situation appears in the stereotype theory where a series of natural group algebras is constructed including the following four examples.

On each locally compact group $G$ one can consider the algebra ${\mathcal {C}}(G)$ of all continuous functions $u:G\to {\mathbb {C} }$ with the pointwise multiplication. Being endowed with the topology of uniform convergence on compact sets $T\subseteq G$ , it becomes a stereotype algebra. Its stereotype dual space ${\mathcal {C}}^{\star }(G)$ , which consists of Radon measures with compact support on $G$ , is a stereotype algebra with respect to the operation of convolution:^[9] $\alpha *\beta (u)=\int _{G}\left(\int _{G}u(s\cdot t)\,\alpha (ds)\right)\beta (dt),\quad \alpha ,\beta \in {\mathcal {C}}^{\star }(G),\ u\in {\mathcal {C}}(G).$

The algebra

{\mathcal {C}}^{\star }(G)

is called the stereotype group algebra of measures on the locally compact group

G

.^[10]

On each real Lie group $G$ one can consider the algebra ${\mathcal {E}}(G)$ of all smooth functions $u:G\to {\mathbb {C} }$ with the pointwise multiplication, and the topology of uniform convergence with all derivatives on compact sets $T\subseteq G$ . Again, it is a stereotype algebra. Its stereotype dual space ${\mathcal {E}}^{\star }(G)$ , which consists of distributions with compact support on $G$ , is a stereotype algebra with respect to the operation of convolution of distributions. The algebra ${\mathcal {E}}^{\star }(G)$ is called the stereotype group algebra of distributions on the real Lie group $G$ .

On each Stein group^[11] $G$ one can consider the algebra ${\mathcal {O}}(G)$ of all holomorphic functions $u:G\to {\mathbb {C} }$ with the pointwise multiplication and the topology of uniform convergence on compact sets $T\subseteq G$ . Again, this is a stereotype algebra. Its stereotype dual space ${\mathcal {O}}^{\star }(G)$ , which consists of holomorphic fuhctionals on $G$ , is a stereotype algebra with respect to the operation of convolution of functionals. The algebra ${\mathcal {O}}^{\star }(G)$ is called the stereotype group algebra of analytic functionals on the Stein group $G$ .

On each affine algebraic group $G$ one can consider the algebra ${\mathcal {P}}(G)$ of all polynomials (or regular functions) $u:G\to {\mathbb {C} }$ with the pointwise multiplication and the strongest locally convex topology. This is again a stereotype algebra, and its stereotype dual space ${\mathcal {P}}^{\star }(G)$ , which consists of currents on $G$ , is a stereotype algebra with respect to the operation of convolution of currents. The algebra ${\mathcal {P}}^{\star }(G)$ is called the stereotype group algebra of currents on the affine algebraic group $G$ .

The representation^[12] $\delta :G\to {\mathcal {C}}^{\star }(G)$ , $\delta (x)(u)=u(x)$ , $x\in G$ , $u\in {\mathcal {C}}^{\star }(G)$ is called the representation as delta-functionals.

The representations $\delta :G\to {\mathcal {E}}^{\star }(G)$ , $\delta :G\to {\mathcal {O}}^{\star }(G)$ , $\delta :G\to {\mathcal {P}}^{\star }(G)$ , are defined similarly.

The following two results distinguish the stereotype group algebras among the other models of group algebras in analysis.

Theorem (universal property).^[13]

Universal property of stereotype group algebras.

For any stereotype algebra $A$ the formula

\pi =\varphi \circ \delta

establishes a one-to-one correspondence between

the continuous representations^[12] $\pi :G\to A$ of any given locally compact group $G$ in the stereotype algebra $A$ and the morphisms of stereotype algebras $\varphi :{\mathcal {C}}^{\star }(G)\to A$ ,

the smooth^[14] representations^[12] $\pi :G\to A$ of any given real Lie group $G$ in the stereotype algebra $A$ and the morphisms of stereotype algebras $\varphi :{\mathcal {E}}^{\star }(G)\to A$ ,

the holomorphic^[15] representations^[12] $\pi :G\to A$ of any given Stein group $G$ in the stereotype algebra $A$ and the morphisms of stereotype algebras $\varphi :{\mathcal {O}}^{\star }(G)\to A$ ,

the polynomial (regular)^[16] representations^[12] $\pi :G\to A$ of any given affine algebraic group $G$ in the stereotype algebra $A$ and the morphisms of stereotype algebras $\varphi :{\mathcal {P}}^{\star }(G)\to A$ .

Theorem.^[17] The group algebras ${\mathcal {C}}^{\star }(G)$ , ${\mathcal {E}}^{\star }(G)$ , ${\mathcal {O}}^{\star }(G)$ , ${\mathcal {P}}^{\star }(G)$ are Hopf algebras in the monoidal category (Ste, $\circledast$ , ${\mathbb {C} }$ ) of stereotype spaces.

Notes

^ The unit $\eta :K\to K^{G}$ for this multiplication is defined by the formula
$\eta (\lambda )=\lambda \cdot 1,\quad \lambda \in K,$
where 1 means the identity function on $G$ :
$1(t)=1,\quad t\in G.$
^ The counit $\varepsilon :K^{G}\to K$ for this comultiplication is defined by the formula
$\varepsilon (u)=u(1_{G}),\quad u\in K^{G},$
where $1_{G}$ means the unit in $G$ .
^ In particular, the multiplication $\nabla :K_{G}\otimes K_{G}\to K_{G}$ turns out to be the usual convolution of functionals $(\alpha ,\beta )\in K_{G}\times K_{G}\mapsto \alpha *\beta \in K_{G}$ : $\nabla (\alpha \otimes \beta )(u)=\alpha *\beta (u)=\sum _{s,t\in G}\alpha _{s}\cdot \beta _{t}\cdot u(s\cdot t),\quad \alpha ,\beta \in K_{G},\quad u\in K^{G},$ or $\nabla (\alpha \otimes \beta )=\alpha *\beta =\sum _{s,t\in G}\alpha _{s}\cdot \beta _{t}\cdot \delta _{s\cdot t},$ where $\alpha =\sum _{s\in G}\alpha _{s}\cdot \delta _{s},\quad \beta =\sum _{t\in G}\beta _{t}\cdot \delta _{t}$ are the expansions of $\alpha$ and $\beta$ along the basis $\{\delta _{t};\ t\in G\}$ in the vector space $K_{G}$ .
^ The unit $\eta :K\to K_{G}$ for this multiplication is defined by the formula
$\eta (\lambda )=\lambda \cdot \delta _{1_{G}},\quad \lambda \in K,$
where $1_{G}$ means the unit in $G$
^ The counit $\varepsilon :K_{G}\to K$ for this comultiplication is defined by the formula
$\varepsilon (\delta _{t})=1,\quad t\in G.$
^ Lang 2002, Chapter XVIII § 1.
^ ^a ^b Vinberg 2003, 12.4.
^ Lang 2002, Chapter XVIII.
^ Akbarov 2003, p. 272.
^ If $G$ is an infinite locally compact group then the algebra ${\mathcal {C}}^{\star }(G)$ of measures on $G$ is not a Fréchet algebra. In the case when $G$ is compact, ${\mathcal {C}}^{\star }(G)$ is a Smith space. If $G$ is $\sigma$ -compact, then ${\mathcal {C}}^{\star }(G)$ is a Brauner space.
^ A Stein group is a complex Lie group $G$ which is a Stein manifold.
^ ^a ^b ^c ^d ^e We use here the definition given above: a map $\pi :G\to A$ of a group $G$ into a unital associative algebra $A$ over $K$ is called a representation of the group $G$ in the algebra $A$ , if it preserves the unit and the multiplication:
$\pi (1_{G})=1_{A},\quad \pi (s\cdot t)=\pi (s)\cdot \pi (t),\quad s,t\in G.$
^ Akbarov 2003, p. 275.
^ A map $\xi :M\to X$ of a smooth manifold $M$ into a stereotype space $X$ is said to be smooth if for each functional $f\in X^{\star }$ the composition $f\circ \xi$ is a smooth function on $M$ , and the map $f\in X^{\star }\mapsto f\circ \xi \in {\mathcal {E}}(M)$ is continuous.
^ A map $\xi :M\to X$ of a Stein manifold $M$ into a stereotype space $X$ is said to be holomorphic if for each functional $f\in X^{\star }$ the composition $f\circ \xi$ is a holomorphic function on $M$ , and the map $f\in X^{\star }\mapsto f\circ \xi \in {\mathcal {O}}(M)$ is continuous.
^ A map $\xi :M\to X$ of an affine algebraic variety $M$ over ${\mathbb {C} }$ into a stereotype space $X$ is said to be polynomial (or regular) if for each functional $f\in X^{\star }$ the composition $f\circ \xi$ is a polynomial on $M$ , and the map $f\in X^{\star }\mapsto f\circ \xi \in {\mathcal {P}}(M)$ is continuous.
^ Akbarov 2009, p. 507.

References

Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra". Journal of Mathematical Sciences. 113 (2): 179–349. doi:10.1023/A:1020929201133. {{cite journal}}: Invalid |ref=harv (help)
Akbarov, S.S. (2009). "Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity". Journal of Mathematical Sciences. 162 (4): 459–586. arXiv:0806.3205. doi:10.1007/s10958-009-9646-1. {{cite journal}}: Invalid |ref=harv (help)

[1] The unit $\eta :K\to K^{G}$ for this multiplication is defined by the formula
$\eta (\lambda )=\lambda \cdot 1,\quad \lambda \in K,$
where 1 means the identity function on $G$ :
$1(t)=1,\quad t\in G.$

[2] The counit $\varepsilon :K^{G}\to K$ for this comultiplication is defined by the formula
$\varepsilon (u)=u(1_{G}),\quad u\in K^{G},$
where $1_{G}$ means the unit in $G$ .

[3] In particular, the multiplication $\nabla :K_{G}\otimes K_{G}\to K_{G}$ turns out to be the usual convolution of functionals $(\alpha ,\beta )\in K_{G}\times K_{G}\mapsto \alpha *\beta \in K_{G}$ : $\nabla (\alpha \otimes \beta )(u)=\alpha *\beta (u)=\sum _{s,t\in G}\alpha _{s}\cdot \beta _{t}\cdot u(s\cdot t),\quad \alpha ,\beta \in K_{G},\quad u\in K^{G},$ or $\nabla (\alpha \otimes \beta )=\alpha *\beta =\sum _{s,t\in G}\alpha _{s}\cdot \beta _{t}\cdot \delta _{s\cdot t},$ where $\alpha =\sum _{s\in G}\alpha _{s}\cdot \delta _{s},\quad \beta =\sum _{t\in G}\beta _{t}\cdot \delta _{t}$ are the expansions of $\alpha$ and $\beta$ along the basis $\{\delta _{t};\ t\in G\}$ in the vector space $K_{G}$ .

[4] The unit $\eta :K\to K_{G}$ for this multiplication is defined by the formula
$\eta (\lambda )=\lambda \cdot \delta _{1_{G}},\quad \lambda \in K,$
where $1_{G}$ means the unit in $G$

[5] The counit $\varepsilon :K_{G}\to K$ for this comultiplication is defined by the formula
$\varepsilon (\delta _{t})=1,\quad t\in G.$

[FOOTNOTELang2002Chapter_XVIII_§_1-6] Lang 2002, Chapter XVIII § 1.

[FOOTNOTEVinberg200312.4-7] Vinberg 2003, 12.4.

[FOOTNOTELang2002Chapter_XVIII-8] Lang 2002, Chapter XVIII.

[FOOTNOTEAkbarov2003272-9] Akbarov 2003, p. 272.

[10] If $G$ is an infinite locally compact group then the algebra ${\mathcal {C}}^{\star }(G)$ of measures on $G$ is not a Fréchet algebra. In the case when $G$ is compact, ${\mathcal {C}}^{\star }(G)$ is a Smith space. If $G$ is $\sigma$ -compact, then ${\mathcal {C}}^{\star }(G)$ is a Brauner space.

[Stein-group-11] A Stein group is a complex Lie group $G$ which is a Stein manifold.

[representation-12] We use here the definition given above: a map $\pi :G\to A$ of a group $G$ into a unital associative algebra $A$ over $K$ is called a representation of the group $G$ in the algebra $A$ , if it preserves the unit and the multiplication:
$\pi (1_{G})=1_{A},\quad \pi (s\cdot t)=\pi (s)\cdot \pi (t),\quad s,t\in G.$

[FOOTNOTEAkbarov2003275-13] Akbarov 2003, p. 275.

[smooth-maps-14] A map $\xi :M\to X$ of a smooth manifold $M$ into a stereotype space $X$ is said to be smooth if for each functional $f\in X^{\star }$ the composition $f\circ \xi$ is a smooth function on $M$ , and the map $f\in X^{\star }\mapsto f\circ \xi \in {\mathcal {E}}(M)$ is continuous.

[holomorphic-maps-15] A map $\xi :M\to X$ of a Stein manifold $M$ into a stereotype space $X$ is said to be holomorphic if for each functional $f\in X^{\star }$ the composition $f\circ \xi$ is a holomorphic function on $M$ , and the map $f\in X^{\star }\mapsto f\circ \xi \in {\mathcal {O}}(M)$ is continuous.

[polynomial-maps-16] A map $\xi :M\to X$ of an affine algebraic variety $M$ over ${\mathbb {C} }$ into a stereotype space $X$ is said to be polynomial (or regular) if for each functional $f\in X^{\star }$ the composition $f\circ \xi$ is a polynomial on $M$ , and the map $f\in X^{\star }\mapsto f\circ \xi \in {\mathcal {P}}(M)$ is continuous.

[FOOTNOTEAkbarov2009507-17] Akbarov 2009, p. 507.

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